Introduction - KTH · variable which is measurable with respect to the future (or terminal) ¾-ﬂeld after S, i.e. the one generated by fX (t_S) : ¿ • t • Tg. For this type

1

FEYNMAN-KAC FORMULAS, BACKWARD STOCHASTICDIFFERENTIAL EQUATIONS AND MARKOV PROCESSES:

LONG VERSION

JAN A. VAN CASTEREN

This article is written in honor of G. Lumer whom I consider as my semi-group teacher

Abstract. In this paper we explain the notion of stochastic backward differentialequations and its relationship with classical (backward) parabolic differential equa-tions of second order. The paper contains a mixture of stochastic processes likeMarkov processes and martingale theory and semi-linear partial differential equationsof parabolic type. Some emphasis is put on the fact that the whole theory generalizesFeynman-Kac formulas. A new method of proof of the existence of solutions is given.All the existence arguments are based on rather precise quantitative estimates.

1. Introduction

Backward stochastic differential equations, in short BSDE’s, have been well stud-ied during the last ten years or so. They were introduced by Pardoux and Peng[19], who proved existence and uniqueness of adapted solutions, under suitable square-integrability assumptions on the coefficients and on the terminal condition. Theyprovide probabilistic formulas for solution of systems of semi-linear partial differentialequations, both of parabolic and elliptic type. The interest for this kind of stochasticequations has increased steadily, this is due to the strong connections of these equa-tions with mathematical finance and the fact that they gave a generalization of thewell known Feynman-Kac formula to semi-linear partial differential equations. In thepresent paper we will concentrate on the relationship between time-dependent strongMarkov processes and abstract backward stochastic differential equations. The equa-tions are phrased in terms of a martingale problem, rather than a stochastic differentialequation. They could called weak backward stochastic differential equations. Empha-sis is put on existence and uniqueness of solutions. The paper [26] deals with the samesubject, but it concentrates on comparison theorems and viscosity solutions. The proofof the existence result is based on a theorem which is related to a homotopy argumentas pointed out by the authors of [9]. It is more direct than the usual approach, which

1This is an extended version of [23] which contains hardly any proofs.Date: January 10, 2007.2000 Mathematics Subject Classification. 60H99, 35K20.Key words and phrases. Backward Stochastic Differential Equation, Markov process, Parabolic

equations of second order.The author is obliged to the University of Antwerp (UIA) and FWO Flanders (Grant number

1.5051.04N) for their financial and material support support. He was also very fortunate to havediscussed part of this material with Karel in’t Hout (University of Antwerp), who provided somereferences with a crucial result about a surjectivity property of one-sided Lipschitz mappings: seeTheorem 1 in Croezeix et al [9]. Some aspects concerning this paper were at issue during a conser-vation with Etienne Pardoux (CMI, Universite de Provence, Marseille); the author is grateful for hiscomments and advice.

1

2 JAN A. VAN CASTEREN

uses, among other things, regularizing by convolution products. It also gives ratherprecise quantitative estimates.

For examples of strong solutions which are driven by Brownian motion the reader isreferred to e.g. section 2 in Pardoux [18]. If the coefficients x 7→ b(s, x) and x 7→ σ(s, x)of the underlying (forward) stochastic differential equation are linear in x, then thecorresponding forward-backward stochastic differential equation is related to optionpricing in financial mathematics. The backward stochastic differential equation mayserve as a model for a hedging strategy. For more details on this interpretation see e.g.El Karoui and Quenez [15], pp. 198–199. E. Pardoux and S. Zhang [20] use BSDE’s togive a probabilistic formula for the solution of a system of Parabolic or elliptic semi-linear partial differential equation with Neumann boundary condition. In this paperwe want to consider the situation where the family of operators L(s), 0 ≤ s ≤ T ,generates a time-inhomogeneous Markov process

(Ω,FτT ,Pτ,x) , (X(t) : T ≥ t ≥ 0) , (E, E) (1.1)

in the sense that

d

dsEτ,x [f (X(s))] = Eτ,x [L(s)f (X(s))] , f ∈ D (L(s)) , τ ≤ s ≤ T.

We consider the operators L(s) as operators on (a subspace of) the space of boundedcontinuous functions on E, i.e. on Cb(E) equipped with the supremum norm: ‖f‖∞ =supx∈E |f(x)|, f ∈ Cb(E). With the operators L(s) we associate the squared gradientoperator Γ1 defined by

Γ1 (f, g) (τ, x) = lims↓τ

1

s− τEτ,x [(f (X(s))− f (X(τ))) (g (X(s))− g (X(τ)))] , (1.2)

for f , g ∈ D (Γ1). These squared gradient operators are also called energy operators:see e.g. Barlow, Bass and Kumagai [5]. We assume that every operator L(s), 0 ≤s ≤ T , generates a diffusion in the sense of the following definition. In the sequelit is assumed that the family of operators L(s) : 0 ≤ s ≤ T possesses the propertythat the space of functions u : [0, T ] × E → R with the property that the function

(s, x) 7→ ∂u

∂s(s, x) + L(s)u (s, ·) (x) belongs to C0 ([0, T ]× E) := C0 ([0, T ]× E;R) is

dense in the space C0 ([0, T ]× E). This subspace of functions is denoted by D(L), andthe operator L is defined by Lu(s, x) = L(s)u (s, ·) (x), u ∈ D(L). It is also assumedthat the family A is a core for the operator L. We assume that the operator L, orthat the family of operators L(s) : 0 ≤ s ≤ T, generates a diffusion in the sense ofthe following definition.

1.1. Definition. A family of operators L(s) : 0 ≤ s ≤ T is said to generate a diffusionif for every C∞-function Φ : Rn → R, with Φ(0, . . . , 0) = 0, and every pair (s, x) ∈[0, T ]× E the following identity is valid

L(s) (Φ (f1, . . . , fn) (s, ·)) (x) (1.3)

=n∑

j=1

∂Φ

∂xj

(f1, . . . , fn) L(s)fj(x) +1

2

n∑

j,k=1

∂2Φ

∂xj∂xk

(f1, . . . , fn) (x)Γ1 (fj, fk) (s, x)

for all functions f1, . . . , fn in an algebra of functions A, contained in the domain of theoperator L, which forms a core for L.

BSDE’S AND MARKOV PROCESSES 3

Generators of diffusions for single operators are described in Bakry’s lecture notes[1]. For more information on the squared gradient operator see e.g. [3] and [2] aswell. Put Φ(f, g) = fg. Then (1.3) implies L(s) (fg) (s, ·)(x) = L(s)f(s, ·)(x)g(s, x) +f(s, x)L(s)g(s, ·)(x)Γ1 (f, g) (s, x), provided that the three functions f , g and fg belongto A. Instead of using the full strength of (1.3), i.e. with a general function Φ, we justneed it for the product (f, g) 7→ fg: see Proposition 1.11.

Let m be a reference measure on the Borel field E of E, and let p ∈ [1,∞]. If weconsider the operators L(s), 0 ≤ s ≤ T , in Lp (E, E,m)-space, then we also need someconditions on the algebra A of “core” type in the space Lp (E, E,m). For details thereader is referred to Bakry [1].

By definition the gradient of a function u ∈ D (Γ1) in the direction of v ∈ D (Γ1) isthe function (τ, x) 7→ Γ1 (u, v) (τ, x). For given (τ, x) ∈ [0, T ] × E the functional v 7→Γ1 (u, v) (τ, x) is linear: its action is denoted by ∇L

u (τ, x). Hence, for (τ, x) ∈ [0, T ]×Efixed, we can consider ∇L

u (τ, x) as an element in the dual of D (Γ1). The pair

(τ, x) 7→ (u (τ, x) ,∇L

u (τ, x))

may be called an element in the phase space of the family L(s), 0 ≤ s ≤ T , (see JanPruss [21]), and the process s 7→ (

u (s,X(s)) ,∇Lu (s,X(s))

)will be called an element of

the stochastic phase space. Next let f : [0, T ]×E×R×D (Γ1)∗ → R be a “reasonable”

function, and consider, for 0 ≤ s1 < s2 ≤ T the expression:

u (s2, X (s2))− u (s1, X (s1)) +

∫ s2

s1

f(s,X(s), u (s,X(s)) ,∇L

u (s,X(s)))ds

− u (s2, X (s2)) + u (s1, X (s1)) +

∫ s2

s1

(L(s)u (s,X(s)) +

∂u

∂s(s, X(s))

)ds (1.4)

= u (s2, X (s2))− u (s1, X (s1)) +

∫ s2

s1

f(s,X(s), u (s,X(s)) ,∇L

u (s,X(s)))ds

−Mu (s2) + Mu (s1) , (1.5)

where

Mu (s2)−Mu (s1)

= u (s2, X (s2))− u (s1, X (s1))−∫ s2

s1

(L(s)u (s,X(s)) +

∂u

∂s(s, X(s))

)ds

=

∫ s2

s1

dMu(s). (1.6)

1.2. Definition. The process

(Ω,FτT ,Pτ,x) , (X(t) : T ≥ t ≥ 0) , (E, E) (1.7)

is called a time-inhomogeneous Markov process if

Eτ,x

[f(X(t))

∣∣ Fτs

]= Es,X(s) [f(X(t))] , Pτ,x-almost surely. (1.8)

Here f is a bounded Borel measurable function defined on the state space E andτ ≤ s ≤ t ≤ T .

Suppose that the process X(t) in (1.7) has paths which are right-continuous andhave left limits in E. Then it can be shown that the Markov property for fixed times


carries over to stopping times in the sense that (1.8) may be replaced with

Eτ,x

[Y

∣∣ FτS

]= ES,X(S) [Y ] , Pτ,x-almost surely. (1.9)

Here S : E → [τ, T ] is an Fτt -adapted stopping time and Y is a bounded stochastic

variable which is measurable with respect to the future (or terminal) σ-field after S,i.e. the one generated by X (t ∨ S) : τ ≤ t ≤ T. For this type of result the reader isreferred to Chapter 2 in Gulisashvili et al [11]. Markov processes for which (1.9) holdsare called strong Markov processes.

Next we show that under rather general conditions the process s 7→ Mu(s)−Mu(t),t ≤ s ≤ T , as defined in (1.5) is a Pt,x-martingale.

1.3. Definition. The family of operators L(s), 0 ≤ s ≤ T , is said to generate a time-inhomogeneous Markov process

(Ω,FτT ,Pτ,x) , (X(t) : T ≥ t ≥ 0) , (E, E) (1.10)

if for all functions u ∈ D(L), for all x ∈ E, and for all pairs (τ, s) with 0 ≤ τ ≤ s ≤ Tthe following equality holds:

d

dsEτ,x [u (s,X(s))] = Eτ,x

[∂u

∂s(s, x) + L(s)u (s, ·) (X(s))

]. (1.11)

In the following proposition we write Fts, s ∈ [t, T ], for the σ-field generated by X(ρ),

ρ ∈ [t, s].

1.4. Proposition. Fix t ∈ [τ, T ). Let the function u : [t, T ] × E → R be such

that (s, x) 7→ ∂u

∂s(s, x) + L(s)u (s, ·) (x) belongs to C0 ([t, T ]× E) := C0 ([t, T ]× E;R).

Then the process s 7→ Mu(s)−Mu(t) is adapted to the filtration of σ-fields (Fts)s∈[t,T ].

Proof. Suppose that T ≥ s2 > s1 ≥ t. In order to check the martingale property of theprocess Mu(s)−Mu(t), s ∈ [t, T ], it suffices to prove that

Et,x

[Mu (s2)−Mu (s1)

∣∣ Fts1

]= 0. (1.12)

In order to prove (1.12) we notice that by the time-inhomogeneous Markov property:

Et,x

[Mu (s2)−Mu (s1)

∣∣ Fts1

]= Es1,X(s1) [Mu (s2)−Mu (s1)]

= Es1,X(s1)

[u (s2, X (s2))− u (s1, X (s1))

−∫ s2

s1

(L(s)u (s,X(s)) +

∂u

∂s(s,X(s))

)ds

]

= Es1,X(s1) [u (s2, X (s2))− u (s1, X (s1))]

−∫ s2

s1

Es1,X(s1)

[(L(s)u (s,X(s)) +

∂u

∂s(s, X(s))

)]ds

= Es1,X(s1) [u (s2, X (s2))− u (s1, X (s1))]−∫ s2

s1

d

dsEs1,X(s1) [u (s,X(s))] ds

= Es1,X(s1) [u (s2, X (s2))− u (s1, X (s1))]

− Es1,X(s1) [u (s2, X (s2))− u (s1, X (s1))] = 0. (1.13)

The equality in (1.13) establishes the result in Proposition 1.4. ¤


As explained in Definition 1.1 it is assumed that the subspace D(L) contains analgebra of functions which forms a core for the operator L.

1.5. Proposition. Let the family of operators L(s), 0 ≤ s ≤ T , generate a time-inhomogeneous Markov process

(Ω,FτT ,Pτ,x) , (X(t) : T ≥ t ≥ 0) , (E, E) (1.14)

in the sense of Definition 1.3: see equality (1.11). Then the process X(t) has a modi-fication which is right-continuous and has left limits.

In view of Proposition 1.5 we will assume that our Markov process has left limitsand is continuous from the right.

Proof. Let the function u : [0, T ] × E → R belong to the space D(L). Then theprocess s 7→ Mu(s)−Mu(t), t ≤ s ≤ T , is a Pt,x-martingale. Let D[0, T ] be the set ofnumbers of the form k2−nT , k = 0, 1, 2, . . . , 2n. By a classical martingale convergencetheorem (see e.g. Chapter II in Revuz and Yor [22]) it follows that the following limit

lims↑t, s∈D[0,T ]

u (s,X(s)) exists Pτ,x-almost surely for all 0 ≤ τ < t ≤ T and for all x ∈ E.

In the same reference it is also shown that the limit lims↓t, s∈D[0,T ]

u (s,X(s)) exists Pτ,x-

almost surely for all 0 ≤ τ ≤ t < T and for all x ∈ E. Since the locally compact space[0, T ]×E is second countable it follows that the exceptional sets may be chosen to beindependent of (τ, x) ∈ [0, T ]×E, of t ∈ [τ, T ], and of the function u ∈ D(L). Since byhypothesis the subspace D(L) is dense in C0 ([0, T ]× E) it follows that the left-handlimit at t of the process s 7→ X(s), s ∈ D[0, T ] ∩ [τ, t], exists Pτ,x-almost surely forall (t, x) ∈ (τ, T ] × E. It also follows that the right-hand limit at t of the processs 7→ X(s), s ∈ D[0, T ]∩ (t, T ], exists Pτ,x-almost surely for all (t, x) ∈ [τ, T )×E. Thenwe modify X(t) by replacing it with X(t+) = lims↓t, s∈D[0,T ]∩(τ,T ] X(s), t ∈ [0, T ), andX(T+) = X(T ). It also follows that the process t 7→ X(t+) has left limits in E. ¤

The hypotheses in the following Proposition 1.6 are the same as those in Proposition1.5.

1.6. Proposition. Let the continuous function u : [0, T ] × E → R be such that forevery s ∈ [t, T ] the function x 7→ u(s, x) belongs to D (L(s)) and suppose that thefunction (s, x) 7→ [L(s)u (s, ·)] (x) is bounded and continuous. In addition suppose thatthe function s 7→ u(s, x) is continuously differentiable for all x ∈ E. Then the processs 7→ Mu(s) − Mu(t) is a Ft

s-martingale with respect to the probability Pt,x. If v isanother such function, then the (right) derivative of the quadratic co-variation of themartingales Mu and Mv is given by:

d

dt〈Mu,Mv〉 (t) = Γ1 (u, v) (t,X(t)) .

In fact the following identity holds as well:

Mu(t)Mv(t)−Mu(0)Mv(0)

=

∫ t

0

Mu(s)dMv(s) +

∫ t

0

Mv(s)dMu(s) +

∫ t

0

Γ1 (u, v) (s,X(s)) ds. (1.15)


Here Fts, s ∈ [t, T ], is the σ-field generated by the state variables X(ρ), t ≤ ρ ≤ s.

Instead of F0s we usually write Fs, s ∈ [0, T ]. The formula in (1.15) is known as the

integration by parts formula for stochastic integrals.

Proof. We outline a proof of the equality in (1.15). So let the functions u and v be asin Proposition 1.6. Then we have

Mu(t)Mv(t)−Mu(0)Mv(0)

=2n−1∑

k=0

Mu

(k2−nt

) (Mv

((k + 1)2−nt

)−Mv

(k2−nt

))

+2n−1∑

k=0

(Mu

((k + 1)2−nt

)−Mu

(k2−nt

))Mv

(k2−nt

)

+2n−1∑

k=0

(Mu

((k + 1)2−nt

)−Mu

(k2−nt

)) (Mv

((k + 1)2−nt

)−Mv

(k2−nt

)).

(1.16)

The first term on the right-hand side of (1.16) converges to∫ t

0Mu(s)dMv(s), the second

term converges to∫ t

0Mv(s)dMu(s). Using the identity in (1.6) for the function u and

a similar identity for v we see that the third term on the right-hand side of (1.16)

converges to∫ t

0Γ1 (u, v) (s,X(s)) ds.

