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Coordinate-free Stochastic Differential
Equations as Jets
John ArmstrongDept. of MathematicsKing’s College [email protected]
Damiano BrigoDept. of Mathematics
Imperial College [email protected]
First version: October 1, 2015. This version: May 4, 2016
Abstract
We explain how Ito Stochastic Differential Equations on manifoldsmay be defined as 2-jets of curves. We use jets as a natural languageto express geometric properties of SDEs and show how jets can lead tointuitive representations of Ito SDEs, including three different types ofdrawings. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic differential geometry is not necessary andelaborate on the relationships with the jets approach. We consider thetwo calculi as being simply different coordinate systems for the same un-derlying coordinate-free stochastic differential equation. If the extrinsicapproach to differential geometry is adopted, then Stratonovich calculusmay appear to be necessary when studying SDEs on submanifolds but infact one can use the Ito/2-jets framework proposed here by recalling thatthe curvature of the 2-jet follows the curvature of the manifold. We arguethat the choice between Ito and Stratonovich is a modelling choice dic-tated by the type of problem one is facing and the related desiderata. Wealso discuss the forward Kolmogorov equation and the backward diffusionoperator in geometric terms, and consider percentiles of the solutions ofthe SDE and their properties, leading to fan diagrams and their relation-ship with jets. In particular, the median of a SDE solution is associatedto the drift of the SDE in Stratonovich form for small times. Finally, weprove convergence of the 2-jet scheme to classical Ito SDEs solutions.
Keywords: Jets, 2-jets, stochastic differential equations, manifolds, SDEson manifolds, stochastic differential geometry, mean square convergence, nu-merical schemes, Euler scheme, 2-jet scheme, SDEs drawings, Ito calculus,Stratonovich calculus, Fan Diagram, SDE median, SDE mode, SDE quantile,Forward Kolmogorov Equation, Fokker Planck Equation, 2-Jet Scheme Conver-gence, Coordinate-Free SDEs.
AMS classification codes: 58A20, 39A50, 58J65, 60H10, 60J60, 65D18
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1 Introduction
This paper examines the geometry of stochastic differential equations using acoordinate free approach. Our aims are threefold:
(i) show how to visualize stochastic differential equations (SDEs);
(ii) show that a coordinate free formulation of stochastic differential equationson manifolds is both possible and enlightening;
(iii) understand the geometric meaning of the coefficients of Ito and Stratonovichstochastic differential equations.
Most texts on SDEs take an extremely algebraic approach to the subject.By contrast, this paper considers the question of how to draw an SDE in a waythat makes Ito’s lemma intuitively clear. In particular we will illustrate a wayof drawing an SDE on a rubber sheet such that if the sheet is stretched, thediagram transforms according to Ito’s lemma. In other words given an SDEin Rn we give a method of drawing SDEs such that for all f : Rn → Rn thefollowing diagram commutes:
SDE for X SDE for f(X)
Picture of SDE for X in Rn f(Picture of SDE for X)
Ito’s lemma
Draw Drawf
(1)
We will address this question in Section 2 in the case of an SDE drivenby a single Brownian motion. We will also show how our diagrams can beinterpreted in terms of a convergent discrete time numerical scheme and willprove its convergence in the more general case of a SDE driven by a vectorBrownian motion. Importantly, our discrete-time scheme can be formulated incoordinate free language using the geometric notion of jets. Thus our methodof drawing SDEs is general enough to make it possible to draw an SDE on amanifold and it naturally gives rise to a coordinate free method of formulatingSDEs on manifolds.
It is well known that vectors on a manifold can be understood from a numberof different perspectives. Firstly tangent vectors can be defined in terms of localcoordinates and their transformation rules. Secondly one can define vectors interms of operators, specifically as linear functions on the space of germs ofsmooth functions that also obey the Leibniz rule. A third definition of a vectoris as a first order approximation to a curve on a manifold. Fourth, one canunderstand a vector field as an infinitesimal diffeomorphism of a manifold. Sincein addition vector fields can be interpreted as ordinary differential equations(ODEs) on a manifold, this gives four ways of understanding ODEs.
In this paper we will study the parallel interpretations for SDEs. The firstapproach was used by Ito in [14] to define SDEs on manifolds. The secondapproach is coordinate free and is related to understanding SDEs in terms of
2
diffusion operators (see [11] for the case of Rn). The third approach correspondsto our interpretation of SDEs in terms of jets. The fourth approach correspondsto Stratonovich calculus. Many texts such as [7, 6, 21], use Stratonovich calculusto define SDEs on manifolds. The point we wish to emphasize is that this isby no means the only way to the differential geometry of SDEs in a coordinateindependent manner.
The relationship between the jet formulation and the differential operatorformulation of SDEs is the topic of Section 3. We will use the language ofjets to give geometric expressions for many important concepts that arise instochastic analysis. These geometric representations are in many ways moreelegant than the traditional representations in terms of the coefficients of SDEs.In particular we will give coordinate free formulations of the following: Ito’slemma; the diffusion operators; Ito SDEs on manifolds and Brownian motionon Riemannian manifolds.
Also in Section 3, we show how our definition of SDE’s driven by a singleBrownian motion can be generalized to processes driven by multiple Brownianmotions. We show how to draw such SDE’s and illustrate this using the Hestonstochastic volatility model (two-dimensional diffusion) and Brownian motion onthe torus.
In Section 4 we consider how our formulation is related to the Stratonovichformulation of SDEs. We will prove that sections of the bundle of n-jetsof curves in a manifold correspond naturally to n-tuples of vector fields inthe manifold. This generalizes the correspondence between Ito calculus andStratonovich calculus to n-jets. This correspondence also shows that Ito cal-culus and Stratonovich calculus can both be interpreted as simply a choice ofcoordinate system for the space of sections of the bundle of n-jets of curves.Moreover, our result provides a geometric interpretation of the Ito-Stratonovichtransformation, expressing the relationship between Ito and Stratonovich cal-culus.
In Section 5 we consider an alternative approach to drawing SDEs in onedimension. We will see that the 2-jet defining an Ito SDE can be interpreted asdefining a fan diagram showing the limiting trajectories of certain percentiles ofthe probability distributions associated with the SDE solution process. More-over we will show that the drift vector of the Stratonovich formulation can besimilarly interpreted as a short-time approximation to the median. This obser-vation is related to the coordinate independence of 2-jets and vector fields andthe coordinate independence of the notion of median. By contrast the mean ofa process is coordinate dependent and its short time behaviour is given by thedrift vector in the Ito formulation. We will also consider short time behaviourof the mode.
We conclude by remarking that our work has numerous applications:
(i) Differential geometry of SDEs. Coordinate free calculations in differentialgeometry are often simpler, shorter and more insightful than their localcoordinate counterparts. As an application of our theory we were able todefine a novel notion of projection for SDEs ([2, 1]). The local coordinate
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formulation of the projection of an SDE is difficult to compute and it isdifficult to even check that it has the correct properties under coordinatechange. The coordinate free formulation by contrast is short, elegant andmanifestly invariant.
(ii) Pedagogy. For the many students who prefer to understand mathematicsgeometrically rather than algebraically, a diagram of an Ito SDE will bevery helpful.
(iii) Qualitative analysis of SDEs. One can gain considerable understandingof an ODE simply by drawing the corresponding vector field. The sameapplies to SDEs by drawing the 2-jet.
2 SDEs as fields of curves driven by a singleBrownian motion
2.1 Drawing and simulating SDEs as “fields of curves”
Suppose that at every point x in Rn we have an associated smooth curve:
γx : R→ Rn with γx(0) = x
As an example we might define γEx on R2 as follows:
γE(x1,x2)(t) = (x1, x2) + t(−x2, x1) + 3t2(x1, x2)
This specific case is chosen for a number of reasons. We have a radially outwardt2 component and an orthogonal counterclockwise circular component t. Thisallows us to draw a good picture of the trajectories and also to lead to a goodtransformation when using polar coordinates. This specific example has zeroderivatives with respect to t from the third derivative on. We stop at secondorder in t since this will be enough to converge to classical stochastic calculus,as we will see. However, one should also check that for well behaved curves,higher order terms would not lead to limits that are different from the limitobtained from terms up to order two. Indeed, further on in this paper we willconsider also more general γx curves with nonzero higher order derivatives, andwe will show that under some assumptions on the third derivatives of t 7→ γx(t),only the first and second order terms contribute to form the limits convergingto classical stochastic calculus, and we may ignore terms of order higher thantwo.
