Optimal control for jump processes, HJB equationsartax.karlin.mff.cuni.cz/~dvorm3bm/1011z/Petrasek.pdfOptimal control for jump processes, HJB equations Jakub Petr asek Introduction

Optimal controlfor jump

processes,HJB equations

Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up

Dynamicprogrammingprinciple

Application

Formulation ofproblem

Application ofdynamicprogrammingprinciple

Comparison withno-jump case

Maximmumprinciple

Model set-up

Optimalitycondition

Relation todynamicprogramming

Bibliography

Optimal control for jump processes,HJB equations

Jakub Petrasek

Department of Probability and Mathematical StatisticsFaculty of Mathematics and Physics, Charles University in Prague

[email protected]

Seminar in Stochastic Modelling in Economics and Finance

January 10, 2011



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Outline

1 IntroductionMotivationLevy processes

2 Optimal ControlModel set-upDynamic programming principle

3 ApplicationFormulation of problemApplication of dynamic programming principleComparison with no-jump case

4 Maximmum principleModel set-upOptimality conditionRelation to dynamic programming



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Outline





1



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Economic model

At each t ≥ 0 an agent owns a capital Xt by investing in two assets

a riskfree bond that pays interest rate r ,

dBt = rBtdt.

a risky asset with dynamics (geometric Brownian motion +jumps)

dSt = S(t−)

(αdt + σdWt +

∫ ∞−1

zN(dt,dz)

).

At each t ≥ 0 an agent controls

the number of stocks ∆t in his portfolio,

and possibly consumption Ct .

2



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Admissible strategies

Definition

A strategy (∆t , t ≥ 0) is admissible if

1 it is predictable,

2 the portfolio (Xt , t ≥ 0) is self-financing, i.e.

dXt = ∆tdSt + (Xt −∆tSt)dBt .

We denote the set of admissible strategies by A(x) for a given capitalX0 = x .

We can substitute for dSt and dBt and obtain

dXt = ∆tSt−

((α− r)dt + σdWt +

∫ ∞−1

zN(dt,dz)

)+ rXtdt.

3



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Objective of investment

The objective of an agent is to maximize his utility from investmentsby using admissible strategies ∆t(∆t ,Ct). His aim is

1 to maximize his consumption over infinite horizon

sup(∆t ,Ct)∈A(x)

∫ ∞0

e−βtE U(Ct)dt, (1.1)

where β is a discount factor

2 to maximize his terminal utility of terminal wealth in a givenhorizon T

sup∆t∈A(x)

E U(XT ), (1.2)

3 combination of the first two


E U(XT ) +

∫ T

0

e−βtE U(Ct)dt

. (1.3)

4



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Levy process - definition

Definition

Let(

Ω,F , Ftt≥0 ,P)

be a filtered probability space. An adapted

process Lt is called a Levy process if it is continuous in probabilityand has stationary, independent increments.

TheoremLet Lt be a Levy process. Then Lt has the decomposition

Lt = b + σWt +

∫|z|≤1

zN(t, dz) +

∫|z|>1

zN(t, dz), 0 ≤ t <∞.(1.4)

where b ∈ R, σ ≥ 0, (N) N is a (compensated) Poisson randommeasure with a Levy measure ν, all adapted to filtration Ftt≥0.

5



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Levy process - Ito formula

Theorem (Ito formula)

Suppose Lt ∈ R is an Levy process of the form

dLt = bdt + σdWt +

∫ ∞−1

zN(t, dz).

Let f ∈ C1,2(R+ × R) and define Yt = f (t, Lt). Then Yt is again anLevy process and

dYt = ft(t, Lt)dt + fx(t, Lt) (bdt + σdWt) +1

2fxx(t, Lt)σ2dt

+

∫ ∞−1

(f (t, Lt− + z)− f (t, Lt−)) N(dt, dz)

+

∫ ∞−1

(f (t, Lt− + z)− f (t, Lt−)− fx(t, Lt−)z) ν(dz)dt.

6



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Levy process - Generator

Definition

Suppose f : R2 → R. Then the generator A of process Lt (from theprevious theorem) is defined as

Af (s, x) = limt→0+

1

tE [f (s + t, Ls+t)− f (s, x)] ,

where Ls = x .