This completes the proof Proposition 1.6. ¤1.1. Remark. The quadratic variation process of the (local) martingale s 7→ Mu(s) isgiven by the process s 7→ Γ1 (u (s, ·) , u (s, ·)) (s,X(s)), and therefore

Es1,x

[∣∣∣∣∫ s2

s1

dMu(s)

∣∣∣∣2]

= Es1,x

[∫ s2

s1

Γ1 (u (s, ·) , u (s, ·)) (s, X(s)) ds

]< ∞

under appropriate conditions on the function u. Informally we may think of the fol-lowing representation for the martingale difference:

Mu (s2)−Mu (s1) =

∫ s2

s1

∇Lu (s,X(s)) dW (s). (1.17)

Here we still have to give a meaning to the stochastic integral in the right-hand side of(1.17). If E is an infinite-dimensional Banach space, then W (t) should be some kindof a cylindrical Brownian motion. It is closely related to a formula which occurs inMalliavin calculus: see Nualart [16] and [17].

1.2. Remark. It is perhaps worthwhile to observe that for Brownian motion (W (s),Px)the martingale difference Mu (s2) − Mu (s1), s1 ≤ s2 ≤ T , is given by a stochasticintegral:

Mu (s2)−Mu (s1) =

∫ s2

s1

∇u (τ,W (τ)) dW (τ).

Its increment of the quadratic variation process is given by

〈Mu,Mu〉 (s2)− 〈Mu,Mu〉 (s1) =

∫ s2

s1

|∇u (τ,W (τ))|2 dτ.


Next suppose that the function u solves the equation:

f(s, x, u (s, x) ,∇L

u (s, x))

+ L(s)u (s, x) +∂

∂su (s, x) = 0. (1.18)

If moreover, u (T, x) = ϕ (T, x), x ∈ E, is given, then we have

u (t,X(t)) = ϕ (T,X(T )) +

∫ T

t

f(s,X(s), u (s,X(s)) ,∇L

u (s,X(s)))ds−

∫ T

t

dMu(s),

(1.19)with Mu(s) as in (1.6). From (1.19) we get

u (t, x) = Et,x [u (t,X(t))] (1.20)

= Et,x [ϕ (T, X(T ))] +

∫ T

t

Et,x

[f

(s,X(s), u (s,X(s)) ,∇L

u (s,X(s)))]

ds.

1.7. Theorem. Let u : [0, T ]×E → R be a continuous function with the property thatfor every (t, x) ∈ [0, T ]×E the function s 7→ Et,x [u (s,X(s))] is differentiable and that

d

dsEt,x [u (s,X(s))] = Et,x

[L(s)u (s,X(s)) +

∂

∂su (s,X(s))

], t < s < T.

Then the following assertions are equivalent:

(a) The function u satisfies the following differential equation:

L(t)u (t, x) +∂

∂tu (t, x) + f

(t, x, u (t, x) ,∇L

u (t, x))

= 0. (1.21)

(b) The function u satisfies the following type of Feynman-Kac integral equation:

u (t, x) = Et,x

[u (T, X(T )) +

∫ T

t

f(τ, X(τ), u (τ,X(τ)) ,∇L

u (τ, X(τ)))dτ

]. (1.22)

.(c) For every t ∈ [0, T ] the process

s 7→ u (s,X(s))− u (t,X(t)) +

∫ s

t

f(τ,X(τ), u (τ, X(τ)) ,∇L

u (τ, X(τ)))dτ

is an Fts-martingale with respect to Pt,x on the interval [t, T ].

(d) For every s ∈ [0, T ] the process

t 7→ u (T, X(T ))− u (t,X(t)) +

∫ T

t

f(τ,X(τ), u (τ, X(τ)) ,∇L

u (τ,X(τ)))dτ

is an FtT -backward martingale with respect to Ps,x on the interval [s, T ].

1.3. Remark. Suppose that the function u is a solution to the following terminal valueproblem:

L(s)u (s, ·) (x) +∂

∂su (s, x) + f

(s, x, u (s, x) ,∇L

u (s, x))

= 0;

u(T, x) = ϕ(T, x).(1.23)


Then the pair(u (s,X(s)) ,∇L

u (s,X(s)))

can be considered as a weak solution to abackward stochastic differential equation. More precisely, for every s ∈ [0, T ] theprocess

t 7→u (T, X(T ))− u (t,X(t)) +

∫ T

t

f(τ,X(τ), u (τ, X(τ)) ,∇L

u (τ,X(τ)))dτ

is an FtT -backward martingale with respect to Ps,x on the interval [s, T ]. The symbol

∇Luv (s, x) stands for the functional v 7→ ∇L

uv (s, x) = Γ1(u, v)(s, x), where Γ1 is thesquared gradient operator:

Γ1(u, v)(s, x) (1.24)

= limh↓0

1

hEs,x [(u (s,X(s + h))− u (s,X(s))) (f (s,X(s + h))− f (s,X(s)))] .

Possible choices for the function f are

f(s, x, y,∇L

u

)= −V (s, x)y and (1.25)

f(s, x, y,∇L

u

)=

1

2

∣∣∇Lu (s, x)

∣∣2 − V (s, x) =1

2Γ1 (u, u) (s, x)− V (s, x). (1.26)

The choice in (1.25) turns equation (1.23) into the following heat equation:

∂

∂su (s, x) + L(s)u (s, ·) (x)− V (s, x)u(s, x) = 0;

u (T, x) = ϕ(T, x).(1.27)

The function v(s, x) defined by the Feynman-Kac formula

v(s, x) = Es,x

[e−

∫ Ts V (ρ,X(ρ))dρϕ (T, X(T ))

](1.28)

is a candidate solution to equation (1.27).The choice in (1.26) turns equation (1.23) into the following Hamilton-Jacobi-Bellmannequation:

∂

∂su (s, x) + L(s)u (s,X(s))− 1

2Γ1 (u, u) (s, x) + V (s, x)=0;

u (T, x) = − log ϕ(T, x),(1.29)

where − log ϕ(T, x) replaces ϕ(T, x). The function SL defined by the genuine non-linearFeynman-Kac formula

SL(s, x) = − logEs,x

[e−

∫ Ts V (ρ,X(ρ))dρϕ (T,X(T ))

](1.30)

is a candidate solution to (1.29). Often these “candidate solutions” are viscosity solu-tions. However, this will be the main topic of [26].

1.4. Remark. Let u(t, x) satisfy one of the equivalent conditions in Theorem 1.7. PutY (τ) = u (τ, X(τ)), and let M(s) be the martingale determined by M(0) = Y (0) =u (0, X(0)) and by

M(s)−M(t) = Y (s) +

∫ s

t

f(τ,X(τ), Y (τ),∇L

u (τ, X(τ)))dτ.

Then the expression∇Lu (τ, X(τ)) only depends on the martingale part M of the process

s 7→ Y (s). This entitles us to write ZM(τ) instead of ∇Lu (τ,X(τ)). The interpretation


of ZM(τ) is then the linear functional N 7→ d

dτ〈M, N〉 (τ), where N is a Pτ,x-martingale

in M2 (Ω,F0T ,Pt,x). Here a process N belongs to M2 (Ω,F0

T ,Pt,x) whenever N is mar-tingale in L2 (Ω,F0

T ,Pt,x). Notice that the functional ZM(τ) is known as soon as themartingale M ∈ M2 (Ω,F0

T ,Pt,x) is known. From our definitions it also follows that

M(T ) = Y (T ) +

∫ T

0

f (τ, X(τ), Y (τ), ZM(τ)) dτ,

where used the fact that Y (0) = M(0).

1.5. Remark. Let the notation be as in Remark 1.4. Then the variables Y (t) and ZM(t)only depend on the space variable X(t), and as consequence the martingale incrementsM (t2) − M (t1), 0 ≤ t1 < t2 ≤ T , only depend on Ft1

t2 = σ (X(s) : t1 ≤ s ≤ t2). InSection 2 we give Lipschitz type conditions on the function f in order that the BSDE

Y (t) = Y (T ) +

∫ T

t

f (s,X(s), Y (s), ZM(s)) ds + M(t)−M(T ), τ ≤ t ≤ T, (1.31)

possesses a unique pair of solutions

(Y,M) ∈ L2 (Ω,FτT ,Pτ,x)×M2 (Ω,Fτ

T ,Pτ,x) .

Here M2 (Ω, FtT ,Pt,x) stands for the space of all (Ft

s)s∈[t,T ]-martingales in L2 (Ω,FtT ,Pt,x).

Of course instead of writing “BSDE” it would be better to write “BSIE” for BackwardStochastic Integral Equation. However, since in the literature people write “BSDE”even if they mean integral equations we also stick to this terminology. Supposethat the σ (X(T ))-measurable variable Y (T ) ∈ L2 (Ω, Fτ

T ,Pτ,x) is given. In fact wewill prove that the solution (Y,M) of the equation in (1.31) belongs to the spaceS2

(Ω, Fτ

T ,Pτ,x;Rk)×M2

(Ω,Ft

T ,Pt,x;Rk). For more details see the definitions 1.8 and

3.1, and Theorem 4.1.

1.6. Remark. Let M and N be two martingales in M2 [0, T ]. Then, for 0 ≤ s < t ≤ T ,

|〈M, N〉 (t)− |〈M,N〉 (s)||2 ≤ (〈M, M〉 (t)− 〈M,M〉 (s)) (〈N, N〉 (t)− 〈N, N〉 (s)) ,

and consequently∣∣∣∣d

ds〈M,N〉 (s)

∣∣∣∣2

≤ d

ds〈M,M〉 (s) d

ds〈N,N〉 (s).

Hence, the inequality∫ T

0

∣∣∣∣d

ds〈M,N〉 (s)

∣∣∣∣ ds ≤∫ T

0

(d

ds〈M, M〉 (s)

)1/2 (d

ds〈N, N〉 (s)

)1/2

ds (1.32)

follows. The inequality in (1.32) says that the quantity

∫ T

0

∣∣∣∣d

ds〈M,N〉 (s)

∣∣∣∣ ds is dom-

inated by the Hellinger integral H (M,N) defined by the right-hand side of (1.32).

For a proof we refer the reader to [26].

1.7. Remark. Instead of considering ∇Lu (s, x) we will also consider the bilinear mapping

Z(s) which associates with a pair of local semi-martingales (Y1, Y2) a process which is


to be considered as the right derivative of the covariation process: 〈Y1, Y2〉 (s). Wewrite

ZY1(s) (Y2) = Z(s) (Y1, Y2) =d

ds〈Y1, Y2〉 (s).

The function f (i.e. the generator of the backward differential equation) will thenbe of the form: f (s,X(s), Y (s), ZY (s)); the deterministic phase

(u(s, x),∇Lu(s, x)

)is

replaced with the stochastic phase (Y (s), ZY (s)). We should find an appropriate sto-chastic phase s 7→ (Y (s), ZY (s)), which we identify with the process s 7→ (Y (s), MY (s))in the stochastic phase space S2 ×M2, such that

Y (t) = Y (T ) +

∫ T

t

f (s, X(s), Y (s), ZY (s)) ds−∫ T

t

dMY (s), (1.33)

where the quadratic variation of the martingale MY (s) is given by

d 〈MY ,MY 〉 (s) = ZY (s) (Y ) ds = Z(s) (Y, Y ) ds = d 〈Y, Y 〉 (s).This stochastic phase space S2 ×M2 plays a role in stochastic analysis very similar tothe role played by the first Sobolev space H1,2 in the theory of deterministic partialdifferential equations.

1.8. Remark. In case we deal with strong solutions driven by standard Brownian motionthe martingale difference MY (s2)−MY (s1) can be written as

∫ s2

s1ZY (s)dW (s), provided

that the martingale MY (s) belongs to M2 (Ω,G0T ,P). Here G0

T is the σ-field generatedby W (s), 0 ≤ s ≤ T . If Y (s) = u (s,X(s)), then this stochastic integral satisfies:∫ s2

s1

ZY (s)dW (s) = u (s2, X (s2))− u (s1, X (s1))−∫ s2

s1

(L(s) +

∂

∂s

)u (s,X (s)) ds.

(1.34)Such stochastic integrals are for example defined if the process X(t) is a solution to astochastic differential equation (in Ito sense):

X(s) = X(t) +

∫ s

t

b (τ, X(τ)) dτ +

∫ s

t

σ (τ, X(τ)) dW (τ), t ≤ s ≤ T. (1.35)

Here the matrix (σjk (τ, x))dj,k=1 is chosen in such a way that

ajk(τ, x) =d∑

`=1

σj` (τ, x) σk` (τ, x) = (σ(τ, x)σ∗(τ, x))jk .

The process W (τ) is Brownian motion or Wiener process. It is assumed that operatorL(τ) has the form

L(τ)u(x) = b (τ, x) · ∇u(x) +1

2

d∑

j,k=1

ajk (τ, x)∂2

∂xjxk

u(x). (1.36)

Then from Ito’s formula together with (1.34), (1.35) and (1.36) it follows that theprocess ZY (s) has to be identified with σ (s,X(s))∗∇u (s, ·) (X(s)). For more detailssee e.g. Pardoux and Peng [19] and Pardoux [18].


1.9. Remark. Backward doubly stochastic differential equations (BDSDEs) could havebeen included in the present paper: see Boufoussi, Mrhardy and Van Casteren [7]. Inour notation a BDSDE may be written in the form:

Y (t)− Y (T ) =

∫ T

t

f

(s,X(s), Y (s), N 7→ d

ds〈M, N〉 (s)

)ds

+

∫ T

t

g

(s,X(s), Y (s), N 7→ d

ds〈M, N〉 (s)

)d←−B (s)

+ M(t)−M(T ). (1.37)

Here the expression∫ T

t

g

(s,X(s), Y (s), N 7→ d

ds〈M, N〉 (s)

)d←−B (s)

represents a backward Ito integral. The symbol 〈M, N〉 stands for the covariationprocess of the (local) martingales M and N ; it is assumed that this process is absolutelycontinuous with respect to Lebesgue measure. Moreover,

(Ω,FτT ,Pτ,x) , (X(t) : T ≥ t ≥ 0) , (E, E)

is a Markov process generated by a family of operators L(s), 0 ≤ s ≤ T , and Fτt =

σ X(s) : τ ≤ s ≤ t. The process X(t) could be the (unique) weak or strong solutionto a (forward) stochastic differential equation (SDE):

X(t) = x +

∫ t

τ

b (s,X(s)) ds +

∫ t

τ

σ (s,X(s)) dW (s). (1.38)

Here the coefficients b and σ have certain continuity or measurability properties, andPτ,x is the distribution of the process X(t) defined as being the unique weak solution tothe equation in (1.38). We want to find a pair (Y, M) ∈ S2 (Ω,Fτ

t ,Pτ,x)×M2 (Ω,Fτt ,Pτ,x)

which satisfies (1.37).

1.10. Remark. Part of this work was presented at a Colloquium at the University ofGent, October 14, 2005, at the occasion of the 65th birthday of Richard Delanghe andappeared in a preliminary form in [25]. Some results were also presented at the Univer-sity of Clausthal, at the occasion of Michael Demuth’s 60th birthday September 10–11,2006, and at a Conference in Marrakesh, Morocco, “Marrakesh World Conference onDifferential Equations and Applications”, June 15–20, 2006. This work was also partof a Conference on “The Feynman Integral and Related Topics in Mathematics andPhysics: In Honor of the 65th Birthdays of Gerry Johnson and David Skoug”, Lincoln,Nebraska, May 12–14, 2006. Finally, another preliminary version was presented duringa Conference on Evolution Equations, in memory of G. Lumer, at the Universities ofMons and Valenciennes, August 28–September 1, 2006.

We first give some definitions. Fix (τ, x) ∈ [0, T ]× E. In the definitions 1.8 and 1.9the probability measure Pτ,x is defined on the σ-field Fτ

T . In Definition 3.1 we returnto these notions. The following definition and implicit results described therein showsthat, under certain conditions, by enlarging the sample space a family of processes maybe reduced to just one process without losing the S2-property.


1.8. Definition. Fix (τ, x) ∈ [0, T ] × E. An Rk-valued process Y is said to be-long to the space S2

(Ω,Fτ

T ,Pτ,x;Rk)

if Y (t) is Fτt -measurable (τ ≤ t ≤ T ) and if

Eτ,x

[sup

τ≤t≤T|Y (t)|2

]< ∞. It is assumed that Y (s) = Y (τ), Pτ,x-almost surely, for s ∈

[0, τ ]. The process Y (s), s ∈ [0, T ], is said to belong to the space S2unif

(Ω,Fτ

T ,Pτ,x;Rk)

if

sup(τ,x)∈[0,T ]×E

Eτ,x

[sup

τ≤t≤T|Y (t)|2

]< ∞,

and it belongs to S2loc,unif

(Ω,Fτ

T ,Pτ,x;Rk)

provided that

sup(τ,x)∈[0,T ]×K

Eτ,x

[sup

τ≤t≤T|Y (t)|2

]< ∞

for all compact subsets K of E.

If the σ-field Fτt and Pτ,x are clear from the context we write S2

([0, T ],Rk

)or some-

times just S2.

1.9. Definition. Let the process M be such that the process t 7→ M(t) −M(τ), t ∈[τ, T ], is a Pτ,x-martingale with the property that the stochastic variable M(T )−M(τ)belongs to L2 (Ω,Fτ

T ,Pτ,x). Then M is said to belong to the space M2(Ω, Fτ

T ,Pτ,x;Rk).