We will use the superscript E to indicate this example process throughout.We have plotted γE in figure 1.
To be precise we have taken a grid of points in R2 which are marked as dotsin the figure. We have then drawn the curve γEx at each grid point x for theparameter values t in (−0.1, 0.1). In general when drawing such a figure fora general γ, one should use the same range t ∈ (−ε, ε) for every curve in thefigure, but one is free to choose ε to make the diagram visually appealing. (In
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just the same way when drawing vector fields, one chooses a sensible scale foreach vector).
-2 -1 0 1 2 3
-2
-1
0
1
2
3
Figure 1: A plot of γE
Given such a γ, a starting point x0 (the deterministic origin in our example,X0 = 0), a Brownian motion Wt and a time step δt we can define a discretetime stochastic process using the following recurrence relation:
X0 := x0, Xt+δt := γXt(Wt+δt −Wt) (2)
In Figure 2 we have plotted the trajectories of process for γE , the startingpoint (1, 0), a fixed realization of Brownian motion and a number of differenttime steps. Rather than just plotting a discrete set of points for this discretetime process, we have connected the points using the curves in γEXt .
Notice that since the δWt = Wt+δt−Wt are normally distributed with stan-dard deviation
√δt we can interpret the trajectories as being randomly gener-
ated trajectories that move from Xt to Xt+δt by following the curve s 7→ γXt(s)from s = 0 to s = εt
√δt where the εt are independent normally distributed
random variables.As the figure suggests, these discrete time stochastic processes (2) converge
in some sense to a limit as the time step tends to zero.We will use the following notation for the limiting process:
Coordinate free SDE: Xt γXt(dWt), X0 = x0. (3)
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δt = 0.2× 2−5 δt = 0.2× 2−7
δt = 0.2× 2−9 δt = 0.2× 2−11
Figure 2: Discrete time trajectories for γE for a fixed Wt and X0 with differentvalues for δt
For the time being, let us simply treat equation (3) as a short-hand way ofsaying that equation (2) converges in some sense to a limit. Note that it will notconverge for arbitrary γ’s but it does converge for nice γ such as γE or γ’s withsufficiently good regularity. The reader familiar with Ito calculus will want toknow how this notation corresponds to Ito stochastic differential equations andin precisely what sense and under what circumstances equation (2) convergesto a limit. These questions are addressed in Section. 2.2.
An important feature of equation (2) is that it makes no reference to thevector space structure of Rn for our state space X. We have maintained this inthe formal notation used in equation (3). By avoiding using the vector spacestructure on Rn we will be able to obtain a coordinate free understanding ofstochastic differential equations.
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Example 1. For a fixed α ∈ N, in a given coordinate system on R, we candefine curves at each point in R by:
γαx (s) = x+ sα
Let us compute the limit of the discrete time process corresponding to thesecurves. In the case α = 1, we have trivially that the Xt = x0 +Wt. By equation(2) we have that:
Xnδt = x0 +
n∑i=1
(W(i+1)δt −Wiδt)α
= x0 + (δt)α2
n∑i=1
εαi
where the εi are independently normally distributed with mean 0 and standarddeviation 1. Fixing a terminal time T so that δt = T
n we have:
XT = x0 + (T/n)α2
n∑i=1
εαi .
By the central limit theorem we see that if α = 2 this converges in distributionto a Dirac mass centered on x0 + T . If α ≥ 3 we see that this converges to aDirac mass centered on x0.
2.2 SDEs as fields of curves up to order 2: 2-jets
Let us now invoke explicitly the Rn structure of the state space by choosing aspecific coordinate system and consider the (component-wise) Taylor expansionof γx. We have:
γx(t) = x+ γ′x(0)t+1
2γ′′x(0)t2 +Rxt
3, Rx =1
6γ′′′x (ξ), ξ ∈ [0, t],
where Rxt3 is the remainder term in Lagrange form. Substituting this Taylor
expansion in our Equation (2) we obtain
δXt = γ′Xt(0)δWt +1
2γ′′Xt(0)(δWt)
2 +RXt(δWt)3, X0 = x0. (4)
Example 1 suggests that we can replace the term (δWt)2 with δt and we can
ignore terms of order (δWt)3 and above. So we expect that under reasonable
conditions, in the chosen coordinate system, the recurrence relation given by(2) will converge to the same limit as the numerical scheme:
δXt = γ′Xt(0)δWt +1
2γ′′Xt(0)δt, X0 = x0.
Defining a(X) := γ′′X(0)/2 and b(X) := γ′X(0) we have that this last equationcan be written as
δXt = a(Xt)δt+ b(Xt)δWt. (5)
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It is well known that this last scheme (Euler scheme) does converge in someappropriate sense to a limit ([15]) and that this limit is given by the solution tothe Ito stochastic differential equation:
dXt = a(Xt) dt+ b(Xt)dWt, X0 = x0. (6)
Thus, given a coordinate system, we may think of equation (2) as defininga numerical scheme for approximating the Ito SDE (6). In this context we call(2) the 2-jet scheme. A rigorous proof of the convergence of the 2-jet schemein mean square (L2(P)) to the solution of the Ito SDE, based on appropriatebounds on the derivatives of the curves γx, is given in Appendix A.
At this point one may wonder in which sense Equation (2) and its limit arecoordinate free. It is important to note that the coefficients of equation (6)only depend upon the first two derivatives of γ. We say that two smooth curvesγ : R → Rn have the same k-jet (k ∈ N, k > 0) if they are equal up to orderO(tk) in a given coordinate system. If this holds in a given coordinate system,it will hold in all coordinate systems. The k-jet can then be defined for exampleas the equivalence class of all curves that are equal up to order O(tk) in oneand hence all coordinate systems, similarly to what is done to define tangentvectors, leading to a coordinate free definition. Using this terminology, we saythat the coefficients of equation (6) (and (5)) are determined by the 2-jet of acurve γ : R→ Rn in a specific coordinate system.
Jets are defined in much more general terms on manifolds in differential ge-ometry, for example they can be defined as equivalence classes of maps withthe same Taylor expansion up to a given order in given (and hence all) charts.They represent abstract Taylor expansions that allow one to approach differ-ential equations on manifolds in a coordinate-free manner. Also, jets and therelated notions of obstruction and symbol are powerful tools in the analysis ofdifferential equations in a geometric setting.
In the light of the above convergence result, we can say that in pictures suchas Figure 1 one should avoid interpreting any details other than the first twoderivatives of the curve. One way of doing this is by insisting that we draw thequadratic curves that best fit the curves γ rather than the actual curve γ itself.
Notice that vectors can be defined in the same way as 1-jets of smooth curves.In just the same way as we draw quadratic curves in Figure 1, one normallychooses to draw straight lines in a diagram of a vector field.
In summary: vector fields are fields of 1-jets and they represent ODE’s;diagrams such as Figure 1 are pictures of fields of 2-jets and they represent ItoSDEs.
Given a curve γx, we will write j2(γx) for the two jet associated with γx.This is formally defined to be the equivalence class of all curves which are equalto γx up to O(t3).
Since we will show that, under reasonable regularity conditions, the limit ofthe symbolic equation (3) depends only on the 2-jet of the driving curve, wemay rewrite equation (3) as:
Coordinate-free 2-jet SDE: Xt j2(γXt)(dWt), X0 = x0. (7)
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This may be interpreted either as a coordinate free notation for the classicalIto SDE given by equation (6) or as a shorthand notation for the limit of theprocess given by the discrete time equation (2).
2.3 Coordinate-free Ito formula with jets
Suppose that f is a smooth mapping from Rn to itself and suppose that Xsatisfies (2). It follows that f(X) satisfies:
(f(X))t+δt = f ◦ γXt(δWt).
Taking the limit as δt tends to zero we have:
Lemma 1. [Ito’s lemma — coordinate free formulation] If the process Xt sat-isfies
Xt j2(γXt)(dWt)
then f(Xt) satisfiesf(X)t j2(f ◦ γXt)(dWt).
If one prefers the more traditional format for SDEs given in (6) we sim-ply need to calculate the derivatives of f ◦ γ. In a chosen coordinate system,let us write γiX for the i-th component of γX with respect to the coordinatesx1, x2, . . . xn for Rn. Two applications of the chain rule give us:
(f ◦ γX)′(t) =
n∑i=1
∂f
∂xi(γX(t))
dγXdt
(f ◦ γX)′′(t) =
n∑j=1
n∑i=1
∂2f
∂xi∂xj(γX(t))
dγiXdt
dγjXdt
+
n∑i=1
∂f
∂xi(γX(t))
d2γXdt2
We conclude that our lemma is equivalent to the classical Ito’s lemma.We can now interpret Ito’s lemma geometrically as the statement that the
transformation rule for jets under a change of coordinates is given by composi-tion of functions.