Theorem

Suppose f ∈ C1,2(R+ × R). Then Af (s, x) exists and

Af (s, x) = ft(s, x) + fx(s, x)b +1

2fxx(s, x)σ2

+

∫ ∞−1

(f (s, x + z)− f (s, x)− fx(s, x)z) ν(dz).

7



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Outline





8



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

State process

Yt = Y(u)t is a stochastic process (on filtered probability space) with

dynamics

dYt = b(Yt , ut)dt + σt(Yt , ut)dWt +

∫Rγ(Yt− , ut− , z)N(dt,dz),

Y0 = y ∈ Rk

where

b : Rk ×U → Rk , σ : Rk ×U → Rk×m, γ : Rk ×U ×Rk → Rk×l

are given functions (time homogenous), W is Wiener process (on the

given probability space), N compensated Poisson random measureand U ⊂ Rp given set.

u(t) = u(t, ω) : R+ × Ω→ U

is predictable control process and Yt = Y(u)t is a controlled

jump-diffusion.9



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Performance criterion

For a fixed T (possibly T =∞) we define

J(u)(y) = E

[∫ T

0

f (Yt , ut)dt + g(YT )

],

wheref : S × U → R, g : Rk → R

are given continuous functions, S is called solvency region.

DefinitionControl u is admissible, denote u ∈ A if the state process has aunique, strong solution for all x ∈ S and

E

[∫ T

0

f (Yt , ut)dt + g(YT )

]<∞.

10



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Value function

Our goal is to find the value function v and an optimal controlu∗ ∈ A such that

v(x) = J(u∗) = supu∈A

J(u)(x).

We consider Markov controls u(t) = u(Yt−), then

Av(y) =k∑

i=1

bi (y , u(y))vxi (y) +k∑

i,j=1

(σσT

)ij

(y , u(y))vxixj (y)

+k∑

i=1

∫R

[v(y + γj(y , u(y), zj))− v(y)

−∇v(y)γj(y , u(y), zj)] νj(dzj)dt.

11



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Revision

If we start the state process from any t ∈ [0,T − h] it holds

v(Yt) ≥ E

[∫ t+h

t

f (Ys , u(Ys))ds + v(Y(u)t+h)

](2.1)

with equality for u∗ = u. We know that

E tv(Y(u)t+h) = v(Yt) +

∫ t+h

t

A(u)v(Ys)ds

and by substitution into (2.1) we obtain

0 ≥ E

[∫ t+h

t

(f (Ys , u(Ys)) + A(u)v(Ys)

)ds

]

or in differential0 ≥ f (y , u(y)) + A(u)v(y),

for any u and equality holds for u = u∗.12



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

HJB for optimal control of jump diffusion

Lemma (Verification lemma)

Let v ∈ C1,2 satisfies the following

1 limt→T v(Yt) = g(YT )

2 For any u ∈ A(x)

f (y , u(y)) + A(u)v(y) ≤ 0.

3 There is u ∈ A(y) such that

f (y , u(y)) + A(u)v(y) = 0.

Thenu = u∗.

andv(y) = v(y) = J(u∗)(y), for any y ∈ S.

13



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Remarks

Verification theorem holds also for random time T however withadditional requirements.

Example

T = inf t > 0,Yt /∈ S

The Hamilton-Jacobi-Bellman equation provides ”only”sufficient condition for an optimum, but not necessary, which isprovided by Pontryagin Maximum principle.

14



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Outline





15



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Investor’s question

We refer back to the motivation example. An investor puts hismoney into risky St and riskless Bt asset. His portfolio Xt evolves

dXt = ∆tSt−


∫ ∞−1

zN(dt,dz)

)+rXtdt−ctXtdt.

and he wants to maximize utility from his consumption


∫ ∞0

e−βtE U(Ct)dt, (3.1)

Investor knows that his utility is given by the power utility function,i.e.

U(x) =x1−p

1− p, p > 0, p 6= 1,

= log(x), p = 1.