By the Burkholder-Davis-Gundy inequality (see inequality (3.6) below) it follows that

Eτ,x

[sup

τ≤t≤T|M(t)−M(τ)|2

]is finite if and only if M(T )−M(τ) belongs to the space

L2 (Ω,FτT ,Pτ,x). Here an Fτ

t -adapted process M(·) −M(τ) is called a Pτ,x-martingaleprovided that Eτ,x [|M(t)−M(τ)|] < ∞ and Eτ,x

[M(t)−M(τ)

∣∣ Fτs

]= M(s)−M(τ),

Pτ,x-almost surely, for T ≥ t ≥ s ≥ τ . The martingale difference s 7→ M(s) −M(0),s ∈ [0, T ], is said to belong to the space M2

unif

(Ω, Fτ

T ,Pτ,x;Rk)

if


Eτ,x

[sup


]< ∞,

and it belongs to M2loc,unif

(Ω,Fτ

T ,Pτ,x;Rk)

provided that

sup(τ,x)∈[0,T ]×K

Eτ,x

[sup


]< ∞

for all compact subsets K of E. From the Burkholder-Davis-Gundy inequality (seeinequality (3.6) below) it follows that the process M(s) −M(0) belongs to the spaceM2

unif

(Ω,Fτ

T ,Pτ,x;Rk)

if and only if


Eτ,x

[|M(T )−M(τ)|2] = sup(τ,x)∈[0,T ]×E

Eτ,x [〈M, 〉 (T )− 〈M,M〉 (τ)] < ∞.

Here 〈M, M〉 stand for the quadratic variation process of the process t 7→ M(t)−M(0).

The notions in the definitions 1.8 and (1.9) will exclusively be used in case the familyof measures Pτ,x : (τ, x) ∈ [0, T ]× E constitute the distributions of a Markov processwhich was defined in Definition 1.2.

Again let the Markov process, with right-continuous sample paths and with leftlimits,

(Ω,FτT ,Pτ,x) , (X(t) : T ≥ t ≥ 0) , (E, E) (1.39)


be generated by the family of operators L(s) : 0 ≤ s ≤ t: see definitions 1.2, equality(1.8), and 1.3, equality (1.10).

Next we define the family of operators Q (t1, t2) : 0 ≤ t1 ≤ t2 ≤ T by

Q (t1, t2) f(x) = Et1,x [f (X (t2))] , f ∈ C0 (E) , 0 ≤ t1 ≤ t2 ≤ T. (1.40)

Fix ϕ ∈ D(L). Since the process t 7→ Mϕ(t) −Mϕ(s), t ∈ [s, T ], is a Ps,x-martingalewith respect to the filtration (Fs

t )t∈[s,T ], and X(t) = x Pt,x almost surely, the followingequality follows:

∫ t

s

Es,x [L(ρ)ϕ (ρ, ·) (X(ρ))] dρ + Et,x [ϕ (t,X(t))]− Es,x [ϕ (t,X(t))]

= ϕ(t, x)− ϕ(s, x)−∫ t

s

Es,x

[∂ϕ

∂ρ(ρ,X(ρ))

]dρ. (1.41)

The fact that a process of the form t 7→ Mϕ(t)−Mϕ(s), t ∈ [s, T ], is a Ps,x-martingalefollows from Proposition 1.4. In terms of the family of operators

Q (t1, t2) : 0 ≤ t1 ≤ t2 ≤ Tthe equality in (1.41) can be rewritten as

∫ t

s

Q (s, ρ) L(ρ)ϕ (ρ, ·) (x) dρ + Q(t, t)ϕ (t, ·) (x)−Q(s, t)ϕ (t, ·) (x)

= ϕ(t, x)− ϕ(s, x)−∫ t

s

Q (s, ρ)∂ϕ

∂ρ(ρ, ·) (x)dρ. (1.42)

From (1.42) we infer that

L(s)ϕ(s, ·)(x) = − limt↓s

Q(t, t)ϕ (t, ·) (x)−Q(s, t)ϕ (t, ·) (x)

t− s. (1.43)

Equality (1.42) also yields the following result. If ϕ ∈ D(L) is such that

L(ρ)ϕ (ρ, ·) (y) = −∂ϕ

∂ρ(ρ, y),

thenϕ (s, x) = Q (ρ, t) ϕ (t, ·) (x) = Es,x [ϕ (t,X(t))] . (1.44)

Since 0 ≤ s ≤ t ≤ T are arbitrary from (1.44) we see

Q (s, t′) ϕ (t′, ·) (x) = Q (s, t) Q (t, t′) ϕ (t′, ·) (x) 0 ≤ s ≤ t ≤ t′ ≤ T, x ∈ E. (1.45)

If in (1.45) we (may) choose the function ϕ (t′, y) arbitrary, then the family Q(s, t),0 ≤ s ≤ t ≤ T , automatically is a propagator in the space C0(E) in the sense thatQ (s, t) Q (t, t′) = Q (s, t′), 0 ≤ s ≤ t ≤ t′ ≤ T . For details on propagators or evolutionfamilies see [11].

1.11. Remark. In the sequel we want to discuss solutions to equations of the form:

∂

∂tu (t, x) + L(t)u (t, ·) (x) + f

(t, x, u (t, x) ,∇L

u (t, x))

= 0. (1.46)

For a preliminary discussion on this topic see Theorem 1.7. Under certain hypotheses onthe function f we will give some existence and uniqueness results. Let m be (equivalentto) the Lebesgue measure in Rd. In a concrete situation where every operator L(t) is a


genuine diffusion operator in L2(Rd, m

)we consider the following Backward Stochastic

Differential equation

u (s,X(s)) = Y (T, X(T )) +

∫ T

s

f(ρ,X(ρ), u (ρ, X(ρ)) ,∇L

u (ρ,X(ρ)))

dρ

−∫ T

s

∇Lu (ρ,X(ρ)) dW (ρ) . (1.47)

Here we suppose that the process t 7→ X(t) is a solution to a genuine stochasticdifferential equation driven by Brownian motion and with one-dimensional distribution

u(t, x) satisfying L(t)u (t, ·) (x) =∂u

∂t(t, x). In fact in that case we will not consider

the equation in (1.47), but we will try to find an ordered pair (Y, Z) such that

Y (s) = Y (T ) +

∫ T

s

f (ρ, X(ρ), Y (ρ) , Z (ρ)) dρ−∫ T

s

〈Z (ρ) , dW (ρ)〉 . (1.48)

If the pair (Y, Z) satisfies (1.48), then u (s, x) = Es,x [Y (s)] satisfies (1.46). MoreoverZ(s) = ∇L

u (s,X(s)) = ∇Lu (s, x), Ps,x-almost surely. For more details see section 2 in

Pardoux [18].

1.12. Remark. Some remarks follow:

(a) In section 2 weak solutions to BSDEs are studied.(b) In section 7 of [26] and in section 2 of Pardoux [18] strong solutions to BSDEs

are discussed: these results are due to Pardoux and collaborators.(c) BSDEs go back to Bismut [6].

(d) If L(s)u(s, x) =1

2

d∑

j,k=1

aj,k(s, x)∂2u

∂xjxk

(s, x) +d∑

j=1

bj(s, x)∂u

∂xj

(s, x), then

Γ1 (u, v) (s, x) =d∑

j,k=1

aj,k(s, x)∂u

∂xj

(s, x)∂v

∂xk

(s, x).

As a corollary to theorems 1.7 and 3.6 we have the following result.

1.10. Corollary. Suppose that the function u solves the following

∂u

∂s(s, y) + L(s)u(s, ·) (y) + f

(s, y, u(s, y),∇L

u(s, y))

= 0;

u (T, X(T )) = ξ ∈ L2 (Ω,FτT ,Pτ,x) .

(1.49)

Let the pair (Y,M) be a solution to

Y (t) = ξ +

∫ T

t

f (s,X(s), Y (s), ZM(s)) ds + M(t)−M(T ), (1.50)

with M(τ) = 0. Then

(Y (t),M(t)) = (u (t,X(t)) ,Mu(t)) ,

where

Mu(t) = u (t,X(t))− u (τ, X(τ))−∫ t

τ

L(s)u (s, ·) (X(s)) ds−∫ t

τ

∂u

∂s(s, X(s)) ds.


Notice that the processes s 7→ ∇Lu (s,X(s)) and s 7→ ZMu(s) may be identified and

that ZMu(s) only depends on (s,X(s)). The decomposition

u (t,X(t))−u (τ, X(τ)) =

∫ t

τ

(∂u

∂s(s,X(s)) + L(s)u (s, ·) (X(s))

)ds+Mu(t)−Mu(τ)

(1.51)splits the process t 7→ u (t,X(t)) − u (τ,X(τ)) into a part which is bounded variation(i.e. the part which is absolutely continuous with respect to Lebesgue measure on [τ, T ])and a Pτ,x-martingale part Mu(t)−Mu(τ) (which in fact is a martingale difference part).

If L(s) = 12∆, then X(s) = W (s) (standard Wiener process or Brownian motion)

and (1.51) can be rewritten as

u (t, W (t))− u (τ,W (τ)) =

∫ t

τ

(∂u

∂s(s,W (s)) +

1

2∆u (s, ·) (W (s))

)ds

+

∫ t

τ

∇u (s, ·) (W (s)) dW (s) (1.52)

where∫ t

τ∇u (s, ·) (W (s)) dW (s) is to be interpreted as an Ito integral.

1.13. Remark. Suggestions for further research:

(a) Find “explicit solutions” to BSDEs with a linear drift part. This should be atype of Cameron-Martin formula or Girsanov transformation.

(b) Treat weak (and strong) solutions BDSDEs in a manner similar to what ispresented here for BSDEs.

(c) Treat weak (strong) solutions to BSDEs generated by a function f which is notnecessarily of linear growth but for example of quadratic in one or both of itsentries Y (t) and ZM(t).

(d) Can anything be done if f depends not only on s, x, u(s, x), ∇u (s, x), but alsoon L(s)u (s, ·) (x)?

1.11. Proposition. Let the functions f , g ∈ D(L) be such that their product fg alsobelongs to D(L). Then Γ1 (f, g) is well defined and for (s, x) ∈ [0, T ]×E the followingequality holds:

L(s) (fg) (s, ·) (x)− f(s, x)L(s)g (s, ·) (x)− L(s)f (s, ·) (x)g(s, x) = Γ1 (f, g) (s, x).(1.53)

Proof. Let the functions f and g be as in Proposition 1.11. For h > 0 we have:

(f (X(s + h))− f (X(s))) (g (X(s + h))− g (X(s)))

= f (X(s + h)) g (X(s + h))− f (X(s)) g (X(s)) (1.54)

− f (X(s)) (g (X(s + h))− g (X(s)))− (f (X(s + h))− f (X(s))) g (X(s)) .

Then we take expectations with respect to Es,x, divide by h > 0, and pass to the limitas h ↓ 0 to obtain equality (1.53) in Proposition 1.11. ¤

2. A probabilistic approach: weak solutions

In this section and also in sections 3 we will study BSDE’s on a single probabilityspace. In the sections 4 and 4 we will consider Markov families of probability spaces.In the present section we write P instead of P0,x, and similarly for the expectations E


and E0,x. Here we work on the interval [0, T ]. Since we are discussing the martingaleproblem and basically only the distributions of the process t 7→ X(t), t ∈ [0, T ], thesolutions we obtain are of weak type. In case we consider strong solutions we applya martingale representation theorem (in terms of Brownian Motion). In Section 4we will also use this result for probability measures of the form Pτ,x on the interval[τ, T ]. In this section we consider a pair of Ft = F0

t -adapted processes (Y, M) ∈L2

(Ω,FT ,P;Rk

)× L2(Ω, FT ,P : Rk

)such that Y (0) = M(0) and such that

Y (t) = Y (T ) +

∫ T

t

f (s,X(s), Y (s), ZM(s)) ds + M(t)−M(T ) (2.1)

where M is a P-martingale with respect to the filtration Ft = σ (X(s) : s ≤ t). In [26]we will employ the results of the present section with P = Pτ,x, where (τ, x) ∈ [0, T ]×E.

2.1. Proposition. Let the pair (Y, M) be as in (2.1), and suppose that Y (0) = M(0).Then

Y (t) = M(t)−∫ t

0

f (s,X(s), Y (s), ZM(s)) ds, and (2.2)

Y (t) = E[Y (T ) +

∫ T

t

f (s,X(s), Y (s), ZM(s)) ds∣∣ Ft

]; (2.3)

M(t) = E[Y (T ) +

∫ T

0


]. (2.4)

The equality in (2.2) shows that the process M is the martingale part of the semi-martingale Y .

Proof. The equality in (2.3) follows from (2.1) and from the fact that M is a martingale.Next we calculate

E[Y (T ) +

∫ T

0


]

= E[Y (T ) +

∫ T

t


]+

∫ t

0

f (s,X(s), Y (s), ZM(s)) ds

= Y (t) +

∫ t

0


(employ (2.1))

= Y (T ) +

∫ T

t

f (s,X(s), Y (s), ZM(s)) ds + M(t)−M(T )

+

∫ t

0


= Y (T ) +

∫ T

0

f (s,X(s), Y (s), ZM(s)) ds + M(t)−M(T )

= M(T ) + M(t)−M(T ) = M(t). (2.5)


The equality in (2.5) shows (2.4). Since

M(T ) = Y (T ) +

∫ T

0


the equality in (2.2) follows. ¤In the following theorem we write z = ZM(s) and y belongs to Rk.

2.2. Theorem. Suppose that there exist finite constants C1 and C2 such that

〈y2 − y1, f (s, x, y2, z)− f (s, x, y1, z)〉 ≤ C1 |y2 − y1|2 ; (2.6)

|f (s, x, y, ZM2(s))− f (s, x, y, ZM1(s))|2 ≤ C22

d

ds〈M2 −M1,M2 −M1〉 (s). (2.7)

Then there exists a unique pair of adapted processes (Y, M) such that Y (0) = M(0)and such that the process M is the martingale part of the semi-martingale Y :

Y (t) = M(t)−M(T ) + Y (T ) +

∫ T

t


= M(t)−∫ t

0

f (s,X(s), Y (s), ZM(s)) ds. (2.8)

Proof. The uniqueness follows from Corollary 3.4 of Theorem 3.3 below. In the exis-tence part of the proof of Theorem 2.2 we will approximate the function f by Lipschitzcontinuous functions fδ, 0 < δ < (2C1)

−1, where each function fδ has Lipschitz con-stant δ−1, but at the same time inequality (2.7) remains valid for fixed second variable(in an appropriate sense). It follows that for the functions fδ (2.7) remains valid andthat (2.6) is replaced with

|fδ (s, x, y2, z)− fδ (s, x, y1, z)| ≤ 1

δ|y2 − y1| . (2.9)

In the uniqueness part of the proof it suffices to assume that (2.6) holds. In Theorem3.6 we will see that the monotonicity condition (2.6) also suffices to prove the existence.For details the reader is referred to the propositions 3.7 and 3.8, Corollary 3.9, and toProposition 3.10. In fact for M ∈ M2 fixed, and the function y 7→ f (s, x, y, ZM(s))satisfying (2.6) the function y 7→ y − δf (s, x, y, ZM(s)) is surjective as a mappingfrom Rk to Rk and its inverse exists and is Lipschitz continuous with constant 2. TheLipschitz continuity is proved in Proposition 3.8. The surjectivity of this mapping is aconsequence of Theorem 1 in [9]. As pointed out by Crouzeix et al the result followsfrom a non-trivial homotopy argument. A relatively elementary proof of Theorem 1 in[9] can be found for a continuously differentiable function in Hairer and Wanner [12]:see Theorem 14.2 in Chapter IV. For a few more details see Remark 3.2. Let fs,M bethe mapping y 7→ f (s, y, ZM(s)), and put

fδ (s, x, y, ZM(s)) = f(s, x, (I − δfs,x,M)−1 , ZM(s)

). (2.10)

Then the functions fδ, 0 < δ < (2C1)−1, are Lipschitz continuous with constant δ−1.

Proposition 3.10 treats the transition from solutions of BSDE’s with generator fδ withfixed martingale M ∈ M2 to solutions of BSDE’s driven by f with the same fixedmartingale M . Proposition 3.7 contains the passage from solutions (Y, N) ∈ S2×M2 toBBSDE’s with generators of the form (s, y) 7→ f (s, y, ZM(s)) for any fixed martingale


M ∈ M2 to solutions for BSDE’s of the form (2.8) where the pair (Y, M) belongs toS2 ×M2. By hypothesis the process s 7→ f (s, x, Y (s), ZM(s)) satisfies (2.6) and (2.7).Essentially speaking a combination of these observations show the result in Theorem2.2. ¤2.1. Remark. In the literature functions with the monotonicity property are also calledone-sided Lipschitz functions. In fact Theorem 2.2, with f(t, x, ·, ·) Lipschitz continuousin both variables, will be superseded by Theorem 3.5 in the Lipschitz case and byTheorem 3.6 in case of monotonicity in the second variable and Lipschitz continuity inthe third variable. The proof of Theorem 2.2 is part of the results in Section 3. Theorem4.1 contains a corresponding result for a Markov family of probability measures. Itsproof is omitted, it follows the same lines as the proof of Theorem 3.6.