Since we now understand the geometric content of Ito’s lemma, we can drawa picture to illustrate it. Consider the transformation φ : R2/{0} → [−π, π]×Rby
(θ, s) = φ(x1, x2) =
(arctan(x2/x1), log(
√x2
1 + x22)
),
or equivalentlyφ(exp(s) cos(θ), exp(s) sin(θ)) = (θ, s),
applied to our example process γE . This can be viewed as a transformation ofthe complex plane φ(z) = i log(z). We use φ to transform the fourth picture
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in Figure 2 in two different ways. Firstly we apply directly φ to each point ofFigure 2 to obtain a new point to be inserted in a new figure. This is doneby using image manipulation software. In other words we stretch the imagewithout any consideration of it’s mathematical structure. The result of this isshown in the upper part of Figure 3.
The process j2(φ ◦ γE) plotted using image manipulation software
The process j2(φ ◦ γE) plotted by applying Ito’s lemma
Figure 3: Two plots of the process j2(φ ◦ γE) in the plane (θ, s).
As an alternative approach, we transform our equation using Ito’s lemmaapplied to the function φ. This results in the equation for θ and s:
d(θ, s) =
(0,
7
2
)dt+ (1, 0) dWt.
We can then use this equation to plot the process (θ, s) directly by simulatingthe process in discrete time as before. The result is shown in the lower part ofFigure 3.
As one can see the two approaches to plotting the transformed process giveessentially identical results, showing an example of our earlier diagram (1) atwork. The differences one can see are: the lower quality in the upper image,obtained by transforming pixels rather than using vector graphics; the gridpoints at which the 2-jets are plotted are changed; small differences in thesimulated path since we have only simulated discrete time paths.
We have assumed in our presentation so far that the 2-jet j2(γx) is associatedin a deterministic and time independent manner with the point x. However, the
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theory can be generalized to time dependent and stochastic choices of 2-jets.
3 Differential Geometry of Ito SDEs
We can now define an Ito SDE on a manifold, M , driven by a single Brownianmotion W by choosing a 2-jet corresponding to the curve γx at each point xin the manifold with γx(0) = x. Our aim in this section is to show that thisperspective allows one to give coordinate free definitions of the key concepts instochastic differential geometry.
As we have already observed our discrete time formulation (equation (2)) ofIto SDEs on manifolds is already coordinate free in that it makes no mentionof the vector space structure of Rn. Given a field of 2-jets of curves at eachpoint of a manifold M we can write down a corresponding Ito SDE using thenotation of equation (7). This may be interpreted either as indicating the limitof the discrete time process or as a short-hand for a classical Ito SDE. Thereformulation of Ito’s lemma in the language of jets shows that this secondinterpretation will be independent of the choice of coordinates. The only issueone needs to consider are the bounds needed to ensure existence of solutions.The details of transferring the theory of existence and uniqueness of solutionsof SDEs to manifolds are considered in, for example, [7], [6], [14] and [8].
The point we wish to emphasize is that the coordinate free formulation ofan SDE given in equation (7) can be interpreted just as easily on a manifold ason Rn. This leads us to the following
Definition 1. (Coordinate free Ito SDEs driven by 1-dimensional Brow-nian motion). A Ito SDE on the manifold M is a section of the bundle of 2-jetsof curves over M together with a one-dimensional Brownian motion W .
Let us see how this simplifies the study of the differential geometry of SDEs.Suppose now that f is a function mapping M to R. We can define a dif-
ferential operator acting on functions in terms of a 2-jet associated with γx asfollows.
Definition 2. (Backward diffusion operator via 2-jets). The Backwarddiffusion operator for the Ito SDE defined by the 2-jet associated with the curveγx is defined on suitable functions f as
Lγx(f) :=1
2(f ◦ γx)′′(0).
In contexts where the SDE is understood we will simply write L.Note that this definition is simply the drift term of the Ito SDE for f(X)
computed using Ito’s lemma. To first order, the drift measures how the expec-tation of a variable changes over time. Thus, with δ denoting the forward tincrement as usual:
(δE[f(Xt)|Xs = x]) |s=t = (Lγx(f))δt+O(δt2) (8)
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Before proceeding to define the forward diffusion operator, let us brieflyrecall the theory of the bundle of densities on a manifold.
Recall that given a vector space V and a group homomorphism:
τ : GL(n,R)→ Aut(V )
we can define an associated bundle V over M . The fibres of the bundle overa point p are given by equivalence classes of charts φ : M → Rn and vectorsv ∈ V under the relation:
(φ1, v1) ∼ (φ2, v2) ⇔ τ((φ1)∗ ◦ (φ2)−1
∗)p
(v2) = v1.
Each chart φ : M → Rn defines local coordinates on V. We use this to definethe smooth structure on V. This generalizes straightforwardly [17] to allow oneto associate a vector bundle with any principal G-bundle and a representationof the Lie group G. In our special case, the principal G-bundle we are using isthe frame bundle of the manifold.
Now consider the representation:
τ(g) = |det(g)| ∈ Aut(R).
This defines a bundle over a manifold M called the bundle of densities. Thisbundle is denoted Vol. The usual transformation formula for probability densi-ties under changes of coordinates tells us that a probability density over M is asection of Vol.
Integration defines a pairing between functions and densities on M by:∫: Γ(R)× Γ(Vol)→ R
by
∫(f, ρ) :=
∫fρ.
On non-compact manifolds one must either insist that one of f or ρ is compactlysupported or consider decay rates of f and ρ to ensure this is well defined.
Note that from a probabilistic point of view, this pairing is interpreted astaking expectations.
Using integration by parts, we can define L∗ to be the formal adjoint of Lwith respect to this pairing. This is called the forward diffusion operator.
Even if one does not know the initial state X0 but knows its probabilitydensity, ρ, one may integrate equation (8) to obtain:
∂
∂t
∫M
f(ρ) =
∫M
(Lf)(ρ)
Since this holds for all smooth compactly supported functions f we can deduce:
∂ρ
∂t= L∗(ρ).
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We conclude that the Fokker–Planck equation follows from Ito’s lemma forfunctions.
Notice that both L and L∗ are linear second order operators. The keydifference is that they have different domains. L acts on functions; L∗ actson densities. This gives a geometric explanation as to why L appears in theFeynman–Kac equation which tells us about the evolution of expectations offunctions whereas L∗ appears in the Fokker–Planck equation which tells usabout the evolution of probability densities.
Notice also that our definitions of the diffusion operators are coordinate free.Our discussion can be generalized to SDEs driven by d independent Brownian
motions Wαt with α ∈ {1, 2, . . . , d}.
Definition 3. (SDEs in Rn driven by vector-Brownian motion as 2-jets). Consider jets of functions of the form:
γx : Rd → Rn
Just as before we can consider difference equations of the form:
Xt+δt := γXt(δW 1
t , . . . , δWdt
)or, if writing δWα
t = εα√δt, with ε independent normals,
Xt+δt := γXt
(ε1√δt, . . . , εd
√δt).
Again, the limiting behaviour of such difference equations will only depend uponthe 2-jet j2(γx) and can be denoted by (7), where it is now understood that dWt
is the vector Brownian motion increment. We can state the following theorem,showing that the 2-jet scheme converges in L2(P) to the classic Ito SDE withthe same coefficients in a given coordinate system. This theorem proves one ofthe main results of this paper: a Ito SDE can be represented in a coordinatefree manner simply as a 2-jet driven by Brownian motion.
Theorem 1. Convergence of the 2-jet schemes to Ito SDEs. Let γx :Rd → Rn be a smoothly varying family of functions whose first and secondderivatives in Rd satisfy Lipschitz conditions (and hence linear growth bounds).In other words we require that there exists a positive constant K such that forall x, y ∈ Rn and α, β ∈ {1, 2, . . . , d} we have:∣∣∣∣ ∂γx∂uα
∣∣∣u=0− ∂γy∂uα
∣∣∣u=0
∣∣∣∣ ≤ K|x− y|∣∣∣∣ ∂2γx∂uα∂uβ
∣∣∣u=0− ∂γy∂uα∂uβ
∣∣∣u=0
∣∣∣∣ ≤ K|x− y|(and hence
∣∣∣∣ ∂γx∂uα
∣∣∣u=0
∣∣∣∣2 ≤ K2(1 + |x|2
)∣∣∣∣ ∂γx∂uα∂uβ
∣∣∣u=0
∣∣∣∣2 ≤ K2(1 + |x|2
) ).