16



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Change of notation

New processes

θt =∆tSt−Xt−

is the proportion of capital invested in risky asset at

time t,

ct = Ct

Xt−denotes the consumption proportion.

Dynamics of investor’s portfolio:

dXt =θtXt−


∫ ∞−1

zN(dt,dz)

)+ rXtdt − ctXtdt.

(3.2)

with X (0) = x , θt ∈ Ft− , ct ∈ Ft− .

17



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Computation of generator

We would like to apply the verificatin lemma on the controlledprocess Yt = (t,Xt)T , with Y0 = (0, x)T .

Generator of v(Yt)

A(u)v(y) = vt + ((α− r)θ + r − c) xvx +1

2σ2θ2x2vxx

+

∫ ∞−1

(v(t, x + xθz)1−p − v(t, x)− θzvx

)ν(dz).

’Consumption’

f (y , u(y)) = e−βtU(cx).

18



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

PDE

We guess the form of the value function, v(t, x) = Ke−βtx1−p

A(u)v(y) = Ke−βtx1−p [−β + ((α− r)θ + r − c) (1− p)

− 1

2σ2θ2p(1− p)

+

∫ ∞−1

((1 + θz)1−p − 1− θz(1− p)

)ν(dz)

]= Ke−βtx1−p [−β + (r − c) (1− p) + h(θ)] .

A(u)v(y) + f (y , u(y))

= Ke−βtx1−p

[−β + (r − c)(1− p) + h(θ) +

c1−p

K (1− p)

].(3.3)

19



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

HJB equations

We apply the verification theorem. We demand

supu∈A

A(u)v(y) + f (y , u(y))

= 0.

We differentiate formula (3.3) with respect to c and θ.

Optimal proportion

0 = Λ(θ) = (α− r)− σ2θp +

∫ ∞−1

(1− (1 + θz)−p

)zν(dz).

Optimal consumption

0 = −(1− p) +c−p

K⇒ c∗ = (K (1− p))−1/p

20



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

HJB equations

Constant K

Finally we substitute θ∗ and c∗ into equation (3.3) and demandequality to zero

0 = A(u∗)v(y) + f (y , u∗(y))

0 = Ke−βtx1−p [−β + r(1− p) + h(θ∗)

− K−1/p(1− p)−1/p+1 + (K (1− p))−1/p]

= Ke−βtx1−p[−β + r(1− p) + h(θ∗)− p (K (1− p))−1/p

]

⇒ nontrivial solution is

K =1

1− p[β − r(1− p)− h(θ∗)p]−p

.

21



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

HJB equations

Theorem (Optimal Proportion and Consumption)

Assume the portfolio (3.2) and the objective. Let

Λ(θ∗) = 0

andβ − r(1− p)− h(θ∗) > 0.

Then

θ∗ is the optimal proportion,

c∗ = (K (1− p))−1/p

v(0, x) = Kx1−p is the value function,

where

K =1

1− p[β − r(1− p)− h(θ∗)p]−p

. (3.4)

22



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Merton proportion and consumption

Merton investment proportion

θ0 =α− r

pσ2,

Merton consumption proportion

c0 = A(p) =β − r(1− p)

p− 1

2

(α− r)2

σ2

1− p

p.

Let all the no-jump variables be indexed by 0.

???What is the effect of jumps on optimal values?

23



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Merton proportion and consumption

Merton investment proportion

θ0 =α− r

pσ2,

Merton consumption proportion

c0 = A(p) =β − r(1− p)

p− 1

2

(α− r)2

σ2

1− p

p.

Let all the no-jump variables be indexed by 0.

???What is the effect of jumps on optimal values?

23



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Optimal (jumps included) proportion andconsumption

Optimal proportion θ∗ solves the equation

Λ(θ∗) = (α− r)− σ2θp +

∫ ∞−1

(1− (1 + θz)−p

)zν(dz) = 0.

Optimal consumption

c∗ = (K (1− p))−1/p

for a constant K given by equation (3.4).

24



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Function Λ

We know that for Λ0(θ) solves the Merton problem and can see that

1

Λ(0) = α− r ,

Λ(θ) is a decreasing function of θ.