3. Existence and Uniqueness of solutions to BSDE’s

The equation in (1.46) can be phrased in a semi-linear setting as follows. Find afunction u (t, x) which satisfies the following partial differential equation:

∂u

∂s(s, x) + L(s)u (s, x) + f

(s, x, u(s, x),∇L

u (s, x))

= 0;

u(T, x) = ϕ (T, x) , x ∈ E.(3.1)

Here ∇Lf2

(s, x) is the linear functional f1 7→ Γ1 (f1, f2) (s, x) for smooth enough func-

tions f1 and f2. For s ∈ [0, T ] fixed the symbol ∇Lf2

stands for the linear mappingf1 7→ Γ1 (f1, f2) (s, ·). One way to treat this kind of equation is considering the follow-ing backward problem. Find a pair of adapted processes (Y, ZY ), satisfying

Y (t)− Y (T )−∫ T

t

f (s,X(s), Y (s), Z(s) (·, Y )) ds = M(t)−M(T ), (3.2)

where M(s), t0 < t ≤ s ≤ T , is a forward local Pt,x-martingale (for every T > t > t0).The symbol ZY1 , Y1 ∈ S2

([0, T ],Rk

), stands for the functional

ZY1 (Y2) (s) = Z(s) (Y1(·), Y2(·)) =d

ds〈Y1(·), Y2(·)〉 (s), Y2 ∈ S2

([0, T ],Rk

). (3.3)

If the pair (Y, ZY ) satisfies (3.2), then ZY = ZM . Instead of trying to find the pair(Y, ZY ) we will try to find a pair (Y, M) ∈ S2

([0, T ],Rk

)×M2([0, T ],Rk

)such that

Y (t) = Y (T ) +

∫ T

t

f (s,X(s), Y (s), ZM(s)) ds + M(t)−M(T ).

Next we define the spaces S2([0, T ],Rk

)and M2

([0, T ],Rk

): compare with the defini-

tions 1.8 and 1.9.

3.1. Definition. Let (Ω, F,P) be a probability space, and let Ft, t ∈ [0, T ], be afiltration on F. Let t 7→ Y (t) be an stochastic process with values in Rk which isadapted to the filtration Ft and which is P-almost surely continuous. Then Y is saidto belong to S2

([0, T ],Rk

)provided that

E

[sup

t∈[0,T ]

|Y (t)|2]

< ∞.


3.2. Definition. The space of Rk-valued martingales in L2(Ω,F,P;Rk

)is denoted

by M2([0, T ],Rk

). So that a continuous martingale t 7→ M(t) − M(0) belongs to

M2([0, T ],Rk

)if

E[|M(T )−M(0)|2] < ∞. (3.4)

Since the process t 7→ |M(t)|2 − |M(0)|2 − 〈M,M〉 (t) + 〈M, M〉 (0) is a martingaledifference we see that

E[|M(T )−M(0)|2] = E [〈M, M〉 (T )− 〈M, M〉 (0)] , (3.5)

and hence a martingale difference t 7→ M(t) − M(0) in L2(Ω,F,P;Rk

)belongs to

M2([0, T ],Rk

)if and only if E [〈M, M〉 (T )− 〈M,M〉 (0)] is finite. By the Burkholder-

Davis-Gundy this is the case if and only if E[sup0<t<T |M(t)−M(0)|2] < ∞.

To be precise, let M(s), t ≤ s ≤ T , be a continuous local L2-martingale taking valuesin Rk. Put M∗(s) = supt≤τ≤s |M(τ)|. Fix 0 < p < ∞. The Burkholder-Davis-Gundyinequality says that there exist universal finite and strictly positive constants cp andCp such that

cpE[(M∗(s))2p] ≤ E [〈M(·),M(·)〉p (s)] ≤ CpE

[(M∗(s))2p] , t ≤ s ≤ T. (3.6)

If p = 1, then cp = 14, and if p = 1

2, then cp = 1

8

√2. For more details and a proof see

e.g. Ikeda and Watanabe [13].

The following theorem will be employed to prove continuity of solutions to BSDE’s.It also implies that BSDE’s as considered by us possess at most unique solutions.The variables (Y, M) and (Y ′, M ′) attain their values in Rk endowed with its Eu-

clidean inner-product 〈y′, y〉 =∑k

j=1 y′jyj, y′, y ∈ Rk. Processes of the form s 7→f (s, Y (s), ZM(s)) are progressively measurable processes whenever the pair (Y, M) be-longs to the space mentioned in (3.7) mentioned in next theorem.

3.3. Theorem. Let the pairs (Y, M) and (Y ′,M ′), which belong to the space

L2([0, T ]× Ω,F0

T , dt× P)×M2(Ω, F0

T ,P), (3.7)

be solutions to the following BSDE’s:

Y (t) = Y (T ) +

∫ T

t

f (s, Y (s), ZM(s)) ds + M(t)−M(T ), and (3.8)

Y ′(t) = Y ′(T ) +

∫ T

t

f ′ (s, Y ′(s), ZM ′(s)) ds + M ′(t)−M ′(T ) (3.9)

for 0 ≤ t ≤ T . In particular this means that the processes (Y,M) and (Y ′,M ′) areprogressively measurable and are square integrable. Suppose that the coefficient f ′ satis-fies the following monotonicity and Lipschitz condition. There exist some positive andfinite constants C ′

1 and C ′2 such that the following inequalities hold for all 0 ≤ t ≤ T :

〈Y ′(t)− Y (t), f ′ (t, Y ′(t), ZM ′(t))− f ′ (t, Y (t), ZM ′(t))〉 ≤ (C ′1)

2 |Y ′(t)− Y (t)|2 , (3.10)

and

|f ′ (t, Y (t), ZM ′(t))− f ′ (t, Y (t), ZM(t))|2 ≤ (C ′2)

2 d

dt〈M ′ −M, M ′ −M〉 (t). (3.11)


Then the pair (Y ′ − Y,M ′ −M) belongs to S2(Ω,F0

T ,P;Rk) ×M2

(Ω, F0

T ,P;Rk), and

there exists a constant C ′ which depends on C ′1, C ′

2 and T such that

E[

sup0<t<T

|Y ′(t)− Y (t)|2 + 〈M ′ −M, M ′ −M〉 (T )

]

≤ C ′E[|Y ′(T )− Y (T )|2 +

∫ T

0

|f ′ (s, Y (s), ZM(s))− f (s, Y (s), ZM(s))|2 ds

]. (3.12)

3.1. Remark. From the proof it follows that for C ′ we may choose C ′ = 260eγT , whereγ = 1 + 2 (C ′

1)2 + 2 (C ′

2)2.

By taking Y (T ) = Y ′(T ) and f (s, Y (s), ZM(s)) = f ′ (s, Y (s), ZM(s)) it also im-plies that BSDE’s as considered by us possess at most unique solutions. A preciseformulation reads as follows.

3.4. Corollary. Suppose that the coefficient f satisfies the monotonicity condition(3.10) and the Lipschitz condition (3.11). Then there exists at most one pair (Y,M) ∈L2 ([0, T ]× Ω,F0

T , dt× P) × M2 (Ω,F0T ,P) which satisfies the backward stochastic dif-

ferential equation (3.8).

Proof of Theorem 3.3. Put Y = Y ′ − Y and M = M ′ − M . From Ito’s formula itfollows that∣∣Y (t)

∣∣2 +⟨M, M

⟩(T )− ⟨

M, M⟩(t)

=∣∣Y (T )

∣∣2 + 2

∫ T

t

⟨Y (s), f ′ (s, Y ′(s), ZM ′(s))− f ′ (s, Y (s), ZM ′(s))

⟩ds

+ 2

∫ T

t

⟨Y (s), f ′ (s, Y (s), ZM ′(s))− f ′ (s, Y (s), ZM(s))

⟩

+ 2

∫ T

t

⟨Y (s), f ′ (s, Y (s), ZM(s))− f (s, Y (s), ZM(s))

⟩ds

− 2

∫ T

t

⟨Y (s), dM(s)

⟩. (3.13)

Inserting the inequalities (3.10) and (3.11) into (3.13) shows:∣∣Y (t)

∣∣2 +⟨M, M

⟩(T )− ⟨

M, M⟩(t)

≤∣∣Y (T )

∣∣2 + 2 (C ′1)

2

∫ T

t

∣∣Y (s)∣∣2 ds + 2C ′

2

∫ T

t

∣∣Y (s)∣∣(

d

ds

⟨M, M

⟩(s)

)1/2

ds

+ 2

∫ T

t

∣∣Y (s)∣∣ |f ′ (s, Y (s), ZM(s))− f (s, Y (s), ZM(s))| ds

− 2

∫ T

t

⟨Y (s), dM(s)

⟩. (3.14)

The elementary inequalities 2ab ≤ 2C ′2a

2 +b2

2C ′2

and 2ab ≤ a2 + b2, 0 ≤ a, b ∈ R, apply

to the effect that∣∣Y (t)

∣∣2 +1

2

(⟨M, M

⟩(T )− ⟨

M, M⟩(t)

)


≤∣∣Y (T )

∣∣2 +(1 + 2 (C ′

1)2+ 2 (C ′

2)2)∫ T

t

∣∣Y (s)∣∣2 ds

+

∫ T

t

|f ′ (s, Y (s), ZM(s))− f (s, Y (s), ZM(s))|2 ds

− 2

∫ T

0

⟨Y (s), dM(s)

⟩+ 2

∫ t

0

⟨Y (s), dM(s)

⟩. (3.15)

For a concise formulation of the relevant inequalities we introduce the following func-tions and the constant γ:

AY (t) = E[∣∣Y (t)

∣∣2],

AM(t) = E[⟨

M, M⟩(T )− ⟨

M, M⟩(t)

],

C(t) = E[|f ′ (s, Y (s), ZM(s))− f (s, Y (s), ZM(s))|2

],

B(t) = AY (T ) +

∫ T

t

C(s)ds = B(T ) +

∫ T

t

C(s)ds, and

γ = 1 + 2 (C ′1)

2+ 2 (C ′

2)2. (3.16)

Using the quantities in (3.16) and remembering the fact that the final term in (3.15)represents a martingale difference, the inequality in (3.15) implies:

AY (t) +1

2AM(t) ≤ B(t) + γ

∫ T

t

AY (s)ds. (3.17)

Using (3.17) and employing induction with respect to n yields:

AY (t) +1

2AM(t) ≤ B(t) +

∫ T

t

n∑

k=0

γk+1(T − s)k

k!B(s)ds +

∫ T

t

γn+2(T − s)n+1

(n + 1)!AY (s)ds.

(3.18)Passing to the limit for n →∞ in (3.18) results in:

AY (t) +1

2AM(t) ≤ B(t) + γ

∫ T

t

eγ(T−s)B(s)ds. (3.19)

Since B(t) = AY (T ) +∫ T

tC(s)ds we infer from (3.19):

AY (t) +1

2AM(t) ≤ eγ(T−t)

(AY (T ) +

∫ T

t

C(s)ds

). (3.20)

By first taking the supremum over 0 < t < T and then taking expectations in (3.15)gives:

E[

sup0<t<T

∣∣Y (t)∣∣2

]≤ E

[∣∣Y (T )∣∣2

]+

(1 + 2 (C ′

1)2+ 2 (C ′

2)2) ∫ T

0

E[∣∣Y (s)

∣∣2]ds

+

∫ T

0

E[|f ′ (s, Y (s), ZM(s))− f (s, Y (s), ZM(s))|2

]ds

+ 2E[

sup0<t<T

∫ t

0

⟨Y (s), dM(s)

⟩]. (3.21)


The quadratic variation of the martingale t 7→ ∫ t

0

⟨Y (s), dM(s)

⟩is given by the increas-

ing process t 7→ ∫ t

0

∣∣Y (s)∣∣2 d

⟨M, M

⟩(s). From the Burkholder-Davis-Gundy inequality

(3.6) we know that

E[

sup0<t<T

∫ t

0

⟨Y (s), dM(s)

⟩] ≤ 4√

2E

[(∫ T

0

∣∣Y (s)∣∣2 d

⟨M, M

⟩(s)

)1/2]

. (3.22)

For more details on the Burkholder-Davis-Gundy inequality, see e.g. Ikeda and Watan-abe [13]. Again we use an elementary inequality 4

√2ab ≤ 1

4a2 + 32b2 and plug it into

(3.22) to obtain

E[

sup0<t<T

∫ t

0

⟨Y (s), dM(s)

⟩] ≤ 4√

2E

[sup

0<t<T

∣∣Y (t)∣∣(∫ T

0

d⟨M, M

⟩(s)

)1/2]

≤ 1

4E

[sup

0<t<T

∣∣Y (t)∣∣2

]+ 32E

[⟨M, M

⟩(T )

]. (3.23)

From (3.20) we also infer

γ

∫ T

0

AY (s)ds ≤ γ

∫ T

0

eγ(T−s)

(AY (T ) +

∫ T

s

C(ρ)dρ

)ds

=(eγT − 1

)AY (T ) +

∫ T

0

(eγT − eγρ

)C(ρ)dρ. (3.24)

Inserting the inequalities (3.23) and (3.24) into (3.21) yields:

E[

sup0<t<T

∣∣Y (t)∣∣2

](3.25)

≤ eγTE[∣∣Y (T )

∣∣2]

+ eγT

∫ T

0

C(s)ds +1

2E

[sup

0<t<T

∣∣Y (t)∣∣2

]+ 64E

[⟨M, M

⟩(T )

].

From (3.20) we also get

E[⟨

M, M⟩(T )

]= AM(0)

≤ 2eγT

(AY (T ) +

∫ T

0

C(s)ds

)= 2eγT

(E

[∣∣Y (T )∣∣2

]+

∫ T

0

C(s)ds

). (3.26)

A combination of (3.26) and (3.25) results in

E[

sup0<t<T

∣∣Y (t)∣∣2

]≤ 258eγT

(E

[∣∣Y (T )∣∣2

]+

∫ T

0

C(s)ds

). (3.27)

Adding the right- and left-hand sides of (3.25) and (3.26) proves Theorem 3.3 with theconstant C ′ given by C ′ = 260eγT , where γ = 1 + 2 (C ′

1)2 + 2 (C ′

2)2. ¤

In the definitions 3.1 and 3.2 the spaces S2([0, T ],Rk

)and M2

([0, T ],Rk

)are defined.

In Theorem 3.6 we will replace the Lipschitz condition (3.28) in Theorem 3.5 for thefunction Y (s) 7→ f (s, Y (s), ZM(s)) with the (weaker) monotonicity condition (3.50).Here we write y for the variable Y (s) and z for ZM(s). It is noticed that we considera probability space (Ω, F,P) with a filtration (Ft)t∈[0,T ] = (F0

t )t∈[0,T ] where FT = F.


3.5. Theorem. Let f : [0, T ]×Rk× (M2)∗ → Rk be a Lipschitz continuous in the sense

that there exists finite constants C1 and C2 such that for any two pairs of processes(Y, M) and (U,N) ∈ S2

([0, T ],Rk

)×M2([0, T ],Rk

)the following inequalities hold for

all 0 ≤ s ≤ T :

|f (s, Y (s), ZM(s))− f (s, U(s), ZM(s))| ≤ C1 |Y (s)− U(s)| , and (3.28)

|f (s, Y (s), ZM(s))− f (s, Y (s), ZN(s))| ≤ C2

(d

ds〈M −N, M −N〉 (s)

)1/2

. (3.29)

Suppose that E[∫ T

0|f(s, 0, 0)|2 ds

]< ∞. Then there exists a unique pair (Y, M) ∈

S2([0, T ],Rk

)×M2([0, T ],Rk

)such that

Y (t) = ξ +

∫ T

t

f (s, Y (s), ZM(s)) ds + M(t)−M(T ), (3.30)

where Y (T ) = ξ ∈ L2(Ω,FT ,Rk

)is given and where Y (0) = M(0).

For brevity we write

S2 ×M2 = S2([0, T ],Rk

)×M2([0, T ],Rk

)= S2

(Ω,F0

T ,P;Rk)×M2

(Ω,F0

T ,P;Rk).

In fact we employ this theorem with the function f replaced with fδ, 0 < δ < (2C1)−1,

where fδ is defined by

fδ (s, y, ZM(s)) = f(s, (I − δfs,M)−1 , ZM(s)

). (3.31)

Here fs,M(y) = f (s, y, ZM(s)). If the function f is monotone (or one-sided Lipschitz)in the second variable with constant C1, and Lipschitz in the second variable withconstant C2, then the function fδ is Lipschitz in y with constant δ−1.

Proof. The proof of the uniqueness part follows from Corollary 3.4.In order to prove existence we proceed as follows. By induction we define a sequence

(Yn,Mn) in the space S2 ×M2 as follows.