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Note that we are using the letter u to denote the coordinates for Rd. Suppose inaddition that we have a uniform bound on the third derivatives:∣∣∣∣ ∂3γx
∂uα∂uβ∂uγ
∣∣∣u=0
∣∣∣∣ ≤ KLet T be a fixed time and let T N := {0, δt, 2 δt, . . . , N δt = T}, be a set ofdiscrete time points. Let Xt denote the 2-jet scheme defined by:
Xt+ε = γXt(Wt+ε −Wt), t ∈ T N , ε ∈ [0, δt], X0 = x0, (9)
then this converges in L2(P) to the classical Ito solution of the correspondingSDE, namely
Xt = X0 +
∫ t
0
a(Xs) ds+
d∑α=1
∫ t
0
bα(Xs) dWαs , t ∈ [0, T ]
where
a(x) :=1
2
d∑α=1
∂2γx∂uα∂uα
∣∣∣u=0
and
bα(x) :=∂γx∂uα
∣∣∣u=0
.
More precisely we have the estimate:
supt∈[0,T ]
E{|XN
t − Xt|2}≤ C δt = C
T
N(10)
for some constant C independent of N . We denote the coordinate free equationobtained as limit of (9) by
Xt j2(γXt)(dWt).
The proof is given in the appendix.
Thus the limiting SDE associated with our difference equation is:
dXt =1
2
∑α
∑β
∂2γXt∂xα∂xβ
dWαt dW β
t +∑α
∂γXt∂xα
dWαt (11)
Here xα are the standard orthonormal coordinates for Rd. Again our equationshould be interpreted with the convention that dWα
t dW βt = gαβE dt where gE is
equal to 1 if α equals β and 0 otherwise.
Notice that in the above definition we choose to write gE instead of using aKronecker δ because one might want to choose non orthonormal coordinates forRd and so it is useful to notice that gE represents the symmetric 2-form defining
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the Euclidean metric on Rd. Using as Kronecker δ would incorrectly suggestthat this term transforms as an endomorphism rather than as a symmetric 2-form. An advantage of our notation is that we can use the Einstein summationconvention. For example equation (11) can then be simplified to:
dXit =
1
2∂α∂βγ
idWαt dW β
t + ∂αγi dWα
t
=1
2∂α∂βγ
igαβE dt+ ∂αγi dWα
t
(12)
In the n-dimensional case we define the backward diffusion operator by:
Lγxf :=1
2∆E(f ◦ γx) =
1
2∂α∂β(f ◦ γx)gαβE . (13)
Here ∆E is the Laplacian defined on Rd. Lγx acts on functions defined on M .We define L∗ to be its formal adjoint which acts on densities defined on M .
Again inspired by the Rn case, we may state the following
Definition 4. (Coordinate free Ito SDEs driven by vector Brownianmotion). A Ito SDE on a manifold M is a section of the bundle of 2-jets ofmaps Rd →M together with d Brownian motions W i
t , i = 1, . . . , d.
3.1 Weak and strong equivalence
We see that both the Ito SDE (12) and the backward diffusion operator use onlypart of the information contained in the 2-jet. Specifically only the diagonalterms of ∂α∂βγ
i (those with α = β) influence the SDE and even for these termsit is only their average value that is important. The same consideration appliesto the backward diffusion operator. This motivates the following
Definition 5. We say that two 2-jets γ1x and γ2
x
γix : R→M
are weakly equivalent ifLγ1
x= Lγ2
x.
We say that γ1 and γ2 are strongly equivalent if in addition
j1(γ1) = j1(γ2).
We see that the SDEs defined by the two sets of 2-jets are equivalent if the2-jets are strongly equivalent. This means that given the same realization ofthe driving Brownian motions Wα
t the solutions of the SDEs will be almostsurely the same (under reasonable assumptions to ensure pathwise uniquenessof solutions to the SDEs).
When the 2-jets are weakly equivalent, the transition probability distribu-tions resulting from the dynamics of the related SDEs are the same even though
15
the dynamics may be different for any specific realisation of the Brownian mo-tions. For this reason one can define a diffusion process on a manifold as asmooth selection of a second order linear operator L at each point that deter-mines the transition of densities. A diffusion can be realised locally as an SDE,but not necessarily globally.
Recall that the top order term of a quasi linear differential operator is calledits symbol. In the case of a second order quasi linear differential operator Dwhich maps R-valued functions to R-valued functions, the symbol defines asection of S2T , the bundle of symmetric tensor products of tangent vectors,which we will call gD.
In local coordinates, if the top order term of D is:
Df = aij∂i∂jf + lower order
then gD is given by:gD(Xi, Xj) = aijXiXj
We are using the letter g to denote the symbol for a second order operator be-cause, in the event that g is positive definite and d = dimM , g defines a Rieman-nian metric on M . In these circumstances we will say that the SDE/diffusion isnon-singular. Thus we can associate a canonical Riemannian metric gL to anynon-singular SDE/diffusion.
Definition 6. A non-singular diffusion on a manifold M is called a RiemannianBrownian motion if
L(f) =1
2∆gL(f).
Note that given a Riemannian metric h on M there is a unique RiemannianBrownian motion (up to diffusion equivalence) with gL = h. This is easilychecked with a coordinate calculation.
This completes our definitions of the key concepts in stochastic differentialgeometry and indicates some of the important connections between stochasticdifferential equations, Riemannian manifolds, second order linear elliptic oper-ators and harmonic maps.
We emphasize that all our definitions are coordinate free and we have workedexclusively with Ito calculus. It is more conventional to perform stochasticdifferential geometry using Stratonovich calculus. The justification usually givenfor this is that Stratonovich calculus obeys the usual chain rule so the coefficientsof Stratonovich SDEs can be interpreted as vector fields. For example one canimmediately see if the trajectories to a Stratonovich SDE almost surely lie onparticular submanifold by testing if the coefficients of the SDE are all tangentto the manifold. We would argue that the corresponding test for Ito SDEs isalso perfectly simple and intuitive: one checks whether the 2-jets follow thecurvature of the manifold. As is well known, the probabilistic properties of theStratonovich integral are not as nice as the properties of the Ito integral. Inparticular the Ito integral is a martingale whereas the Stratonovich integral isnot. Thus the use of Stratonovich calculus on manifolds comes at a price.
16
3.2 Drawing SDEs driven by multi-dimensional Brownianmotion
We now understand that we can draw an Ito SDE driven by d-dimensionalBrownian motion by drawing a function:
γx : Rd →M
at every point on the manifold M . Of course, in practice one only draws thefunction at a finite set of sample points in M .
To draw a function in a sensible way, one draws the image of the ε-ball ateach point for some chosen ε. One should choose a consistent value for everysample point. The image of a function does not completely define the function,one might also want to draw labelled grid lines on the image in order to showhow the coordinate system on Rd is mapped to the coordinate system on theimage. However, if one is only interested in SDEs up to diffusion equivalenceand one’s SDEs are non-singular then it is easy to check that one can recoverthe diffusion equivalence class of γx from just the image of the ε-ball and theimage of the origin.
Such a drawing is sufficient to encode the SDE, but it actually contains someredundant information about γx beyond just the diffusion equivalence class. Forthis reason, it is best to draw a representative of the diffusion equivalence classγx for which the matrix ∂α∂βγx is equal to (gE)αβ (here we have simply usedthe metric gE to lower the indices of (gE)αβ).
With these conventions understood, in Figure 4 we show a plot of the Hestonstochastic volatility model with drift (see [13]):
dSt = µStdt+√νtStdW
1t
dνt = κ(θ − νt)dt+ ξ√νt (ρdW 1
t +√
1− ρ2dW 2t )
(14)
The drift term in (14) can be seen visually by the fact that the dots showingthe points for which we have plotted the 2-jets are off centre in each ellipse.The metric induced by the Heston equation is shown visually by the ellipsesthemselves. The length of a vector starting at the centre of each ellipse and goingto the boundary of the ellipse is approximately equal to ε and this approximationbecomes increasingly accurate as ε tends to zero.