2 Function (1− (1 + θz)−p) z is positive for z ∈ (−1/θ,∞).

We conclude thatΛ(θ) < Λ0(θ).

Corollary

θ∗ ≤ θ0,

v ≤ v0,

c∗ ≤ c0, p > 1,

c∗ ≥ c0, 0 < p < 1.

25



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Comparison with Merton cont.

x1 (money units in St)

x2 (money units in Bt)

the Merton line (ν = 0)

risk increasing jumps

(x1 = θ01−θ0x2)

S

S

risk decreasing jumps

26



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Approximation of small jumps

Let us suppose that the measure ν has light tails (jumps are small inabsolute value). We can use the taylor expansion

1− (1 + θz)−p = pzθ + o(z2)

and after the substitution into Λ

θ∗ ≈ 1

p

α− r

σ2 +∫∞−1

z2ν(dz),

i.e. for smaller jumps we can approximate Levy process by a

Brownian motion with volatility√σ2 +

∫∞−1

z2ν(dz).

27



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Investor’s question II

An investor wants to maximize his utility from the terminal wealth

sup∆t∈A(x)

E U(XT ) (3.5)

Optimal strategy

It is optimal to put constant proportion θ∗ of his money into the riskyasset, same as in the previous.

28



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Outline





29



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Maximum principle - intro

Alternative approach for solving optimal control.In the deterministic case introduced by Russian mathematicianLev Pontryagin.

State process

Xt = X(u)t with dynamics

dXt = b(t,Xt , ut)dt+σt(t,Xt , ut)dWt+

∫Rγ(t,Xt− , ut− , z)N(dt,dz).

Objective

J(u) = E

[∫ T

0

f (t,Xt , ut)dt + g(XT )

],

for T <∞ deterministic, f continuous, g concave. We want to findan admissible policy u∗ ∈ A such that

J(u∗) = supu∈A

J(u).

30



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Hamiltonian

We define a function, called Hamiltonian

H : [0,T ]× R× U × R× R×R → R

by

H(t, x , u, p, q, r) = b(t,Xt , ut)p + σt(t,Xt , ut)q

+

∫Rγ(t,Xt , ut , z)r(t, z)ν(dz), (4.1)

where R is the set of functions r : [0,T ]× R→ R such that theintegral (4.1) converges. p, q, r satisfies the corresponding adjointbackward stochastic differential equation

dpt = −Hx(t, x , u, p, q, r)dt + qdWt +

∫R

r(t−, z)N(dt,dz),

pT = g ′(XT ). (4.2)

31



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Optimality condition

Let u, u∗ ∈ A and let X ∗t = X(u∗)t , Xt = X

(u)t be the corresponding

state processes. We know that u∗ is optimal if

J(u∗) ≥ J(u), ∀u ∈ A,

and after substitution

J(u∗)−J(u) = E

[∫ T

0

(f (t,X ∗t , u∗t )− f (t,Xt , ut)) dt + g(X ∗T )− g(XT )

]

Assumption

We assume that the integrals in the following derivation are finite.

32



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

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Terminal wealth condition I

Since g is concave

E [g(X ∗T )− g(XT )] ≥ E [(X ∗T − XT ) g ′(X ∗T )]

= E [(X ∗T − XT ) p∗(T )]

= E

[∫ T

0

(X ∗t− − Xt−) dp∗t +

∫ T

0

p∗t−d (X ∗t − Xt)

+

∫ T

0

(σ(t,X ∗t , u∗t )− σ(t,Xt , ut)) q∗t dt

+

∫ T

0

(γ(t,X ∗t , u∗t , z)− γ(t,Xt , ut , z)) r∗(t, z)ν(dz)dt

].

33



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


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Terminal wealth condition II

We substitute into pt and Xt and not rewrite m’gales with zeroexpected value

= E

[∫ T

0

− (X ∗t − Xt) Hx(t,X ∗t , u∗t , p∗t , q∗t , r∗(t, .))dt

+

∫ T

0

p∗t (b(t,X ∗t , u∗t )− b(t,Xt , ut)) dt

+

∫ T

0


+

∫ T

0


].