Yn+1(t) = E[ξ +

∫ T

t

f (s, Yn(s), ZMn(s)) ds∣∣ Ft

], and (3.32)

Mn+1(t) = E[ξ +

∫ T

0


], (3.33)

Then, since the process s 7→ f (s, Yn(s),Mn(s)) is adapted we have:

ξ +

∫ T

t

f (s, Yn(s), ZMn(s)) ds + Mn+1(t)−Mn+1(T )

= ξ +

∫ T

t

f (s, Yn(s), ZMn(s)) ds + E[ξ +

∫ T

0


]

− E[ξ +

∫ T

0

f (s, Yn(s), ZMn(s)) ds∣∣ FT

]

= ξ +

∫ T

t

f (s, Yn(s), ZMn(s)) ds + E[ξ +

∫ T

0


]

− ξ −∫ T

0

f (s, Yn(s), ZMn(s)) ds


= E[ξ +

∫ T

0


]−

∫ t

0

f (s, Yn(s), ZMn(s)) ds

= E[ξ +

∫ T

t


]= Yn+1(t). (3.34)

Suppose that the pair (Yn,Mn) belongs S2 ×M2. We first prove that (Yn+1,Mn+1) isa member of S2 ×M2. Therefore we fix α = 1 + C2

1 + C22 ∈ R where C1 and C2 are as

in (3.28) and (3.29) respectively. From Ito’s formula we get:

e2αt |Yn+1(t)|2 + 2α

∫ T

t

e2αs |Yn+1(s)|2 ds +

∫ T

t

e2αsd 〈Mn+1,Mn+1〉 (s)

= e2αT |Yn+1(T )|2 + 2

∫ T

t

e2αs 〈Yn+1(s), f (s, Yn(s), ZMn(s))− f (s, Yn(s), 0)〉 ds

+ 2

∫ T

t

e2αs 〈Yn+1(s), f (s, Yn(s), 0)− f (s, 0, 0)〉 ds

+ 2

∫ T

t

e2αs 〈Yn+1(s), f (s, 0, 0)〉 ds− 2

∫ T

t

e2αs 〈Yn+1(s), dMn+1(s)〉 . (3.35)

We employ (3.28) and (3.29) to obtain from (3.35):

e2αt |Yn+1(t)|2 + 2α

∫ T

t


∫ T

t


≤ e2αT |Yn+1(T )|2 + 2C2

∫ T

t

e2αs |Yn+1(s)|(

d

ds〈Mn,Mn〉 (s)

)1/2

ds

+ 2C1

∫ T

t

e2αs |Yn+1(s)| |Yn(s)| ds

+ 2

∫ T

t

e2αs |Yn+1(s)| |f (s, 0, 0)| ds− 2

∫ T

t

e2αs 〈Yn+1(s), dMn+1(s)〉 . (3.36)

The elementary inequalities 2ab ≤ 2Cja2 +

b2

2Cj

, a, b ∈ R, j = 0, 1, 2, with C0 = 1, in

combination with (3.36) yields

e2αt |Yn+1(t)|2 + 2α

∫ T

t


∫ T

t


≤ e2αT |Yn+1(T )|2 + 2C22

∫ T

t

e2αs |Yn+1(s)|2 ds +1

2

∫ T

t

e2αsd 〈Mn,Mn〉 (s)

+ 2C21

∫ T

t

e2αs |Yn+1(s)|2 ds +1

2

∫ T

t

e2αs |Yn(s)|2 ds

+

∫ T

t


∫ T

t

e2αs |f (s, 0, 0)|2 ds− 2

∫ T

t

e2αs 〈Yn+1(s), dMn+1(s)〉 ,(3.37)


and hence by the choice of α from (3.37) we infer:

e2αt |Yn+1(t)|2 +

∫ T

t


∫ T

t


+ 2

∫ T

0

e2αs 〈Yn+1(s), dMn+1(s)〉

≤ e2αT |Yn+1(T )|2 +1

2

∫ T

t

e2αsd 〈Mn,Mn〉 (s) +1

2

∫ T

t

e2αs |Yn(s)|2 ds

+

∫ T

t

e2αs |f (s, 0, 0)|2 ds + 2

∫ t

0

e2αs 〈Yn+1(s), dMn+1(s)〉 . (3.38)

The following steps can be justified by observing that the process Yn+1 belongs to thespace L2 (Ω,F0

T ,P), and that sup0≤t≤T |Yn+1(t)| < ∞ P-almost surely. By stoppingthe process Yn+1(t) at the stopping time τN being the first time t ≤ T that |Yn+1(t)|exceeds N . In inequality (3.38) we then replace t by t∧τN and proceed as below with thestopped processes instead of the processes itself. Then we use monotone convergencetheorem to obtain inequality (3.41). By the same approximation argument we may

assume that E[∫ T

te2αs 〈Yn+1(s), dMn+1(s)〉

]= 0. Hence (3.38) implies that

E[e2αt |Yn+1(t)|2 +

∫ T

t


∫ T

t

e2αsd 〈Mn+1,Mn+1〉 (s)]

≤ e2αTE[|Yn+1(T )|2] +

1

2E

[∫ T

t

e2αsd 〈Mn,Mn〉 (s)]

+1

2E

[∫ T

t

e2αs |Yn(s)|2 ds

]

+ E[∫ T

t

e2αs |f (s, 0, 0)|2 ds

]< ∞. (3.39)

Invoking the Burkholder-Davis-Gundy inequality and applying the equality⟨∫ ·

0

e2αs 〈Yn+1(s), dMn+1(s)〉 ,∫ ·

0

e2αs 〈Yn+1(s), dMn+1(s)〉⟩

(t)

=

∫ t

0

e4αs |Yn+1(s)|2 d 〈Mn+1,Mn+1〉 (s)

to (3.38) yields:

E[

sup0<t<T

e2αt |Yn+1(t)|2]

≤ e2αTE[|Yn+1(T )|2] +

1

2E

[∫ T

0


+1

2E

[∫ T

0

e2αs |Yn(s)|2 ds

]

+ E[∫ T

0

e2αs |f (s, 0, 0)|2 ds

]− 2E

[∫ T

0

e2αs 〈Yn+1(s), dMn+1(s)〉]

+ 8√

2E

[(∫ T

0

e4αs |Yn+1(s)|2 d 〈Mn+1,Mn+1〉 (s))1/2

]


(without loss of generality assume that E[∫ T

0e2αs 〈Yn+1(s), dMn+1(s)〉

]= 0)

≤ e2αTE[|Yn+1(T )|2] +

1

2E

[∫ T

0


+1

2E

[∫ T

0

e2αs |Yn(s)|2 ds

]

+ E[∫ T

0

e2αs |f (s, 0, 0)|2 ds

]

+ 8√

2E

[sup

0<t<Teαt |Yn+1(t)|

(∫ T

0

e2αsd 〈Mn+1,Mn+1〉 (s))1/2

]

(8√

2ab ≤ a2

2+ 64b2, a, b ∈ R)

≤ e2αTE[|Yn+1(T )|2] +

1

2E

[∫ T

0


+1

2E

[∫ T

0

e2αs |Yn(s)|2 ds

]

+ E[∫ T

0

e2αs |f (s, 0, 0)|2 ds

]

+1

2E

[sup

0<t<Te2αt |Yn+1(t)|2

]+ 64E

[∫ T

0

e2αsd 〈Mn+1,Mn+1〉 (s)]

(apply (3.39))

≤ 65e2αTE[|Yn+1(T )|2] +

65

2E

[∫ T

0


+65

2E

[∫ T

0

e2αs |Yn(s)|2 ds

]

+ 65E[∫ T

0

e2αs |f (s, 0, 0)|2 ds

]+

1

2E

[sup

0<t<Te2αt |Yn+1(t)|2

]. (3.40)

From (3.40) it follows that

E[

sup0<t<T

e2αt |Yn+1(t)|2]

≤ 130e2αTE[|Yn+1(T )|2] + 130E

[∫ T

0

e2αs |f (s, 0, 0)|2 ds

]

+ 65E[∫ T

0


+ 65E[∫ T

0

e2αs |Yn(s)|2 ds

]< ∞. (3.41)

From (3.39) and (3.41) it follows that the pair (Yn+1,Mn+1) belongs to S2 ×M2.Another application of Ito’s formula shows:

e2αt |Yn+1(t)− Yn(t)|2 + 2α

∫ T

t

e2αs |Yn+1(s)− Yn(s)|2 ds

+

∫ T

t

e2αsd 〈Mn+1 −Mn,Mn+1 −Mn〉 (s)

= e2αT |Yn+1(T )− Yn(T )|2

+ 2

∫ T

t

e2αs⟨Yn+1(s)− Yn(s), f (s, Yn(s), ZMn(s))− f

(s, Yn(s), ZMn−1(s)

)⟩ds


+ 2

∫ T

t

e2αs⟨Yn+1(s)− Yn(s), f


)− f(s, Yn−1(s), ZMn−1(s)

)⟩ds

− 2

∫ T

t

e2αs 〈Yn+1(s)− Yn(s), dMn+1(s)− dMn(s)〉 . (3.42)

From (3.28), (3.29), and (3.42) we infer

e2αt |Yn+1(t)− Yn(t)|2 + 2α

∫ T

t


+

∫ T

t


≤ e2αT |Yn+1(T )− Yn(T )|2

+ 2C2

∫ T

t

e2αs |Yn+1(s)− Yn(s)|(

d

ds〈Mn −Mn−1,Mn −Mn−1〉 (s)

)1/2

ds

+ 2C1

∫ T

t

e2αs |Yn+1(s)− Yn(s)| |Yn(s)− Yn−1(s)| ds

− 2

∫ T

t

e2αs 〈Yn+1(s)− Yn(s), dMn+1(s)− dMn(s)〉

≤ e2αT |Yn+1(T )− Yn(T )|2

+ 2C22

∫ T

t

e2αs |Yn+1(s)− Yn(s)|2 ds +1

2

∫ T

t

e2αsd 〈Mn −Mn−1,Mn −Mn−1〉 (s)

+ 2C21

∫ T

t

e2αs |Yn+1(s)− Yn(s)|2 ds +1

2

∫ T

t

e2αs |Yn(s)− Yn−1(s)|2 ds

− 2

∫ T

t


Since α = 1 + C21 + C2

2 the inequality in (3.43) implies:

e2αt |Yn+1(t)− Yn(t)|2 + 2

∫ T

t


+

∫ T

t


≤ e2αT |Yn+1(T )− Yn(T )|2

+1

2

∫ T

t

e2αsd 〈Mn −Mn−1,Mn −Mn−1〉 (s)

+1

2

∫ T

t


− 2

∫ T

0


+ 2

∫ t

0



Upon taking expectations in (3.44) we see

e2αtE[|Yn+1(t)− Yn(t)|2] + 2E

[∫ T

t


]

+ E[∫ T

t

e2αsd 〈Mn+1 −Mn, Mn+1 −Mn〉 (s)]

≤ e2αTE[|Yn+1(T )− Yn(T )|2]

+1

2E

[∫ T

t

e2αsd 〈Mn −Mn−1,Mn −Mn−1〉 (s)]

+1

2E

[∫ T

t


]. (3.45)

In particular it follows that

2E[∫ T

t


]+ E

[∫ T

t

e2αsd 〈Mn+1 −Mn,Mn+1 −Mn〉 (s)]

≤ 1

2E

[∫ T

t


]+

1

2E

[∫ T

t


,

provided that Yn+1(T ) = Yn(T ). It follows that the sequence (Yn,Mn) converges withrespect to the norm ‖·‖α defined by

∥∥∥∥(

YM

)∥∥∥∥2

α

= E[∫ T

0

e2αs |Y (s)|2 ds +

∫ T

0

e2αsd 〈M,M〉 (s)]

.

Employing a similar reasoning as the one we used to obtain (3.40) and (3.41) from(3.44) we also obtain:

sup0≤t≤T

e2αt |Yn+1(t)− Yn(t)|2

≤ e2αT |Yn+1(T )− Yn(T )|2

+1

2

∫ T

0

e2αsd 〈Mn −Mn−1, Mn −Mn−1〉 (s)

+1

2

∫ T

0


− 2

∫ T

0


+ 2 sup0≤t≤T

∫ t

0


By taking expectations in (3.46), and invoking the Burkholder-Davis-Gundy inequality(3.6) for p = 1

2we obtain;

E[

sup0≤t≤T

e2αt |Yn+1(t)− Yn(t)|2]

≤ e2αTE[|Yn+1(T )− Yn(T )|2]


+1

2E

[∫ T

0


+1

2E

[∫ T

0


]

+ 2E[

sup0≤t≤T

∫ t

0

e2αs 〈Yn+1(s)− Yn(s), dMn+1(s)− dMn(s)〉]

≤ e2αTE[|Yn+1(T )− Yn(T )|2]

+1

2E

[∫ T

0


+1

2E

[∫ T

0


]

+ 8√

2E

[(∫ T

0

e4αs |Yn+1(s)− Yn(s)|2 d 〈Mn+1 −Mn,Mn+1 −Mn〉 (s))1/2

]

(insert the definition of ‖·‖α)

≤ e2αTE[|Yn+1(T )− Yn(T )|2] +

1

2

∥∥∥∥(

Yn − Yn−1

Mn −Mn−1

)∥∥∥∥2

α

+ 8√

2E

[sup

0≤t≤Teαs |Yn+1(s)− Yn(s)|

(∫ T

0

e2αsd 〈Mn+1 −Mn, Mn+1 −Mn〉 (s))1/2

]

(8√

2ab ≤ a2

2+ 64b2, a, b ∈ R)

≤ e2αTE[|Yn+1(T )− Yn(T )|2] +

1

2

∥∥∥∥(

Yn − Yn−1

Mn −Mn−1

)∥∥∥∥2

α

+1

2E

[sup

0≤t≤Te2αs |Yn+1(s)− Yn(s)|2

]

+ 64E[∫ T

0

e2αsd 〈Mn+1 −Mn, Mn+1 −Mn〉 (s)]

. (3.47)

Employing inequality (3.45) (with t = 0) together with (3.47), and the definition ofthe norm ‖·‖α yields the inequality

E[

sup0≤t≤T

e2αt |Yn+1(t)− Yn(t)|2]

+ 129E[∫ T

0


]

+1

2E

[∫ T

0


≤ 131

2e2αTE

[|Yn+1(T )− Yn(T )|2] +131

4

∥∥∥∥(

Yn − Yn−1

Mn −Mn−1

)∥∥∥∥2

α

+1

2E

[sup

0≤s≤Te2αs |Yn+1(s)− Yn(s)|2

]. (3.48)


(In order to justify the transition from (3.46) to (3.48) like in passing from inequality(3.38) to (3.41) a stopping time argument might be required.) Consequently, from(3.48) we see

E[

sup0≤t≤T

e2αt |Yn+1(t)− Yn(t)|2]

+ E[∫ T

0


≤ 131e2αTE[|Yn+1(T )− Yn(T )|2] +

131

2

∥∥∥∥(

Yn − Yn−1

Mn −Mn−1

)∥∥∥∥2

α

. (3.49)

Since by definition Yn(T ) = E[ξ

∣∣ FTT

]for all n ∈ N, this sequence also converges with

respect to the norm ‖·‖S2×M2 defined by

∥∥∥∥(

YM

)∥∥∥∥2

S2×M2

= E[

sup0<s<T

|Y (s)|2]

+ E [〈M, M〉 (T )− 〈M,M〉 (0)] ,

because

Yn+1(0) = Mn+1(0) = E[ξ +

∫ T

0

fn (s, Yn(s), ZMn(s)) ds∣∣ F0

0

], n ∈ N.

This concludes the proof of Theorem 3.5. ¤

In the following theorem we replace the Lipschitz condition (3.28) in Theorem 3.5for the function Y (s) 7→ f (s, Y (s), ZM(s)) with the (weaker) monotonicity condition(3.50). Here we write y for the variable Y (s) and z for ZM(s).

3.6. Theorem. Let f : [0, T ] × Rk × (M2)∗ → Rk be monotone in the variable y and

Lipschitz in z. More precisely, suppose that there exist finite constants C1 and C2 suchthat for any two pairs of processes (Y,M) and (U,N) ∈ S2

([0, T ],Rk

)×M2([0, T ],Rk

)the following inequalities hold for all 0 ≤ s ≤ T :

〈Y (s)− U(s), f (s, Y (s), ZM(s))− f (s, U(s), ZM(s))〉 ≤ C1 |Y (s)− U(s)|2 , (3.50)

|f (s, Y (s), ZM(s))− f (s, Y (s), ZN(s))| ≤ C2

(d

ds〈M −N, M −N〉 (s)

)1/2

, (3.51)

and

|f (s, Y (s), 0)| ≤ f(s) + K |Y (s)| . (3.52)

If E[∫ T

0

∣∣f(s)∣∣2 ds

]< ∞, then there exists a unique pair (Y, M) ∈ S2

([0, T ],Rk

) ×M2

([0, T ],Rk

)such that

Y (t) = ξ +

∫ T

t

f (s, Y (s), ZM(s)) ds + M(t)−M(T ), (3.53)

where Y (T ) = ξ ∈ L2(Ω,FT ,Rk

)is given and where Y (0) = M(0).

In order to prove Theorem 3.6 we need the following proposition, the proof of whichuses the monotonicity condition (3.50) in an explicit manner.


3.7. Proposition. Suppose that for every ξ ∈ L2 (Ω, F0T ,P) and M ∈ M2 there exists

a pair (Y,N) ∈ S2 ×M2 such that

Y (t) = ξ +

∫ T

t

f (s, Y (s), ZM(s)) ds + N(t)−N(T ). (3.54)

Then for every ξ ∈ L2 (Ω, F0T ,P) there exists a unique pair (Y,M) ∈ S2 × M2 which

satisfies (3.53).

The following proposition can be viewed as a consequence of Theorem 12.4 in [12].The result is due to Burrage and Butcher [8] and Crouzeix [10]. The obtained constantsare somewhat different from ours.

3.8. Proposition. Fix a martingale M ∈ M2, and choose δ > 0 in such a way thatδC1 < 1. Here C1 is the constant which occurs in inequality (3.50). Choose, for

given y ∈ Rk, the stochastic variable Y (t) ∈ Rk in such a way that y = Y (t) −δf

(t, Y (t), ZM(t)

). Then the mapping y 7→ f

(t, Y (t), ZM(t)

)is Lipschitz continuous

with a Lipschitz constant which is equal to1

δmax

(1,

δC1

1− δC1

). Moreover, the map-

ping y 7→ I − δf (t, y, ZM(t)) is surjective and has a Lipschitz continuous inverse with

Lipschitz constant1

1− δC1

.