In Figure 5 we plot the corresponding figure for Riemannian Brownian mo-tion on a torus with respect to the metric induced on the torus by the Euclideanmetric on R3. As one can see, all the ellipses appear to be the same size andthe points appear to be in the centre of each ellipse. This first point showsthat the metric induced by Riemannian Brownian motion is indeed equal to themetric induced by the embedding. The second point shows that the forwardand backward operators are equal.
17
0.0 0.5 1.0 1.5 2.0
Ν
0.0
0.5
1.0
1.5
2.0
S
Figure 4: A plot of the Heston model (14) with parameter values ξ = 1, θ = 0.4,κ = 1, µ = 0.1, ρ = 0.5 and where we have plotted the image of the epsilonballs for ε = 0.05
4 Ito & Stratonovich as different coordinates
For simplicity, let us assume in this section that the driver is one dimensional.Thus to define an SDE on a manifold, one must choose a 2-jet of a curve at eachpoint of the manifold. One way to specify a k-jet of a curve at every point inan a neighbourhood is to first choose a chart for the neighbourhood and thenconsider curves of the form:
γx(t) = x+
k∑i=1
ai(x)ti (15)
where ai : Rn → Rn. As we have already seen in Lemma 1, these coefficientfunctions ai depend upon the choice of chart in a relatively complex way. Forexample for 2-jets the coefficient functions are not vectors but instead transformaccording to Ito’s lemma. We will call this the standard representation for afamily of k-jets.
An alternative way to specify the k-jet of a curve at every point is to choosek vector fields A1, . . . , Ak on the manifold. One can then define ΦtAi to be thevector flow associated with the vector field Ai. This allows one to define curves
18
Figure 5: Plot of the SDE defining the Riemannian Brownian motion on thetorus corresponding to the induced metric on the torus
at each point x as follows:
γx(t) = Φtk
Ak(Φt
k−1
Ak−1(. . . (ΦtA1
(x)) . . .)) (16)
where tk denotes the k-th power of t. We will call this the vector representationfor a family of k-jets. It is not immediately clear that all k-jets of curves can bewritten in this way. Let us prove this.
Suppose one chooses a chart and attempts to compute the relationship be-tween the coefficients ai in the standard representation and the componentsof the vector fields Ai in the vector representation. It is clear that the o(t)term a1(x) will depend bijectively and linearly on A1(x). Thus there is abijection between 1-jets written in the form (15) and 1-jets written in theform (16). The o(t2) term will depend linearly upon A2(x) together witha more complex term derived from A1 and its first derivative. Symbolicallya1(x) = ρ(A2(x)) +f(A1(x), (∇A1)(x)) where ρ is a linear bijection determinedby the choice of chart. It follows that there is also a bijection between 2-jetswritten in the standard form and 2-jets written in the vector form. Inductivelywe have:
Theorem 2. Smooth k-jets of a curves can be defined uniquely by a list of kvector fields X1, . . .Xk according to the formula (16)
Notice that the vector representation specifically allows us to define a familyof k-jets varying from point to point. In more technical language, the vectorrepresentation allows us to specify a section of the bundle of k-jets. If one onlyspecifies vectors at a point rather than vector fields, one cannot define the vectorflows and so equation (16) cannot be used to define a k-jet at the point. Thus
19
although there is no natural map from k vectors defined at a point to a k-jet ofa curve, there is a natural map from k vector fields defined in a neighbourhoodto a smoothly varying choice of k-jet at each point.
The standard and vector representations simply give us two different coor-dinate systems for the infinite dimensional space of families of k-jets.
When this general theory about k-jets is applied to stochastic differen-tial equations one sees that two corresponding coordinate representations ofa stochastic differential equation will emerge. Let us calculate in more detailcorrespondence between the two representations.
Lemma 2. Suppose that a family of 2-jets of curves is given in the vectorrepresentation as
γx(t) = Φt2
A (ΦtB(x))
for vector fields A and B. Choose a coordinate chart and let Ai, Bi be thecomponents of the vector fields in this chart. Then the corresponding standardrepresentation for the family of 2-jets is:
γx(t) = x+ a(x)t2 + b(x)t
with
ai = Ai +1
2
∂Bi
∂xjBj
bi = Bi.
Proof. By definition of the flow ΦtB , its components (ΦtB)i satisfy:
∂(ΦtB(x))i
∂t= Bi(ΦtB(x)) (17)
Differentiating this we have:
∂2(ΦtB(x))i
∂t2=∂Bi
∂xj∂(ΦtB)j
∂t
= Bj∂Bi
∂xj.
(18)
We now compute the derivatives of γt(x). We write (Φt2
A )i for the i-th component
of the function Φt2
A .
∂
∂t((Φt
2
A )i(ΦtB(x))) = 2t∂(Φt
2
A )i
∂t(ΦtB) +
∂(Φt2
A )i
∂xj(ΦtB)
∂(ΦtB)j
∂t
Differentiating this again one obtains:
∂2
∂t2((Φt
2
A )i(ΦtB(x))) = 4t2∂2(Φt
2
A )i
∂t2(ΦtB) + 2
∂(Φt2
A )i
∂t(ΦtB) + 4t
∂2(Φt2
A )i
∂xj∂t(ΦtB)
∂(ΦtB)j
∂t
+∂2(Φt
2
A )i
∂xj∂xk(ΦtB)
∂(ΦtB)j
∂t
∂(ΦtB)k
∂t+∂(Φt
2
A )i
∂xj∂ΦtB∂t2
20
At time t = 0 we have that Φ0A is simply the identity. So its partial derivatives
at time 0 are trivial to compute. Hence
∂2
∂t2((Φt
2
A )i(ΦtB(x)))∣∣∣t=0
= 2∂(Φt
2
A )i
∂t(ΦtB) +
∂(ΦtB)i
∂t2
= 2Ai +Bj∂Bi
∂xj
The last line follows from the definition of ΦtA as the flow associated with thevector A together with equation (18). We can now write down the expressionfor a(x). The expression for b(x) follows immediately from equation (17).
As we have already discussed, the standard representation of a 2-jet corre-sponds to conventional Ito calculus. What we have just demonstrated is thatthe vector representation of a 2-jet corresponds to Fisk–Stratonovich–McShanecalculus [9] [18] [22] (Stratonovich from now on). Moreover we have given ageometric interpretation of how the coordinate free notion of a 2-jet of a curveis related to the vector fields defining a Stratonovich SDE.
It is worth now discussing the different stochastic calculi in the light ofthe above result. Despite the initial seminal paper by Ito [14] in a Ito calcu-lus context, where Ito’s formula is used to change coordinates for a SDE ona manifold, Stratonovich calculus has been the main calculus when interfac-ing stochastic analysis with differential geometry [7, 21]. We used Stratonovichcalculus ourselves in our past works on stochastic differential geometry applica-tions to signal processing [3], see also [4], [5]. Stratonovich calculus has becomeso dominant in stochastic differential geometry that some authors have evenasserted that stochastic differential geometry requires the use of Stratonovichcalculus. As Ito’s paper long ago demonstrated, this is not the case. Indeedunder the formalism one can argue that it is still the Ito calculus that “does allthe work” ([21], Chapter V.30, p. 184).
From the point of view of this paper, we consider these two calculi as be-ing simply different coordinate systems for the same underlying coordinate-freestochastic differential equation. As such one should choose the most convenientcoordinate system for the problem at hand. Stratonovich calculus has someclear advantages, most notably the transformation rule for vector fields is sim-pler than that for 2-jets written using the standard representation. Ito calculusalso has some clear advantages, it has simpler probabilistic properties stemmingfrom the fact that the Ito integral is non-anticipative and results in a martingale.
If one likes to think of differential geometry from an extrinsic perspective (i.e.if one doesn’t like to think in terms of charts, but instead in terms of manifoldsembedded in Rn) then Stratonovich calculus may appear to be necessary since anSDE remains confined to a submanifold M almost surely if the A and B vectorfields are tangent to the manifold. This means that it is easy to write down theStratonovich SDE induced on a submanifold from a Stratonovich SDE on Rn.However, this convenience is in no way fundamental. Indeed this phenomenon issimply a consequence of the fact that under these circumstances the curvatureof the 2-jet of the curve follows the curvature of the manifold.