34



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

’Consumption’ condition

By the definition of H

E

[∫ T

0

(f (t,X ∗t , u∗t )− f (t,Xt , ut)) dt

]

= E

[∫ T

0

(H(t,X ∗t , u∗t , p∗t , q∗t , r∗(t, .))

−H(t,Xt , ut , p∗t , q∗t , r∗(t, .))) dt

−∫ T

0

p∗t (b(t,X ∗t , u∗t )− b(t,Xt , ut)) dt

−∫ T

0


−∫ T

0


].

35



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Terminal wealth + ’Consumption’ condition

J(u∗) − J(u) ≥ E

[∫ T

0

(H(t,X ∗t , u∗t , p∗t , q∗t , r∗(t, .))

−H(t,Xt , ut , p∗t , q∗t , r∗(t, .))) dt

−∫ T

0

(X ∗t − Xt) Hx(t,X ∗t , u∗t , p∗t , q∗t , r∗(t, .))dt

]

and if we find condition such that

J(u∗) − J(u) ≥ 0 (4.3)

we know that u∗ is the optimal control.

36



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Theorem

Theorem (Sufficient maximum principle)

Let u∗ ∈ A with corresponding solution X ∗ = X (u∗) and supposethere exists a solution (p∗t , q

∗t , r∗(t, z)) of the corresponding adjoint

equation. Moreover, suppose that

H(t,X ∗t , u∗t , p∗t , q∗t , r∗(t, .)) = sup

u∈UH(t,X ∗t , u, p

∗t , q∗t , r∗(t, .)), t ∈ [0,T ],

andH∗(x) = max

u∈UH(t, x , u, p∗t , q

∗t , r∗(t, .)) (4.4)

exists and is a concave function of x, t ∈ [0,T ] (Arrow condition).Then u∗ is optimal control.

37



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Remarks to theorem

Condition (4.4) is guaranteed by concavity of the functionH(t, x , u, p∗t , q

∗t , r∗(t, .)) in (x , u), t ∈ [0,T ].

To finish the proof, denote

h(t, x , u) = H(t, x , u, p∗t , q∗t , r∗(t, .))

andh∗(t, x) = max

u∈Ah(t, x , u)

Optimality condition (4.3) holds if

0 ≤ h∗(t, x∗)− h(t, x , u)− (x∗ − x)h∗′(t, x∗)

≥ h∗(t, x∗)− h∗(t, x)− (x∗ − x)h∗′(t, x∗) ≥ 0.

because h∗ is concave in x for t ∈ [0,T ].

38



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Relation to dynamic programming

For the dynamic programming we define criterion

J(u)(s, x) = E

[∫ T−s

0

f (t + s,Xt , ut)dt + g(XT−s)

],

v(s, x) = supu∈A

J(u)(s, x).

Theorem

Assume v ∈ C1,3 and that there exists an optimal control u∗t andcorresponding state process X ∗t for the maximum principle problem.Define

pt = vx(t,X ∗t ),

qt = σ(t,X ∗t , u∗t )vxx(t,X ∗t ),

r(t, z) = vx(t,X ∗t + γ(t,X ∗t , u∗t , z))− vx(t,X ∗t ).

Then pt , qt , r(t, .) solve the adjoint equation (4.2).39



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

Bibliography

R. Cont and P. Tankov.Financial modelling with jump processes.Chapman & Hall/CRC Financial Mathematics Series., 2004.

K. Janecek.Advanced topics in financial mathematics.Study material, MFF UK, 2008.

B. Øksendal and A. Sulem.Applied stochastic control of jump diffusions. 2nd ed.Universitext. Berlin: Springer., 2007.

40



Jakub Petrasek

Introduction

Motivation

Levy processes

Optimal Control

Model set-up


Application




Maximmumprinciple

Model set-up

Optimalitycondition


Bibliography

...

Thank you for attention

41

Documents

Optimal control for jump processes, HJB equationsartax.karlin.mff.cuni.cz/~dvorm3bm/1011z/Petrasek.pdfOptimal control for jump processes, HJB equations Jakub Petr asek Introduction