Proof of Proposition 3.8. Let the pair (y1, y2) ∈ Rk×Rk and the pair of Rk×Rk-valued

stochastic variables(Y1(t), Y2(t)

)be such that the following equalities are satisfied:

y1 = Y1(t)− δf(t, Y1(t), ZM(t)

)and y2 = Y2(t)− δf

(t, Y2(t), ZM(t)

). (3.55)

We have to show that there exists a constant C(δ) such that∣∣∣f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)∣∣∣ ≤ C(δ) |y2 − y1| . (3.56)

In order to achieve this we will exploit the inequality:⟨Y2(t)− Y1(t), f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)⟩≤ C1

∣∣∣Y2(t)− Y1(t)∣∣∣2

.

(3.57)Inserting the equalities in (3.55) into (3.57) results in

⟨y2 − y1, f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)⟩

+ δ∣∣∣f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)∣∣∣2

≤ C1 |y2 − y1|2 + 2δC1

⟨y2 − y1, f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)⟩

+ C1δ2∣∣∣f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)∣∣∣2

. (3.58)

Notice that (3.58) is equivalent to:

δ∣∣∣f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)∣∣∣2


≤ C1 |y2 − y1|2 + 2

(δC1 − 1

2

) ⟨y2 − y1, f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)⟩

+ C1δ2∣∣∣f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)∣∣∣2

. (3.59)

Put α =1− |1− 2δC1|

2δC1

. Notice that, since 1 − δC1 > 0, the constant α is positive as

well, α = 1 provided 2δC1 < 1. Since δC1 < 1 and

2∣∣∣⟨y2 − y1, f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)⟩∣∣∣

≤ 1

αδ|y2 − y1|2 + αδ

∣∣∣f(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)∣∣∣2

, (3.60)

the inequality in (3.59) implies

δ∣∣∣f

(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)∣∣∣ ≤ max

(1,

δC1

1− δC1

)|y2 − y1| . (3.61)

The Lipschitz constant is given by C(δ) =1

δmax

(1,

δC1

1− δC1

): compare (3.61) and

(3.56). The surjectivity of the mapping y 7→ y − f (t, y, ZM(t)) is a consequence ofTheorem 1 in Croezeix et al [9]. Denote the mapping y 7→ t (t, y, ZM(t)) by ft,M . Thenfor 0 < 2δC1 < 1 the mapping I − δft,M is invertible. Since

(I − δft,M)−1 = I + δft,M

(t, (I − δft,M)−1 , ZM(t)

),

and since by (3.61) the mapping y 7→ f(t, (I − δft,M)−1 y, ZM(t)

)is Lipschitz con-

tinuous with Lipschitz constant1

δmax

(1,

δC1

1− δC1

)we see that the mapping y 7→

(I − δft,M)−1 y is Lipschitz continuous with constant max

(2,

1

1− δC1

). A somewhat

better constant is obtained by again using (3.57), and replacing

f(t, Y2(t), ZM(t)

)− f

(t, Y1(t), ZM(t)

)

with δ−1 (y2 − y1 − y2 + y1). Then we see:

|y2 − y1|2 − 〈y2 − y1, y2 − y1〉 ≤ δC1 |y2 − y1|2 , (3.62)

and hence

(1− δC1) |y2 − y1|2 ≤ 〈y2 − y1, y2 − y1〉 ≤ |y2 − y1| |y2 − y1| . (3.63)

Altogether this proves Proposition 3.8. ¤

3.9. Corollary. For δ > 0 such that 2δC1 < 1 there exist processes Yδ and Yδ ∈ S2 anda martingale Mδ ∈ M2 such that the following equalities are satisfied:

Yδ(t) = Yδ(t)− δf(t, Yδ(t), ZM(t)

)

= Yδ(T ) +

∫ T

t

f(s, Yδ(s), ZM(s)

)ds + Mδ(t)−Mδ(T ). (3.64)


Proof. From Theorem 1 (page 87) in Crouzeix et al [9] it follows that the mappingy 7→ y − δf (t, y, ZM(t)) is a surjective map from Rk onto itself, provided 0 < δC1 < 1.If y2 and y1 in Rk are such that y2 − δf (t, y2, ZM(t)) = y1 − δf (t, y1, ZM(t)). Then

|y2 − y1|2 = 〈y2 − y1, δf (t, y2, ZM(t))− δf (t, y1, ZM(t))〉 ≤ δC1 |y2 − y1|2 ,

and hence y2 = y1. It follows that the continuous mapping y 7→ y−δf (t, y, ZM(t)) has acontinuous inverse. Denote this inverse by (I − δft,M)−1. Moreover, for 0 < 2δC1 < 1,

the mapping y 7→ f(t, (I − δft,M)−1 , Zm(t)

)is Lipschitz continuous with Lipschitz

constant δ−1, which follows from Proposition 3.8. The remaining assertions in Corollary3.9 are consequences of Theorem 3.5 where the Lipschitz condition in (3.28) was usedwith δ−1 instead of C1. This establishes the proof of Corollary 3.9. ¤3.2. Remark. The surjectivity property of the mapping y 7→ f (s, y, ZM(s)) follows fromTheorem 1 in [9]. The authors use a homotopy argument to prove this theorem forC1 = 0. Upon replacing f (t, y, ZM(t)) with f (t, y, ZM(t)) − C1y the result follows inour version. An elementary proof of Theorem 1 in [9] can be found for a continuouslydifferentiable function in Hairer and Wanner [12]: see Theorem 14.2 in Chapter IV.The author is grateful to Karel in’t Hout (University of Antwerp) for pointing outRunge-Kutta type results and these references.

Proof of Proposition 3.7. The proof of the uniqueness part follows from Corollary 3.4.Fix ξ ∈ L2 (Ω,F0

T ,P), and let the martingale Mn−1 ∈ M2 be given. Then by hypoth-esis there exists a pair (Yn,Mn) ∈ S2 ×M2 which satisfies:

Yn(t) = ξ +

∫ T

t

f(s, Yn(s), ZMn−1(s)

)ds + Mn(t)−Mn(T ). (3.65)

Another use of this hypothesis yields the existence of a pair (Yn+1,Mn+1) ∈ S2 ×M2

which again satisfies (3.65) with n + 1 instead of n. We will prove that the sequence(Yn,Mn) is a Cauchy sequence in the space S2×M2. Put γ = 1+2C1 +2C2

2 . We applyIto’s formula to obtain

eγT |Yn+1(T )− Yn(T )|2 − eγt |Yn+1(t)− Yn(t)|2

= γ

∫ T

t

eγs |Yn+1(s)− Yn(s)|2 ds + 2

∫ T

t

eγs 〈Yn+1(s)− Yn(s), d (Yn+1(s)− Yn(s))〉

+

∫ T

t

eγsd 〈Mn+1 −Mn,Mn+1 −Mn〉 (s)

= γ

∫ T

t

eγs |Yn+1(s)− Yn(s)|2 ds

+ 2

∫ T

t

eγs 〈Yn+1(s)− Yn(s), d (Mn+1(s)−Mn(s))〉

− 2

∫ T

t

eγs 〈Yn+1(s)− Yn(s), f (s, Yn+1(s), ZMn(s))− f (s, Yn(s), ZMn(s))〉 ds

+ 2

∫ T

t

eγs⟨Yn+1(s)− Yn(s), f (s, Yn(s), ZMn(s))− f


)⟩ds

+

∫ T

t



(employ (3.50) and (3.51))

≥ γ

∫ T

t


+ 2

∫ T

t


− 2C1

∫ T

t


− 2C2

∫ T

t

eγs |Yn+1(s)− Yn(s)|(

d

ds〈Mn −Mn−1,Mn −Mn−1〉 (s)

)1/2

ds

+

∫ T

t


(employ the elementary inequality 2ab ≤ 2a2 + 12b2)

≥ (γ − 2C1 − 2C2

2

) ∫ T

t


− 1

2

∫ T

t

eγsd 〈Mn −Mn−1,Mn −Mn−1〉 (s)

+

∫ T

t


+ 2

∫ T

t

eγs 〈Yn+1(s)− Yn(s), d (Mn+1(s)−Mn(s))〉 . (3.66)

From (3.66) we infer the inequality

(γ − 2C1 − 2C2

2

) ∫ T

t

eγs |Yn+1(s)− Yn(s)|2 ds +

∫ T

t


+ eγt |Yn+1(t)− Yn(t)|2 + 2

∫ T

t


≤ eγT |Yn+1(T )− Yn(T )|2 +1

2

∫ T

t

eγsd 〈Mn −Mn−1, Mn −Mn−1〉 (s). (3.67)

By taking expectations in (3.67) we get, since γ = 1 + 2C1 + 2C22 ,

E[∫ T

t


]+ E

[∫ T

t

eγsd 〈Mn+1 −Mn,Mn+1 −Mn〉 (s)]

+ eγtE[|Yn+1(t)− Yn(t)|2]

≤ eγTE[|Yn+1(T )− Yn(T )|2] +

1

2E

[∫ T

t

eγsd 〈Mn −Mn−1,Mn −Mn−1〉 (s)]

. (3.68)

Iterating (3.68) yields:

E[∫ T

t


]+ E

[∫ T

t


+ eγtE[|Yn+1(t)− Yn(t)|2]


≤n∑

k=1

1

2n−keγTE

[|Yk+1(T )− Yk(T )|2] +1

2nE

[∫ T

t

eγsd 〈M1 −M0,M1 −M0〉 (s)]

=1

2nE

[∫ T

t

eγsd 〈M1 −M0,M1 −M0〉 (s)]

(3.69)

where in the last line we used the equalities Yk(T ) = ξ, k ∈ N. From the Burkholder-Davis-Gundy inequality with p = 1

2(see (3.6)) together with (3.69) it follows that

E[

max0≤t≤T

∫ t

0

eγs 〈Yn+1(s)− Yn(s), d (Mn+1 −Mn) (s)〉]

≤ 4√

2E

[(∫ T

0

e2γs |Yn+1(s)− Yn(s)|2 d 〈Mn+1 −Mn,Mn+1 −Mn〉 (s))1/2

]

≤ 4√

2E

[sup

0≤t≤Te

12γs |Yn+1(s)− Yn(s)|

(∫ T

0

eγsd 〈Mn+1 −Mn,Mn+1 −Mn〉 (s))1/2

]

(use the elementary inequality 4√

2ab ≤ 14a2 + 32b2)

≤ 1

4E

[sup

0≤t≤Teγs |Yn+1(s)− Yn(s)|2

]+ 32E

[∫ T

0


≤ 1

4E

[sup


]+

1

2n−5E

[∫ T

0

eγsd 〈M1 −M0,M1 −M0〉 (s)]

.

(3.70)

From (3.67) and (3.70) we obtain

sup0≤t≤T

eγt |Yn+1(t)− Yn(t)|2 + 2

∫ T

0


≤ eγT |Yn+1(T )− Yn(T )|2 +1

2

∫ T

0

eγsd 〈Mn −Mn−1,Mn −Mn−1〉 (s)

+ 2 sup0≤t≤T

∫ t

0

eγs 〈Yn+1(s)− Yn(s), d (Mn+1(s)−Mn(s))〉 . (3.71)

From (3.69) (for n − 1 instead of n), (3.70), and the fact that Yn+1(T ) = Yn(T ) = ξfrom (3.69) we infer the inequalities:

E[

sup0≤t≤T

eγt |Yn+1(t)− Yn(t)|2]

≤ 1

2E

[∫ T

0


+ 2E[

sup0≤t≤T

∫ t

0

eγs 〈Yn+1(s)− Yn(s), d (Mn+1(s)−Mn(s))〉]

≤ 1

2E

[∫ T

0



+1

2E

[sup


]+ 64E

[∫ T

0


≤ 1

2E

[sup


]+

65

2nE

[∫ T

0

eγsd 〈M1 −M0,M1 −M0〉 (s)]

.

(3.72)

From (3.72) we infer the inequality

E[

sup0≤t≤T

eγt |Yn+1(t)− Yn(t)|2]≤ 65

2nE

[∫ T

0

eγsd 〈M1 −M0,M1 −M0〉 (s)]

. (3.73)

(In order to justify the passage from (3.67) to (3.73) like in passing from inequality(3.38) to (3.41) a stopping time argument might be required.) From (3.69) and (3.73)it follows that the sequence (Yn,Mn) converges in the space S2×M2, and that its limit(Y, M) satisfies (3.53) in Theorem 3.6. This completes the proof of Proposition 3.7. ¤3.10. Proposition. Let the notation and hypotheses be as in Theorem 3.6. Let for

δ > 0 with 2δC1 < 1 the processes Yδ, Yδ ∈ S2 and the martingale Mδ ∈ M2 be suchthat the equalities of (3.64) in Corollary 3.9 are satisfied. Then the family

(Yδ,Mδ) : 0 < δ <1

2C1

converges in the space S2 × M2 if δ decreases to 0, provided that the terminal valueξ = Yδ(T ) is given.

Let (Y,M) be the limit in the space S2 ×M2. In fact from the proof of Proposition3.10 it follows that ∥∥∥∥

(Yδ − YMδ −M

)∥∥∥∥S2×M2

= O(δ) (3.74)

as δ ↓ 0, provided that ‖Yδ2(T )− Yδ1(T )‖L2(Ω,F0T ,P) = O (|δ2 − δ1|).

Proof of Proposition 3.10. Let C1 be the constant which occurs in inequality (3.50) inTheorem 3.6, and fix 0 < δ2 < δ1 < (2C1)

−1. Our estimates give quantitative boundsin case we restrict the parameters δ, δ1 and δ2 to the interval

(0, (4C1 + 4)−1). An

appropriate choice for the constant γ in the present proof turns out to be γ = 6 + 4C1

(see e.g. the inequalities (3.76), (3.88), (3.89), and (3.90) below). An appropriatechoice for the positive number a, which may be a function of the parameters δ1 andδ2, in (3.87), (3.88) and subsequent inequalities below is given by a = (δ1 + δ2)

−1. Forconvenience we introduce the following notation: 4Y (s) = Yδ2(s) − Yδ1(s), 4M(s) =

Mδ2(s)−Mδ1(s), 4Y (s) = Yδ2(s)− Yδ1(s), and 4f(s) = fδ2(s)− fδ1(s) where fδ(s) =

f(s, Yδ(s), ZM(s)

). From the equalities in (3.64) we infer

Yδ(t) = Yδ(t)− δfδ(t) = Yδ(T ) +

∫ T

t

fδ(s)ds + Mδ(t)−Mδ(T ). (3.75)

First we prove that the family(Yδ,Mδ) : 0 < δ < (4C1 + 4)−1 is bounded in the space

S2×M2. Therefore we fix γ > 0 and apply Ito’s formula to the process t 7→ eγt |Yδ(t)|2to obtain:

eγT |Yδ(T )|2 − eγt |Yδ(t)|2


= γ

∫ T

t

eγs |Yδ(s)|2 ds + 2

∫ T

t

eγs 〈Yδ(s), dYδ(s)〉+

∫ T

t

eγsd 〈Mδ,Mδ〉 (s)

= γ

∫ T

t

eγs∣∣∣Yδ(s)− δfδ(s)

∣∣∣2

ds− 2

∫ T

t

eγs⟨Yδ(s), fδ(s)

⟩ds

− 2

∫ T

t

eγs⟨Yδ(s)− Yδ(s), fδ(s)

⟩ds +

∫ T

t

eγsd 〈Mδ,Mδ〉 (s)

+ 2

∫ T

t

eγs 〈Yδ(s), dMδ(s)〉

= γ

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds + γ

∫ T

t

eγs∣∣∣δfδ(s)

∣∣∣2

ds

− 2 (1 + γδ)

∫ T

t

eγs⟨Yδ(s), fδ(s)− f (s, 0, ZM(s))

⟩ds

+ 2

∫ T

t

eγs⟨δfδ(s), fδ(s)

⟩ds− 2 (1 + γδ)

∫ T

t

eγs⟨Yδ(s), f (s, 0, ZM(s))

⟩ds

+

∫ T

t

eγsd 〈Mδ,Mδ〉 (s) + 2

∫ T

t


= γ

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds +(γδ2 + 2δ

) ∫ T

t

eγs∣∣∣fδ(s)

∣∣∣2

ds

− 2 (1 + γδ)

∫ T

t

eγs⟨Yδ(s), fδ(s)− f (s, 0, ZM(s))

⟩ds

− 2 (1 + γδ)

∫ T

t

eγs⟨Yδ(s), f (s, 0, ZM(s))− f (s, 0, 0)

⟩ds

− 2 (1 + γδ)

∫ T

t

eγs⟨Yδ(s), f (s, 0, 0)

⟩ds

+

∫ T

t


∫ T

t


(employ the inequalities (3.50), (3.51), and (3.52) of Theorem 3.6)

≥ γ

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds +(γδ2 + 2δ

) ∫ T

t

eγs∣∣∣fδ(s)

∣∣∣2

ds

− 2C1 (1 + γδ)

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds

− 2C2 (1 + γδ)

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣(

d

ds〈M, M〉 (s)

)1/2

ds

− 2 (1 + γδ)

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣ |f (s, 0, 0)| ds

+

∫ T

t


∫ T

t



≥ (γ − 2 (C1 + 1) (1 + γδ))

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds +(γδ2 + 2δ

) ∫ T

t

eγs∣∣∣fδ(s)

∣∣∣2

ds

− C22 (1 + γδ)

∫ T

t

eγsd 〈M, M〉 (s)− (1 + γδ)

∫ T

t

eγs |f (s, 0, 0)|2 ds

+

∫ T

t


∫ T

t

eγs 〈Yδ(s), dMδ(s)〉 . (3.76)

From (3.76) we infer the inequality:

(γ − 2 (C1 + 1) (1 + γδ))

∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds +(γδ2 + 2δ

) ∫ T

t

eγs∣∣∣fδ(s)

∣∣∣2

ds

+

∫ T

t


∫ T

t

eγs 〈Yδ(s), dMδ(s)〉+ eγt |Yδ(t)|2

≤ eγT |Yδ(T )|2 + (1 + γδ)

(C2

2

∫ T

t

eγsd 〈M, M〉 (s) +

∫ T

t

eγs |f (s, 0, 0)|2 ds

). (3.77)

From (3.77) we deduce

(γ − 2 (C1 + 1) (1 + γδ))E[∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds

]+

(γδ2 + 2δ

)E

[∫ T

t

eγs∣∣∣fδ(s)

∣∣∣2

ds

]

+ eγtE[|Yδ(t)|2

]

≤ eγTE[|Yδ(T )|2]

+ (1 + γδ)

(C2

2E[∫ T

t

eγsd 〈M, M〉 (s)]

+ E[∫ T

t

eγs |f (s, 0, 0)|2 ds

]). (3.78)

In particular from (3.78) we see

E[∫ T

t

eγs∣∣∣Yδ(s)

∣∣∣2

ds

]

≤ 1

γ − 2 (C1 + 1) (1 + γδ)eγTE

[|Yδ(T )|2]

+1 + γδ

γ − 2 (C1 + 1) (1 + γδ)

(C2

2E[∫ T

t


+ E[∫ T

t

eγs∣∣f (s)

∣∣2 ds

]).