21
In general, since Stratonovich calculus and Ito calculus are just two co-ordinate systems one would expect to be able to work using either calculusinterchangeably. The most important difference between Stratonovich calculusand Ito calculus arises during the modelling process. It is when choosing whatequation to write down in the first place that the choice between the calculi ismost telling. Note that the modelling process is not a strictly mathematicalprocess: it relies upon the modellers intuition.
From a modelling point of view Ito calculus has a strong advantage for appli-cations to subjects such as mathematical finance, where one considers decisionmakers who cannot use information about the future. The non-anticipativenature of the Ito integral will make it easier to write down models for suchproblems using Ito calculus. On the other hand in subjects such as physics orengineering conservation laws and time-symmetry may be paramount, in whichcase modelling with the Stratonovich calculus may be simpler.
5 Fan Diagrams
In this section we will discuss how statistical properties of stochastic differen-tial equations can be read off from the coefficients of the SDE and how theseproperties can be illustrated using fan diagrams.
A fan diagram is a standard tool in econometrics for illustrating the predic-tions of a model. In Figure 6 we have plotted a fan diagram for a stock pricemodelled by geometric Brownian motion. The negative times in the plot showhistorical values for the stock price. For positive times, we plot a single randomrealization of geometric Brownian motion together with two percentiles thatindicate the range of values attained by other realizations. We have chosen toplot the percentiles Φ(1) ≈ 84% and Φ(−1) ≈ 16%.
Standard statistical properties of a distribution depend upon the coordinatesystem used for the distribution. For example the definition of the mean of aprocess in Rn involves the vector space structure of Rn. For this reason onewould expect the trajectory of the mean of a process to depend upon the vectorspace structure. If one makes a non-linear transformation of Rn the trajectoryof the mean changes. Indeed Ito’s lemma tells us that the trajectory does noteven remain constant to first order under coordinate changes.
Another manifestation of the same phenomenon is the fact that given anR valued random variable X and a non-linear function f , then E(f(X)) 6=f(E(X)). Again this arises because the definition of mean depends upon thevector space structure and f may not respect this.
However, the definition of the α-percentile depends only upon the orderingof R and not its vector space structure. As a result, for continuous monotonic fand X with connected state space, the median of f(X) is equal to f applied tothe median of X. If f is strictly increasing, the analogous result holds for theα percentile. If f is decreasing, the α percentile of f(X) is given by f appliedto the 1− α percentile of X.
This has the implication that the trajectory of the α-percentile of an R valued
22
Figure 6: A fan diagram for a stock price, S modelled using geometric Brownianmotion
stochastic process is invariant under smooth monotonic coordinate changes ofR. In other words, percentiles have a coordinate free interpretation. The meandoes not. This raises the question of how the trajectories of the percentiles canbe related to the coefficients of the stochastic differential equation. We will nowcalculate this relationship.
First we note that all smooth one dimensional Riemannian manifolds areisomorphic. When interpreted in terms of SDEs, this tells us that for any onedimensional SDE with non-vanishing volatility term:
dXt = a(Xt, t) dt+ b(Xt, t)dWt, X0 = x0 (19)
we can find a coordinate system with respect to which the volatility term isequal to one. Such a transformation is known as a Lamperti transformation(see for example [19]) and is given by Zt = φ(t,Xt) where φ(t, x) is a primitiveintegral of 1/b(x, t) with respect to x. Let
dZt = α(Zt, t)dt+ dWt, Z0 = z0 = φ(0, x0)
be the transformed equation. Since one dimensional Riemannian manifolds aretranslation invariant, we have a gauge freedom in defining a Lamperti transfor-mation determined by the base-point of the isomorphism. Define a path z0(t)by the ordinary differential equation:
dz0
dt= α(z0(t), t), z0(0) = z0.
If we set Yt = Zt − z0(t) then Y follows the SDE
dYt = a(Yt, t)dt+ dWt, a(y, t) = α(y + z0(t), t)− α(z0(t), t), Y0 = 0, (20)
23
and moreover the drift of the Y SDE vanishes at 0 for all t, a(0, t) = 0.Summing up, if b(x, t) is nowhere vanishing, there is a unique time depen-
dent transformation that maps the original SDE (19) for X into a SDE withconstant diffusion coefficient, zero initial condition and zero drift at zero. Ac-tually, sufficient conditions for the Lamperti transformed SDE (20) to have aunique strong solutions are more stringent. For example, in the autonomouscase a(x, t) = a(x) and b(x, t) = b(x) one may need a and b bounded from belowand above, with b and its bounds strictly positive, with a ∈ C1 and b ∈ C2.We will assume in the following that sufficient conditions for the solution of theLamperti transformed SDE to exist unique hold.
Thus given a one dimensional SDE with non-vanishing volatility we can al-ways make a transformation such that (subject to bounds) the following propo-sition applies (we denote a by a for simplicity).
Proposition 1. Let a(x, t) be a bounded smooth function on R× [0,+∞) witha(0, t) = 0. Define the operators L and its adjoint L∗ as
Lp =1
2
∂2p
∂x2+ a(x, t)
∂p
∂x− ∂p
∂t, (21)
L∗p =1
2
∂2p
∂x2− ∂(a(x, t)p)
∂x− ∂p
∂t. (22)
Assume further that a(x, t) has additional regularity required to ensure existenceand uniqueness of a fundamental solution for the PDEs Lp = 0 and L∗p = 0.Let Γ(x, t; ξ, τ) be the fundamental solution of Lp = 0. Then for λ ∈ (0, 1) anda fixed terminal time T , Γ satisfies:
Γ(x, t; 0, 0) =1√2πt
exp
(−x
2
2t
)+O
(t+ x2
√t
exp
(−λx
2
2t
))on R× [0, T ], and the fundamental solution of L∗p = 0 satisfies
Γ∗(0, 0;x, t) = Γ(x, t; 0, 0)
(see [10] for the definition of a fundamental solution.)
Proof. Equation L∗p = 0 as the evolution for the density of a solution of a SDEis studied for example in Friedman’s SDE book [11], see Eq. 5.28 in Chapter 6:by Theorem 4.7 in Ch 6 the fundamental solution of L∗p = 0 is the same as thefundamental solution of Lp = 0. We will thus focus on Lp = 0, for which onecan refer to Ch 1 Section 6 of Friedman’s parabolic PDEs book [10]. When a isidentically zero, the fundamental solution of Lp = 0 in (21) is given by:
Z(x, t; ξ, τ) =1√
2π(t− τ)exp
(− (x− ξ)2
2(t− τ)
).
By ([10] Ch 1, Theorem 10, p. 23), the fundamental solution of (21) is given by:
Γ(x, t; ξ, τ) = Z(x, t; ξ, τ) +
∫ t
τ
∫RZ(x, t; η, σ)Φ(η, σ; ξ, τ) dη dσ (23)
24
where
Φ(x, t; ξ, τ) =
∞∑i=1
Φi(x, t; ξ, τ)
where we define Φ1=LZ and for i ≥ 1 we define
Φi+1(x, t; ξ, τ) =
∫ t
τ
∫RL Z(x, t; y, σ)Φi(y, σ, ξ, τ) dy dσ.
Let us define:
Γi(x, t; ξ, τ) :=
∫ t
τ
∫RZ(x, t; η, θ)Φi(η, θ; ξ, τ) dη dθ.
First we bound Γ1.
Γ1(x, t; ξ, τ) =
∫ t
τ
∫RZ(x, t; η, σ)LZ(η, σ; ξ, τ) dη dσ
=
∫ t
τ
∫RZ(x, t; η, σ)a(η, σ)
∂
∂ηZ(η, σ; ξ, τ) dη dσ.
The final line follows because Z is the fundamental solution of (21) when a isequal to zero and so the parts of Γ1 that don’t involve a vanish. Our assumptionsensuring Lipschitz continuity of a(x, t) uniformly in t and the fact that a(0, t) = 0for all t (see (20)) tell us that there is some constant with |a(x, t)| < C|x| forall x and t ∈ [0, T ]. So∣∣∣∣a(η, σ)
∂
∂ηZ(η, σ; 0, 0)
∣∣∣∣ =
∣∣∣∣− 1√2πσ
a(η, σ)η exp
(− η
2
2σ
)∣∣∣∣<
C√2πσ
η2 exp
(− η
2
2σ
)We deduce that
|Γ1(x, t; 0, 0)| ≤ C∫ t
0
∫RZ(x, t; η, σ)η2 1√
2πσexp
(− η
2
2σ
)dη dσ
= Ce−
x2
2t
(t+ x2
)2√
2πt
The final step follows simply by the routine evaluation of the integral. To dothis we use estimates which were used in [10] to prove that our expression for Γexists and is a fundamental solution. First, we have for any λ ∈ (0, 1) that:
|Z(x, t, ξ, τ)| ≤ 1
(t− τ)12
exp
(−λ|x− ξ|
2
2(t− τ)
).
From ([10] Ch 1, 4.14, p. 16) we have that there exist positive constants H andH0 such that for all λ ∈ (0, 1) and m ≥ 2:
|Φm(x, t; ξ, τ)| ≤ H0Hm
ΓE(m)(t− τ)m−
32 exp
(−λ|x− ξ|
2
2(t− τ)
)
25
where ΓE is the gamma function. From ([10] Ch 1, Lemma 3 p. 15) we havethat if r < 3
2 and s < 32 then:
∫ τ
σ
∫R
(t− τ)−r exp
(−h(x− ξ)2
2(t− τ)
)(τ − σ)−s exp
(−h(x− ξ)2
2(t− σ)
)dξ dσ
=
√2π
hB
(3
2− r, 3
2− s)
(t− σ)32−r−s exp
(−h(x− y)2
2(t− σ)
)where B is the beta function. Taking r = 1/2 and s = 3
2 − m we obtain theestimate:
|Γm(x, t; ξ, τ)| ≤ H0Hm
ΓE(m)
√2π
λB(1,m)(t− τ)−
12 +m exp
(−λ(x− ξ)2
2(t− τ)
)=H0H
m
m!
√2π
λ(t− τ)−
12 +m exp
(−λ(x− ξ)2
2(t− τ)
) .
So
∞∑k=2
|Γm(x, t; 0, 0)| ≤∞∑m=2
H0Hm
m!t−
12 +m exp
(−λx
2
2t
)
=
√2π
λH0H
2 exp(Ht)(t)3/2 exp
(−λx
2
2t
)The result follows if we conclude by invoking Theorem 15, Ch 1, p. 28 of [10],or Theorem 4.7 in Ch 6 of [11].
Theorem 3. For sufficiently small t, the α-th percentile of the solutions to (19)is given by:
x0 + b0√tΦ−1(α) +
[a0 −
b0b′0
2(1− Φ−1(α)2)
]t+O(t3/2)
so long as the coefficients of (19) are smooth, the diffusion coefficient b nevervanishes, and sufficient conditions for the Lamperti transformed SDE and forL∗p = 0 to have a unique regular solution hold. In this formula a0 and b0 denotethe values of a(x0, 0) and b(x0, 0) respectively. In particular, the median process
is a straight line up to O(t32 ) with tangent given by the drift of the Stratonovich
version of the Ito SDE (19).The Φ(1) and Φ(−1) percentiles correspond up to
O(t32 ) to the curves γX0
(±√t) where γX0
is any representative of the 2-jet thatdefines the SDE in Ito form.
Proof. We first apply a Lamperti transformation so that the conditions ofProposition 1 apply. Let write y for the coordinates after applying the Lampertitransformation and let us write ρ for the 1-form representing the probabilitydensity. By Proposition 1
26
ρ =
[1√2πt
exp
(−y
2
2t
)+O
(t+ y2
√t
exp
(−λy
2
2t
))]dy
We introduce a new coordinate z = y√t
so that the density can be written:
ρ =
[1√2π
exp
(−z
2
2
)+O
(t(1 + z2) exp
(−λz
2
2
))]dz
Integrating this, we see that for sufficiently small ε we have a uniform estimate:∫ Φ−1(α+ε)
−∞ρ = α+ ε+O(t).
It follows that the α-th percentile in z coordinates is Φ−1(α) + O(t). Hence iny coordinates it is φ−1(α)
√t+O(t3/2)
Let g denote the inverse of the Lamperti transformation that we have taken.Let us write the Taylor series expansion for g around (0,0) as:
g(y, t) = x0 + gyy +1
2gyyy
2 + gtt+O(y2 + t)
By construction the SDE in y coordinates is of the form:
dYt = adt+ dWt
with a(0, t) = 0. When we apply g to this equation we will get equation (19).Using Ito’s lemma we deduce that:
b = gy
a0 = gt +1
2gyy
The first of these equations holds everywhere, the second uses the vanishing of aat 0. To avoid notational ambiguity in partial differentiation, we will temporarilywrite s for the time coordinate when paired with x and t for the time coordinatewhen paired with y. Thus we are making the coordinate transformation:
x = g(y, t)
s = t
We then have that:
∂b
∂x=∂b
∂y
∂y
∂x+∂b
∂s
∂s
∂x
= gyy1
gy
27
So we have that at 0:
gy = b0
gyy = b0b′0
gt =
(a0 −
1
2b0b′0
)Putting these formulae together with our power series expansion for the per-centile in y coordinates into the Taylor series expansion for g gives the desiredresult.
The theorem above has given us the median as a special case, and a linkbetween the median and the Stratonovich version of the SDE. By contrast themean process has tangent given by the drift of the Ito SDE as the Ito integralis a martingale.
For completeness, besides the mean and the median, we wish to consider themode. We claim that under the same conditions as the theorem above, thereare paths mu(t) and ml(t) both satisfying:
mu = x0 + a(x0, 0)t− 3
2b(x0, 0)b′(x0, 0)t+O(t
32 )
ml = x0 + a(x0, 0)t− 3
2b(x0, 0)b′(x0, 0)t+O(t
32 )
such that for sufficiently small t there exists a mode lying in [ml(t),mu(t)] Thisrelationship between the mean, median and mode is approximately seen in manygeneral probability distributions as was observed qualitatively by Pearson [20].
We can consider as an example the case of geometric Brownian motion, wherea(t, x) = µx and b(t, x) = σx with µ and σ deterministic constants, σ > 0. Forthis process the mean is easily computed from its lognormal distirbution asx0 exp(µt). To compute the median, we use Ito’s formula and write the SDEfor the logarithm, which is related to the Lamperti transform for this specificprocess,
d lnXt = (µ− σ2/2)dt+ σdWt, ln(x0).
Now this is a Gaussian process, so that its median is equal to its mean. Thisis easily calculated as lnx0 + (µ − σ2/2)t. Since the median transforms withincreasing transformations, the median of X is just the inverse of the Lampertitransform of the median of lnXt, leading to the exponential x0 exp((µ−σ2/2)t).Finally, for the lognormal distribution we know the mode to be x0 exp((µ −3σ2/2)t. Concluding, mode range, median and mean for the geometric Brownianmotion are sorted in increasing order and are given respectively by
x0 exp((µ− 3σ2/2)t, x0 exp((µ− σ2/2)t), x0 exp(µt).
A first order expansion of the exponentials for small t would give us formulasconsistent with our general result.
28
The geometric Brownian motion is a special case of our more general result.This result gives an alternative way of plotting the two jet that defines a onedimensional stochastic differential equation - at each point x ∈ R one plots thecurve (t, γx(±
√t)). Such a plot is shown for the visually convenient geometric
Brownian motion with vanishing drift µ = 0 in the left hand panel of Figure 7.One interprets this diagram as an infinitesimal fan-diagram showing the Φ(1)and Φ(−1) percentiles.
In the right hand panel of Figure 7 we show how this plot transforms whenone sends (t, St) to (t, log(St)). This is illustrated with solid lines. We also usedashed lines to plot the corresponding diagram for the equation arising fromIto’s lemma:
d(log(St)) = −1
2σ2dt+ σ dWt. (24)
Figure 7: A fan diagram of Geometric Brownian motion (left). The resultof applying (x, y) → (x, log y) to this fan diagram (right, solid line). A fandiagram of equation (24), the Brownian motion with drift associated with thelogS process (right, dashed line).
As predicted by our discussion so far, if one applies the logarithm to thevertical axis of the fan diagram for geometric Brownian motion one will approx-imately obtain the fan diagram for the equation (24). In the parlance of thegeometry of curves, the dashed line is the osculating curve for the solid line inthe family of fan diagrams of Brownian motions with drift.
It is interesting to notice that one can clearly see the drift term in the righthand side of Figure 7. Notice also that this drift arises because the spacingbetween the grid lines on the left hand side of figure 7 increases as one moves upthe page whereas the corresponding grid lines after transformation are evenlyspaced. This can be interpreted as a visual demonstration that the Ito term inthe transformation rule for SDEs is determined by the second derivative of thetransformation.