(3.79)

In addition, from (3.77) we obtain the following inequalities

2

∫ T

0

eγs 〈Yδ(s), dMδ(s)〉+ 2 sup0<t<T

eγt |Yδ(t)|

≤ eγT |Yδ(T )|2 + 2 sup0<t<T

∫ t

0


+ (1 + γδ)

(C2

2

∫ T

t

eγsd 〈M,M〉 (s) +

∫ T

t

eγs |f (s, 0, 0)|2 ds

), (3.80)


and hence by using the Burkholder-Davis-Gundy inequality (3.6) for p = 12:

E[

sup0<t<T

eγt |Yδ(t)|]

≤ eγTE[|Yδ(T )|2] + E

[sup

0<t<T

∫ t

0

eγs 〈Yδ(s), dMδ(s)〉]

+ (1 + γδ)

(C2

2E[∫ T

t

eγsd 〈M,M〉 (s)]

+ E[∫ T

t

eγs |f (s, 0, 0)|2 ds

])

≤ eγTE[|Yδ(T )|2] + 8

√2E

[(∫ T

0

e2γs |Yδ(s)|2 d 〈Mδ,Mδ〉 (s))1/2

]

+ (1 + γδ)

(C2

2E[∫ T

t


+ E[∫ T

t

eγs |f (s, 0, 0)|2 ds

])

≤ eγTE[|Yδ(T )|2] +

1

2E

[sup

0<t<Teγt |Yδ(t)|

]+ 64E

[∫ T

0

eγs |Yδ(s)|2 d 〈Mδ,Mδ〉 (s)]

+ (1 + γδ)

(C2

2E[∫ T

t


+ E[∫ T

t

eγs |f (s, 0, 0)|2 ds

]). (3.81)

From (3.78) and (3.81) we obtain

E[

sup0<t<T

eγt |Yδ(t)|]

≤ 130eγTE[|Yδ(T )|2]

+ 130 (1 + γδ)

(C2

2E[∫ T

t


+ E[∫ T

t

eγs |f (s, 0, 0)|2 ds

]). (3.82)

(In order to justify the passage from (3.80) to (3.82) like in passing from inequality(3.38) to (3.41) a stopping time argument might be required.) Next we notice that

∣∣∣fδ(s)∣∣∣2

≤ 2∣∣f(s)

∣∣2 + 2K2∣∣∣Yδ(s)

∣∣∣2

+ 2C22

d

ds〈M, M〉 (s), (3.83)

and hence

2⟨δ2fδ2(s)− δ1fδ1(s),4f(s)

⟩≥ −2 |δ2 − δ1|

(∣∣∣fδ2(s)∣∣∣2

−∣∣∣fδ1(s)

∣∣∣2)

≥ −4 |δ2 − δ1|(∣∣f(s)

∣∣2 + K2∣∣∣Yδ2(s)

∣∣∣2

+ K2∣∣∣Yδ1(s)

∣∣∣2

+ C22

d

ds〈M,M〉 (s)

). (3.84)

In a similar manner we also get∣∣∣δ2fδ2(s)− δ1fδ1(s)

∣∣∣2

≤ 4(δ22 + δ2

1

) (∣∣f(s)∣∣2 + K2

∣∣∣Yδ2(s)∣∣∣2

+ K2∣∣∣Yδ1(s)

∣∣∣2

+ C22

d

ds〈M, M〉 (s)

). (3.85)

Fix γ > 0, and apply Ito’s lemma to the process t 7→ eγt |4Y (t)|2 to obtain

eγT |4Y (T )|2 − eγt |4Y (t)|2


= γ

∫ T

t

eγs |4Y (s)|2 ds + 2

∫ T

t

eγs 〈4Y (s), d4Y (s)〉+

∫ T

t

eγsd 〈4M,4M〉 (s)

= γ

∫ T

t

eγs∣∣∣4Y (s)− δ2fδ2(s) + δ1fδ1(s)

∣∣∣2

ds− 2

∫ T

t

eγs⟨4Y (s),4f(s)

⟩ds

− 2

∫ T

t

eγs⟨4Y (s)−4Y (s),4f(s)

⟩ds +

∫ T

t

eγsd 〈4M,4M〉 (s)

+ 2

∫ T

t

eγs 〈4Y (s), d4M(s)〉

= γ

∫ T

t

eγs∣∣∣4Y (s)

∣∣∣2

ds + γ

∫ T

t

eγs∣∣∣δ2fδ2(s)− δ1fδ1(s)

∣∣∣2

ds

− 2

∫ T

t

eγs⟨4Y (s),4f(s)

⟩ds− 2γ

∫ T

t

eγs⟨δ2fδ2(s)− δ1fδ1(s),4Y (s)

⟩ds

+ 2

∫ T

t

eγs⟨δ2fδ2(s)− δ1fδ1(s),4f(s)

⟩ds

+

∫ T

t

eγsd 〈4M,4M〉 (s) + 2

∫ T

t

eγs 〈4Y (s), d4M(s)〉 . (3.86)

Employing the inequalities (3.50), (3.84), (3.85) and an elementary one like

2 |〈y1, y2〉| ≤ (a + 1) |y1|2 + (a + 1)−1 |y2|2 , y1, y2 ∈ Rk, a > 0, (3.87)

together with (3.86) we obtain

eγT |4Y (T )|2 − eγt |4Y (t)|2

≥(

γ − 2C1 − γ

a + 1

) ∫ T

t

eγs∣∣∣4Y (s)

∣∣∣2

ds− aγ

∫ T

t

eγs∣∣∣δ2fδ2(s)− δ1fδ1(s)

∣∣∣2

ds

− 8γ |δ2 − δ1|(∫ T

t

eγs∣∣f(s)

∣∣2 ds + C22

∫ T

t

eγsd 〈M,M〉 (s))

− 8γK2 |δ2 − δ1|(∫ T

t

eγs

(∣∣∣Yδ1(s)∣∣∣2

+∣∣∣Yδ2(s)

∣∣∣2)

ds

)

+

∫ T

t

eγsd 〈4M,4M〉 (s) + 2

∫ T

t


≥(

γ − 2C1 − γ

a + 1

) ∫ T

t

eγs∣∣∣4Y (s)

∣∣∣2

ds

− 4γ(2 |δ2 − δ1|+ a

(δ21 + δ2

2

)) (∫ T

t

eγs∣∣f(s)

∣∣2 ds + C22

∫ T

t

eγsd 〈M,M〉 (s))

− 4γK2(2 |δ2 − δ1|+ a

(δ21 + δ2

2

)) (∫ T

t

eγs

(∣∣∣Yδ1(s)∣∣∣2

+∣∣∣Yδ2(s)

∣∣∣2)

ds

)

+

∫ T

t

eγsd 〈4M,4M〉 (s) + 2

∫ T

t

eγs 〈4Y (s), d4M(s)〉 . (3.88)


From (3.88) we obtain

(γa

a + 1− 2C1

) ∫ T

t

eγs∣∣∣4Y (s)

∣∣∣2

ds + eγt |4Y (t)|2

+

∫ T

t

eγsd 〈4M,4M〉 (s) + 2

∫ T

t


≤ eγT |4Y (T )|2

+ 4γ(2 |δ2 − δ1|+ a

(δ21 + δ2

2

)) (∫ T

t

eγs∣∣f(s)

∣∣2 ds + C22

∫ T

t

eγsd 〈M, M〉 (s))

+ 4γK2(2 |δ2 − δ1|+ a

(δ21 + δ2

2

))(∫ T

t

eγs

(∣∣∣Yδ1(s)∣∣∣2

+∣∣∣Yδ2(s)

∣∣∣2)

ds

). (3.89)

From (3.79) and (3.89) we infer

(γa

a + 1− 2C1

)E

[∫ T

t

eγs∣∣∣4Y (s)

∣∣∣2

ds

]+ eγtE

[|4Y (t)|2]

+ E[∫ T

t

eγsd 〈4M,4M〉 (s)]

≤ eγTE[|4Y (T )|2] + γ1 (δ1, δ2) eγTE

[|Yδ1(T )|2] + γ1 (δ2, δ1) eγTE[|Yδ2(T )|2]

+ γ2 (δ1, δ2)

(E

[∫ T

t

eγs∣∣f(s)

∣∣2 ds

]+ C2

2E[∫ T

t

eγsd 〈M,M〉 (s)])

(3.90)

where

γ1 (δ1, δ2) = 4γK2(2 |δ2 − δ1|+ a

(δ21 + δ2

2

)) 1

γ − 2 (C1 + 1) (1 + γδ1);

γ2 (δ1, δ2) = 4γ(2 |δ2 − δ1|+ a

(δ21 + δ2

2

))

×(

1 +K2 (1 + γδ1)

γ − 2 (C1 + 1) (1 + γδ1)+

K2 (1 + γδ2)

γ − 2 (C1 + 1) (1 + γδ2)

)(3.91)

From (3.89) we also get:

sup0<t<T

(eγt |4Y (t)|2) + 2

∫ T

0


≤ eγT |4Y (T )|2

+ 4γ(2 |δ2 − δ1|+ a

(δ21 + δ2

2

))(∫ T

0

eγs∣∣f(s)

∣∣2 ds + C22

∫ T

0

eγsd 〈M, M〉 (s))

+ 4γK2(2 |δ2 − δ1|+ a

(δ21 + δ2

2

)) (∫ T

0

eγs

(∣∣∣Yδ1(s)∣∣∣2

+∣∣∣Yδ2(s)

∣∣∣2)

ds

)

+ 2 sup0<t<T

∫ t

0

eγs 〈4Y (s), d4M(s)〉 . (3.92)


In what follows a stopping time argument might be required. From (3.92), (3.79), theinequality of Burkholder-Davis-Gundy (3.6) for p = 1

2and (3.90) with t = 0 we obtain:

E[

sup0<t<T

(eγt |4Y (t)|2)

]

≤ eγTE[|4Y (T )|2]

+ 4γ(2 |δ2 − δ1|+ a

(δ21 + δ2

2

))(E

[∫ T

0

eγs∣∣f(s)

∣∣2 ds

]+ C2

2E[∫ T

0

eγsd 〈M, M〉 (s)])

+ 4γK2(2 |δ2 − δ1|+ a

(δ21 + δ2

2

)) (E

[∫ T

0

eγs

(∣∣∣Yδ1(s)∣∣∣2

+∣∣∣Yδ2(s)

∣∣∣2)

ds

])

+ 2E[

sup0<t<T

∫ t

0

eγs 〈4Y (s), d4M(s)〉]

≤ eγTE[|4Y (T )|2] + γ1 (δ1, δ2) eγTE

[|Yδ1(T )|2] + γ1 (δ2, δ1) eγTE[|Yδ2(T )|2]

+ γ2 (δ1, δ2)

(C2

2E[∫ T

t


+ E[∫ T

t

eγs∣∣f (s) ds

∣∣2])

+ 8√

2E

[sup

0<t<Teγt |4Y (t)|

(∫ T

0

eγsd 〈4M,4M〉 (s))1/2

]

≤ eγTE[|4Y (T )|2] + γ1 (δ1, δ2) eγTE

[|Yδ1(T )|2] + γ1 (δ2, δ1) eγTE[|Yδ2(T )|2]

+ γ2 (δ1, δ2)

(C2

2E[∫ T

t


+ E[∫ T

t

eγs∣∣f (s) ds

∣∣2])

+1

2E

[sup

0<t<Teγt |4Y (t)|2

]+ 64E

[∫ T

0


. (3.93)

Consequently, from (3.90) and (3.93) we deduce, like in the proof of inequality (3.82),

E[

sup0<t<T

eγt |4Y (t)|2]

≤ 130(eγTE

[|4Y (T )|2] + γ1 (δ1, δ2) eγTE[|Yδ1(T )|2] + γ1 (δ2, δ1) eγTE

[|Yδ2(T )|2])

+ 130γ2 (δ1, δ2)

(C2

2E[∫ T

0


+ E[∫ T

0

eγs∣∣f (s) ds

∣∣2])

. (3.94)

(Again it is noticed that the passage from (3.92) to (3.94) is justified by a stoppingtime argument. The same argument was used several times. The first time we used itin passing from inequality (3.38) to (3.41).) Another appeal to (3.90) and (3.94) shows:

(γa

a + 1− 2C1

)E

[∫ T

t

eγs∣∣∣4Y (s)

∣∣∣2

ds

]

+ E[

sup0<t<T

eγt |4Y (t)|2]

+ E[∫ T

0


≤ 131(eγTE

[|4Y (T )|2] + γ1 (δ1, δ2) eγTE[|Yδ1(T )|2] + γ1 (δ2, δ1) eγTE

[|Yδ2(T )|2])

+ 131γ2 (δ1, δ2)

(C2

2E[∫ T

0


+ E[∫ T

0

eγs∣∣f (s) ds

∣∣2])

. (3.95)


The result in Proposition 3.10 now follows from (3.95) and the continuity of the func-tions y 7→ f (s, y, ZM(s)), y ∈ Rk. The fact that the convergence of the family (Yδ,Mδ),0 < δ ≤ (4C1 + 4)−1 is of order δ, as δ ↓ 0, follows by the choice of our parameters:γ = 4C1 + 4 and a = (δ1 + δ2)

−1. ¤Proof of Theorem 3.6. The proof of the uniqueness part follows from Corollary 3.4.The existence is a consequence of Theorem 3.5, Proposition 3.10 and Corollary 3.9. ¤

The following result shows that in the monotonicity condition we may always assumethat the constant C1 can be chosen as we like provided we replace the equation in (3.30)by (3.96) and adapt its solution.

3.11. Theorem. Let the pair (Y, M) belong to S2([0, T ],Rk

) × M2([0, T ],Rk

). Fix

λ ∈ R and put

(Yλ(t),Mλ(t)) =

(eλtY (t), Y (0) +

∫ t

0

eλsdM(s)

).

Then the pair (Yλ,Mλ) belongs to S2 × M2. Moreover, the following assertions areequivalent:

(i) The pair (Y,M) ∈ S2 ×M2 satisfies Y (0) = M(0) and

Y (t = Y (T ) +

∫ T

t

f (s, Y (s), ZM(s)) ds + M(t)−M(T ).

(ii) The pair (Yλ,Mλ) satisfies Yλ(0) = Mλ(0) and

Yλ(t) = Yλ(T )+

∫ T

t

eλsf(s, e−λsYλ(s), e

−λsZMλ(s)

)ds−λ

∫ T

t

Yλ(s)ds+Mλ(t)−Mλ(T ).

(3.96)

3.3. Remark. Put fλ(s, y, z) = eλsf(s, e−λsy, e−λsz

)−λy. If the function y 7→ f (s, y, z)has monotonicity constant C1, then the function y 7→ fλ (s, y, z) has monotonicityconstant C1− λ. It follows that by reformulating the problem one always may assumethat the monotonicity constant is 0.