29
A Proof of convergence to classical Ito calculus
In this appendix we prove Theorem 1, stating convergence in L2(P) of the 2-jetscheme to the classical Ito solution of the SDE. The techniques used to provealmost-sure convergence of the classical Euler scheme ([12] for example) couldbe adapted to the 2-jet scheme.
Proof. We think of T as fixed and as N increases T N provides a finer discretiza-tion grid approximating [0, T ].
To remove clutter from our equations, during this proof we will adopt thefollowing conventions. C is a constant, independent of N that may change fromline to line. We drop the superscript N from terms such as XN
t . Sums overGreek indices always range from 1 to d. i, j and k are always integers. Termswith integer time subscripts such as Xi are shorthand for Xiδt. Superscript T ’sindicate the vector transpose rather than the terminal time.
Under our hypotheses, we know from [16], Theorem 10.2.2 p. 342 that theEuler scheme:
Xt+ε = Xt + a(Xt)ε+∑α
bα(Xt)(Wαt+ε −Wα
t ) for t ∈ T N , ε ∈ [0, δt]
converges to the solution of the Ito SDE in that:
supt∈[0,T ]
E{|Xt − Xt|2
}≤ C δt .
Our formula for the scheme incorporates interpolation in-between time steps,but the crude scheme is obtained by applying only time steps of size δt.
Since
supt∈[0,T ]
E{|Xt − Xt|2
}≤ supt∈[0,T ]
E{
2|Xt − Xt|2 + 2|Xt − Xt|2}
≤ 2 supt∈[0,T ]
E{|Xt − Xt|2
}+ 2 sup
t∈[0,T ]
E{|Xt − Xt|2
}all we need to conclude with (10) is to show:
supt∈[0,T ]
E{|Xt − Xt|2
}≤ C δt.
Summing the differences of consecutive terms in the Euler scheme with timestep δt we have:
Xk = x0 +
k∑i=1
[a(Xi−1)δt+
∑α
bα(Xi−1)δWαi−1
]where δWk := W(k+1)δt −Wkδt. Using the definition for a we can write this as:
Xk = x0 +
k∑i=1
∑α,β
aα,β(Xi−1)gαβE δt+∑α
bα(Xi−1)δWαi−1
(25)
30
where gE is the metric tensor of the Euclidean metric and where we define
aα,β(x) :=1
2
∂2γx∂uα∂uβ
∣∣∣u=0
, so that a =∑α,β
aα,βgα,β .
Note that we are not using the Einstein summation convention in this proof.Let us write the 2-jet scheme via its Taylor expansion as in (4), namely
Xk = Xk−1 +∑α,β
aα,β(Xk−1)δWαk−1δW
βk−1 +
∑α
bα(Xk−1)δWαk−1 +Rk. (26)
This expression defines the remainder Rk. Note that the remainder dependsalso on N but our notation has suppressed this. Summing the differences ofconsecutive terms, and substituting in the definition of b we obtain:
Xk = x0+
k∑i=1
∑α,β
aα,β(Xi−1)δWαi−1δW
βi−1 +
∑α
bα(Xi−1)δWαi−1 +Ri
. (27)
Subtracting equation (25) from (27) we obtain:
Xk − Xk = Pk +∑α
Qαk +∑α,β
Sα,βk
where we define:
Pk :=
k∑i=1
Ri
Qαk :=
k∑i=1
[b(Xi−1)− b(Xi−1)
]δWα
i−1
Sα,βk :=
k∑i=1
[aα,β(Xi−1)δWα
i−1δWβi−1 − aα,β(Xi−1)gαβE δt
]We have that:
E{|Xk − Xk|2
}≤ C
E {|Pk|2}+∑α
E{|Qαk |2
}+∑α,β
E{|Sα,βk |
2}.
We now wish to obtain bounds for each expectation on the right in terms of δtand the function:
Z(t) = sup0≤s≤t
E{|Xs − Xs|2
}.
Using our bound on the third derivatives of γ, we can bound the remainderterms Ri as follows:
|Ri| ≤ C
(∑α
|δWαi−1|
)3
.
31
Let us write:
M := E
(∑
α
|δWαi−1|
)6 .
We calculate that:
E(|Pk|2) = E{|∑i
Ri|2} ≤ CN2M
≤ C(T
δt
)2
(δt)62
≤ Cδt.
Next we have:
E{|Qαk |2
}≤ E
∣∣∣∣∣k∑i=1
[b(Xi−1)− b(Xi−1)]δWαi−1
∣∣∣∣∣2
≤ CE
∣∣∣∣∣k∑i=1
|Xi−1 − Xi−1|δWαi−1
∣∣∣∣∣2
= CE
{k∑i=1
|Xi−1 − Xi−1|2δt
}
≤ C∫ kδt
0
Z(s) ds.
The second line follows from the first by the Lipschitz condition on the deriva-tives. The third line follows by the discrete Ito isometry or the tower propertyof conditional expectation coupled with independence of Brownian increments.
To bound Sα,β we write it as a sum of two components Sα,β,1 and Sα,β,2
defined as follows:
Sα,β,1k :=
k∑i=1
aα,β(Xi−1)(δWαi−1δW
βi−1 − g
α,βE δt)
Sα,β,2k :=
k∑i=1
[aα,β(Xi−1)− aα,β(Xi−1)]δWαi−1δW
βi−1
By expanding the square term using the definition of Sα,β,1 we find:
E{|Sα,β,1k |2
}=
k∑i=1
k∑j=1
E{aTα,β(Xi−1)aα,β(Xj−1)(δWα
i−1δWβi−1 − g
α,βE δt)(δWα
j−1δWβj−1 − g
α,βE δt)
}
32
When i 6= j the terms on the right hand side vanish. This is because we mayassume WLOG that j > i in which case the last factor of the (i, j)-th term,
δWαj−1δW
βj−1 − g
α,βE δt
is independent of the rest of the term and has expectation 0. We now quote10.2.14 p. 343 in [16] to show:
E{|Xk|2
}≤ C}. (28)
We now compute:
E{|Sα,β,1k |2
}=
k∑i=1
E{|aα,β(Xi−1)|2(δWα
i−1δWβi−1 − g
α,βE δt)2
}≤
k∑i=1
E{|aα,β(Xi−1)|2
}E{
(δWαi−1δW
βi−1 − g
α,βE δt)2
}≤
k∑i=1
C(δt)2E{|aα,β(Xi−1)|2
}≤
k∑i=1
C(δt)2E{
(1 + X2i−1)
}≤
k∑i=1
C(δt)2
≤ Cδt
The second line follows from independence of Brownian motion increments. Thethird from the second by the scaling properties of Brownian motion increments.The fourth from our growth bounds on the second derivatives. The fifth fromthe bound (28).
33
We now compute a bound for the Sα,β,2k term.
E{|Sα,β,2k |2
∣∣∣ ≤ 2
k−1∑i=0
i∑j=0
E{|(aα,β(Xi)− aα,β(Xi))
T (aα,β(Xj)− aα,β(Xj))|
×|δWαi δW
βi δW
αj δW
βj |}
≤ Ck−1∑i=0
i∑j=0
(E{|(aα,β(Xi)− aα,β(Xi))
T (aα,β(Xj)− aα,β(Xj))∣∣}
×E{|δWα
i δWβi δW
αj δW
βj |})
≤ C(δt)2k−1∑i=0
i∑j=0
E{|(aα,β(Xi)− aα,β(Xi))
T
×(aα,β(Xj)− aα,β(Xj))|}
≤ C(δt)2k−1∑i=0
i∑j=0
(E{|(aα,β(Xi)− aα,β(Xi))|2
}+E
{|(aα,β(Xj)− aα,β(Xj))|2
})We have used independence of Brownian motion increments to get from thefirst line to the second. Next we have used the scaling behaviour of Brownianincrements. Next we used Young’s Inequality. Applying now the Lipschitzproperty of aα,β we find:
E{|Sα,β,2k |2
∣∣∣ ≤ C(δt)2k−1∑i=0
i∑j=0
(E{|Xi − Xi|2
}+ E
{|Xj − Xj |2
})≤ Cδt
k−1∑i=0
Z(iδt)
≤ C∫ kδt
0
Z(t) dt
where we took into account the fact that i ≤ N = T/δt, so that i(δt)2 ≤ Cδt.Putting together all our bounds we have:
Z(t) ≤ C(δt+
∫ t
0
Z(s) ds
).
So by the Groenwall inequality, Z(t) ≤ Cδt.
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