Proof of Theorem 3.11. First notice the equality e−λsZMλ(s) = ZM(s): see Remark

1.4. The equivalence of (i) and (ii) follows by considering the equalities in (i) and (ii)in differential form. ¤

4. Backward stochastic differential equations and Markov processes

In this section the coefficient f of our BSDE is a mapping from [0, T ]×E×Rk×(M2)∗

to Rk. Theorem 4.1 below is the analogue of Theorem 3.6 with a Markov family ofmeasures Pτ,x : (τ, x) ∈ [0, T ]× E instead of a single measure. Put

fn(s) = f (s, X(s), Yn(s), ZMn(s)) ,

and suppose that the processes Yn(s) and ZMn(s) only depend of the state-time vari-able (s, X(s)). Put Y (τ, t) f(x) = Eτ,x [f (X(t))], f ∈ C0(E), and suppose thatfor every f ∈ C0(E) the function (τ, x, t) 7→ Y (τ, t)f(x) is continuous on the set(τ, x, t) ∈ [0, T ]× E × [0, T ] : 0 ≤ τ ≤ T. Then it can be proved that the Markovprocess

(Ω,FτT ,Pτ,x) , (X(t) : T ≥ t ≥ 0) , (E, E) (4.1)


has left limits and is right-continuous: see e.g. Theorem 2.22 in [11]. Suppose that thePτ,x-martingale t 7→ N(t) − N(τ), t ∈ [τ, T ], belongs to the space M2

([τ, T ],Pτ,x,Rk

)(see Definition 1.9). It follows that the quantity ZM(s)(N) is measurable with respect toσ

(Fs

s+, N(s+)): see equalities the (4.5), (4.6) and (4.7) below. The following iteration

formulas play an important role:

Yn+1(t) = Et,X(t) [ξ] +

∫ T

t

Et,X(t) [fn(s)] ds,

Mn+1(t) = Et,X(t) [ξ] +

∫ t

0

fn(s)ds +

∫ T

t

Et,X(t) [fn(s)] ds.

Then the processes Yn+1 and Mn+1 are related as follows:

Yn+1(T ) +

∫ T

t

fn(s)ds + Mn+1(t)−Mn+1(T ) = Yn+1(t).

Moreover, by the Markov property, the process

t 7→ Mn+1(t)−Mn+1(τ)

= Eτ,X(τ)

[ξ

∣∣ Fτt

]− Eτ,X(τ) [ξ] + Eτ,X(τ)

[∫ T

τ

fn(s)ds∣∣ Fτ

t

]− Eτ,X(τ)

[∫ T

τ

fn(s)ds

]

= Eτ,X(τ)

[ξ +

∫ T

τ

fn(s)ds∣∣ Fτ

t

]− Eτ,X(τ)

[ξ +

∫ T

τ

fn(s)ds

]

is a Pτ,x-martingale on the interval [τ, T ] for every (τ, x) ∈ [0, T ]× E.In Theorem 4.1 below we replace the Lipschitz condition (3.28) in Theorem 3.5

for the function Y (s) 7→ f (s, Y (s), ZM(s)) with the (weaker) monotonicity condition(4.8) for the function Y (s) 7→ f (s,X(s), Y (s), ZM(s)). Sometimes we write y forthe variable Y (s) and z for ZM(s). Notice that the functional ZMn(t) only dependson Ft

t+ :=⋂

h:T≥t+h>t σ (X(t + h)) and that this σ-field belongs the Pt,x-completionof σ (X(t)) for every x ∈ E. This is the case, because by assumption the processs 7→ X(s) is right-continuous at s = t: see Proposition 1.6. In order to show this wehave to prove equalities of the following type:

Es,x

[Y

∣∣ Fst+

]= Et,X(t) [Y ] , Ps,x-almost surely, (4.2)

for all bounded stochastic variables which are FtT -measurable. By the monotone class

theorem and density arguments the proof of (4.2) reduces to showing these equalitiesfor Y =

∏nj=1 fj (tj, X (tj)), where t = t1 < t2 < · · · < tn ≤ T , and the functions

x 7→ fj (tj, x), 1 ≤ j ≤ n, belong to the space C0(E). So we consider

Es,x

[n∏

j=1

fj (tj, X (tj))∣∣ Fs

t+

]

= f1 (t,X (t))Et,X(t)

[n∏

j=2


t+

]

= f1 (t,X (t)) limh↓0,0<h<t2−t

Es,x

[Es,x

[n∏

j=2


t+h

]∣∣ Fs

t+

]


= f1 (t,X (t)) limh↓0,0<h<t2−t

Es,x

[Et+h,X(t+h)

[n∏

j=2

fj (tj, X (tj))

]∣∣ Fs

t+

]

(the function ρ 7→ Eρ,X(ρ)

[∏nj=2 fj (tj, X (tj))

]is right-continuous)

= f1 (t,X (t))Es,x

[Et,X(t)

[n∏

j=2

fj (tj, X (tj))

]∣∣ Fs

t+

]

= f1 (t,X (t))Et,X(t)

[n∏

j=2

fj (tj, X (tj))

]

= Et,X(t)

[n∏

j=1

fj (tj, X (tj))

], Ps,x-almost surely. (4.3)

Next suppose that the bounded stochastic variable Y is measurable with respect toFt

t+. From (4.2) with s = t it follows that Y = Et,X(t) [Y ], Pt,x-almost surely. Hencesuch a variable Y only depends on the space-time variable (t,X(t)). Since X(t) = xPt,x-almost surely it follows that the variable Et,x

[Y

∣∣ Ftt+

]is Pt,x-almost equal to the

deterministic constant Et,x [Y ]. A similar argument shows the following result. Let0 ≤ s < t ≤ T , and let Y be a bounded Fs

T -measurable stochastic variable. Then thefollowing equality holds Ps,x-almost surely:

Es,x

[Y

∣∣ Fst+

]= Es,x

[Y

∣∣ Fst

](4.4)

In particular it follows that an Fst+-measurable bounded stochastic variable coincides

with the Fst -measurable variable Es,x

[Y

∣∣ Fst

]Ps,x-almost surely for all x ∈ E. Hence

(4.4) implies that the σ-field Fst+ is contained in the Ps,x-completion of the σ-field Fs

t .In addition, notice that the functional ZM(s) is defined by

ZM(s)(N) = limt↓s

〈M, N〉 (t)− 〈M,N〉 (s)t− s

(4.5)

where

〈M, N〉 (t)− 〈M,N〉 (s) = limn→∞

2n−1∑j=0

(M (tj+1,n)−M (tj,n)) (N (tj+1,n)−N (tj,n)) .

(4.6)For this the reader is referred to the remarks 1.4, 1.5, 1.7, and to formula (3.3). Thesymbol tj,n represents the real number tj,n = s + j2−n(t − s). The limit in (4.6)exists Pτ,x-almost surely for all τ ∈ [0, s]. As a consequence the process ZM(s) isFτ

s+-measurable for all τ ∈ [0, s]. It follows that the process N 7→ ZM(s)(N) isPτ,x-almost surely equal to the functional N 7→ Eτ,x

[ZM(s)(N)

∣∣ σ (Fτs , N(s))

]pro-

vided that ZM(s)(N) is σ(Fτ

s+, N(s+))-measurable. If the martingale M is of the

form M(s) = u (s,X(s)) +∫ s

0f(ρ)dρ, then the functional ZM(s)(N) is automatically

σ(Fs

s+, N(s+))-measurable. It follows that, for every τ ∈ [0, s], the following equality

holds Pτ,x-almost surely:

Eτ,x

[ZM(s)(N)

∣∣ σ(Fτ

s+, N(s+))]

= Eτ,x

[ZM(s)(N)

∣∣ σ (Fτs , N(s+))

]. (4.7)


Moreover, in the next Theorem 4.1 the filtered probability measure(Ω,F,

(F0

t

)t∈[0,T ]

,P)

is replaced with a Markov family of measures(Ω,Fτ

T , (Fτt )τ≤t≤T ,Pτ,x

), (τ, x) ∈ [0, T ]× E.

Its proof follows the lines of the proof of Theorem 3.6: it will not be repeated here.Relevant equalities which play a dominant role are the following ones: (3.41), (3.49),(3.82), and (3.95). In these inequalities the measure Pτ,x replaces P and the coefficientf (s, Y (s), ZM(s)) is replaced with f (s,X(s), Y (s), ZM(s)).

4.1. Theorem. Let f : [0, T ]×E×Rk×(M2)∗ → Rk be monotone in the variable y and

Lipschitz in z. More precisely, suppose that there exist finite constants C1 and C2 suchthat for any two pairs of processes (Y,M) and (U,N) ∈ S2

([0, T ],Rk

)×M2([0, T ],Rk

)the following inequalities hold for all 0 ≤ s ≤ T :

〈Y (s)− U(s), f (s,X(s), Y (s), ZM(s))− f (s,X(s), U(s), ZM(s))〉≤ C1 |Y (s)− U(s)|2 , (4.8)

|f (s,X(s), Y (s), ZM(s))− f (s,X(s), Y (s), ZN(s))|

≤ C2

(d

ds〈M −N,M −N〉 (s)

)1/2

, (4.9)

and

|f (s,X(s), Y (s), 0)| ≤ f (s,X(s)) + K |Y (s)| . (4.10)

Fix (τ, x) ∈ [0, T ] × E and let Y (T ) = ξ ∈ L2(Ω, Fτ

T ,Pτ,x;Rk)

be given. In addition,

suppose Eτ,x

[∫ T

τ

∣∣f (s,X(s))∣∣2 ds

]< ∞. Then there exists a unique pair

(Y,M) ∈ S2([τ, T ],Pτ,x,Rk

)×M2([τ, T ],Pτ,x,Rk

)

with Y (τ) = M(τ) such that

Y (t) = ξ +

∫ T

t

f (s,X(s), Y (s), ZM(s)) ds + M(t)−M(T ). (4.11)

Next let ξ = ET,X(T ) [ξ] ∈ ⋂(τ,x)∈[0,T ]×E L2 (Ω,Fτ

T ,Pτ,x) be given. Suppose that the func-

tions (τ, x) 7→ Eτ,x

[|ξ|2] and (τ, x) 7→ Eτ,x

[∫ T

τ

∣∣f (s, X(s))∣∣2 ds

]are locally bounded.

Then there exists a unique pair

(Y,M) ∈ S2loc,unif

([τ, T ],Rk

)×M2loc,unif

([τ, T ],Rk

)

with Y (0) = M(0) such that equation (4.11) is satisfied.Again let ξ = ET,X(T ) [ξ] ∈ ⋂

(τ,x)∈[0,T ]×E L2 (Ω,FτT ,Pτ,x) be given. Suppose that the

functions (τ, x) 7→ Eτ,x

[|ξ|2] and (τ, x) 7→ Eτ,x

[∫ T

τ

∣∣f (s,X(s))∣∣2 ds

]are uniformly

bounded. Then there exists a unique pair

(Y, M) ∈ S2unif

([τ, T ],Rk

)×M2unif

([τ, T ],Rk

)

with Y (0) = M(0) such that equation (4.11) is satisfied.


The notations S2([τ, T ],Pτ,x,Rk

)= S2

(Ω,Fτ

T ,Pτ,x;Rk)

and M2([τ, T ],Pτ,x,Rk

)=

M2(Ω, Fτ

T ,Pτ,x;Rk)

are explained in the definitions 1.8 and 1.9 respectively. The sameis true for the notions

S2loc,unif

([0, T ],Rk

)= S2

loc,unif

(Ω,Fτ

T ,Pτ,x;Rk),

M2loc,unif

([0, T ],Rk

)= M2

loc,unif

(Ω, Fτ

T ,Pτ,x;Rk),

S2unif

([0, T ],Rk

)= S2

unif

(Ω,Fτ

T ,Pτ,x;Rk), and

M2unif

([0, T ],Rk

)= M2

unif

(Ω,Fτ

T ,Pτ,x;Rk).

The probability measure Pτ,x is defined on the σ-field FτT . Since the existence properties

of the solutions to backward stochastic equations are based on explicit inequalities,the proofs carry over to Markov families of measures. Ultimately these inequalitiesimply that boundedness and continuity properties of the function (τ, x) 7→ Eτ,x [Y (t)],0 ≤ τ ≤ t ≤ T , depend the continuity of the function x 7→7→ ET,x [ξ], where ξ is aterminal value function which is supposed to be σ (X(T ))-measurable. In addition,in order to be sure that the function (τ, x) 7→ Eτ,x [Y (t)] is continuous, functions ofthe form (τ, x) 7→ Eτ,x [f (t, u (t,X(t)) , ZM(t))] have to be continuous, whenever thefollowing mappings

(τ, x) 7→ Eτ,x

[∫ T

τ

|u(s,X(s))|2 ds

]and (τ, x) 7→ Eτ,x [〈M, M〉 (T )− 〈M,M〉]

represent finite and continuous functions. In the next example we see how the classicalFeynman-Kac formula is related to backward stochastic differential equations.

4.1. Example. Suppose that the coefficient f has the special form:

f(t, x, r, z) = c(t, x)r + h(t, x)

and that the process s 7→ Xx,t(s) is a solution to a stochastic differential equation:

X t,x(s)−X t,x(t) =

∫ s

t

b(τ,X t,x(τ)

)dτ +

∫ s

t

σ(τ, X t,x(τ)

)dW (τ), t ≤ s ≤ T ;

X t,x(s) = x, 0 ≤ s ≤ t.

In that case, the BSDE is linear:

Y t,x(s) = g(X t,x(T )) +

∫ T

s

[c(r,X t,x(r))Y t,x(s) + h(r,X t,x(r))] dr −∫ T

s

Zt,x(r) dW (r),

hence it has an explicit solution. From an extension of the classical “variation ofconstants formula” (see the argument in the proof of the comparison theorem 1.6 inPardoux [18]) or by direct verification we get:

Y t,x(s) = g(X t,x(T )

)e

∫ Ts c(r,Xt,x(r)) dr +

∫ T

s

h(r,X t,x(r)

)e

∫ rs c(α,Xt,x(α)) dα dr

−∫ T

s

e∫ r

s c(α,Xt,x(α))dαZt,x(r) dW (r).

Now Y t,x(t) = E [Y t,x(t)], so that

Y t,x(t) = E[g(X t,x(T ))e

∫ Tt c(s,Xt,x(s)) ds +

∫ T

t

h(s,X t,x(s)

)e

∫ st c(r,Xt,x(r))drds

],


which is the well-known Feynman-Kac formula. Clearly, solutions to stochastic back-ward stochastic differential equations can be used to represent solutions to classicaldifferential equations of parabolic type, and as such they can be considered as a non-linear extension of the Feynman-Kac formula.

4.2. Example. In this example the family of operators L(s), 0 ≤ s ≤ T , generates aMarkov process in the sense of Definition 1.3: see (1.10). For a “smooth” function vwe introduce the martingales:

Mv,t(s) = v (s,X(s))− v (t,X(t))−∫ s

t

(∂

∂ρ+ L(ρ)

)v (ρ, X(ρ)) dρ. (4.12)

Its quadratic variation part 〈Mv,t〉 (s) := 〈Mv,t,Mv,t〉 (s) is given by

〈Mv,t〉 (s) =

∫ s

t

Γ1 (v, v) (ρ,X(ρ)) dρ.

In this example we will mainly be concerned with the Hamilton-Jacobi-Bellman equa-tion as exhibited in (4.13). We have the following result for generators of diffusions:

it refines Theorem 2.4 in Zambrini [27]. Observe that PMv,t

t,x stands for a Girsanovtransformation of the measure Pt,x.

4.2. Theorem. Let χ : (τ, T ] × E → [0,∞] be a function such that for all τ < t ≤ Tand for sufficiently many functions v

EMv,t

t,x [|log χ (T, X(T ))|] < ∞.

Let SL be a (classical) solution to the following Riccati type equation. For τ < s ≤ Tand x ∈ E the following identity is true:

∂SL

∂s(s, x)− 1

2Γ1 (SL, SL) (s, x) + L(s)SL(s, x) + V (s, x)=0;

SL(T, x) = − log χ(T, x), x ∈ E.(4.13)

Then for any nice real valued v(s, x) the following inequality is valid:

SL(t, x) ≤ EMv,t

t,x

[∫ T

t

(1

2Γ1 (v, v) + V

)(τ, X(τ))dτ

]− EMv,t

t,x [log χ (T,X(T ))] ,

and equality is attained for the “Lagrangian action” v = SL:

SL(t, x) = − logEt,x

[exp

(−

∫ T

t

V (σ,X(σ)) dσ

)χ (T, X(T ))

]. (4.14)

The probability PMv,t

t,x is determined by following equality (4.15). For all finite n-tuples t1, . . . , tn in (t, T ] and all bounded Borel functions fj : [t, T ]×E → R, 1 ≤ j ≤ n,we have:

EMv,t

t,x

[n∏

j=1

fj (tj, X (tj))

]

= Et,x

[exp

(−1

2

∫ T

t

Γ1 (v, v) (τ, X(τ)) dτ −Mv,t(T )

) n∏j=1

fj (tj, X (tj))

]. (4.15)

It is mentioned that Theorem 4.2 is not fully proved. A version where the operatorsL(s) do not depend on s is proved in [24].


Acknowledgement. Part of this work was presented at a Colloquium at the Universityof Gent, October 14, 2005, at the occasion of the 65th birthday of Richard Delangheand appeared in a preliminary form in [25]. Some results were also presented at theUniversity of Clausthal, at the occasion of Michael Demuth’s 60th birthday September10–11, 2006, and at a Conference in Marrakesh, Morocco, “Marrakesh World Confer-ence on Differential Equations and Applications”, June 15–20, 2006. This work wasalso part of a Conference on “The Feynman Integral and Related Topics in Mathemat-ics and Physics: In Honor of the 65th Birthdays of Gerry Johnson and David Skoug”,Lincoln, Nebraska, May 12–14, 2006. Finally, another preliminary version was pre-sented during a Conference on Evolution Equations, in memory of G. Lumer, at theUniversities of Mons and Valenciennes, August 28–September 1, 2006.

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Department of Mathematics and Computer Science, University of Antwerp, Mid-delheimlaan 1, 2020 Antwerp, Belgium

E-mail address: [email protected]

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Introduction - KTH · variable which is measurable with respect to the future (or terminal) ¾-ﬂeld after S, i.e. the one generated by fX (t_S) : ¿ • t • Tg. For this type