Existence and Uniqueness for Stochastic Dynamic Equations

The theory of stochastic dynamic equations extends and unifies the theories of stochastic difference and differential equations. In this paper, we prove the existence and uniqueness of the strong solution of a certain class of stochastic dynamic equations. As a principal tool in the proof, we define and develop the properties of stochastic dynamic integrals with respect to a Brownian motion indexed by a time scale. Finally we illustrate our theory with the examples of stochastic exponential and geometric Brownian motion.


Introduction
The theory of stochastic dynamic equations (SΔE) is a relatively new area of mathematics that extends and unifies the theory of stochastic difference and differential equations.The basic idea is to replace the time domain of the underlying stochastic process with a time scale, T; that is T is an arbitrary nonempty closed subset of the reals R. It is evident that when T = Z we have the theory of stochastic difference equations and when T = R we have the theory of stochastic differential equations.The theory of SΔE was first investigated in (Sanyal, 2008).Subsequently various authors have contributed to this endeavor.Notable contributions include the construction of Brownian motion in (Grow & Sanyal, 2011), the quadratic variation of Brownian motion in (Grow & Sanyal, 2012), the construction of the stochastic dynamic exponential and the explicit solution of geometric Brownian motion in (Bohner & Sanyal, 2010), the existence and uniqueness of random dynamical systems in (Lugan & Lupulescu, 2012), and in the special case of q-calculus time scale T = q Z + = {1, q, q 2 , q 3 , . ..} for q > 1, Itô's lemma and a financial option pricing application in (Haven, 2009(Haven, , 2011)).
In this paper we provide conditions which guarantee the existence and uniqueness of the strong solution of a certain class of stochastic dynamic initial value problems (IVPs) of the form ΔX = a(X, t)Δt + b(X, t)ΔW, X t 0 = X 0 . (1) By a strong solution of (1) on (Ω, F , P) and w.r.t. a fixed Brownian motion W and a fixed initial process X 0 , we mean a stochastic process indexed by a time scale, X = {X s : s ∈ T}, with sample paths that are continuous and with the following properties: 1) X is adapted to the filtration {F s : s ∈ T}, and the integral version of (1) holds almost surely.We define Brownian motion indexed by T as an adapted, continuous stochastic process W = {W t , F t : t ∈ T}, defined on (Ω, F , P) such that 1) W t 0 = 0 a.e.-P; 2) if t 0 ≤ s < t and s, t ∈ T then the increment W t − W s is normally distributed with mean 0 and variance t − s and is independent of F s .
For a generalization of the Kolmogorov-Čentsov theorem and the construction of such a generalized Brownian motion indexed by T, please refer to (Grow & Sanyal, 2011).Here and in the rest of this paper T t denotes [t 0 , t] ∩ T for some t ∈ T. In (2), the first integral is a Δ-integral defined in (Bohner & Peterson, 2001) and the validation for the second integral is provided in Section 3.
The paper is organized as follows.In Section 2 we provide basic definitions regarding time scales, in Section 3 we define the stochastic integral w.r.t. a Brownian motion indexed by a time scale and develop some its properties, and finally in Section 4 we state and prove an existence and uniqueness result concerning (1).

Preliminaries
A time scale, T, is primarily used to unify and extend discrete and continuous analysis (Bohner & Peterson, 2001, 2003).Examples of time scales include R, Z, q N 0 for q > 1, hZ for h > 0, [0, 1] ∪ [2, 3], the Cantor set, etc.Given a time scale, T, we define ρ(t) = sup{s ∈ T: s < t} as the backward shift operator ρ on T κ , σ(t) = inf{s ∈ T: s > t} as the forward shift operator σ on T κ , and μ(t) = σ(t) − t as the graininess function depicting the gaps between two points in a time scale, where the sets T κ and T κ are defined as T κ := T\{sup T} and T κ := T\{inf T}.For the definitions of (i) right dense point, (ii) the set of right dense continuous functions (C rd ), (iii) the set of all regressive functions (R), (iv) the set of positively regressive functions (R + ), (v) Hilger derivative (or Δ-derivative), (vi) Δ-anti-derivative, and (vi) Δ-integral, we refer the readers to (Bohner & Peterson, 2001).For t 0 ∈ T, y 0 ∈ R, and p ∈ R, the IVP has a unique solution given by y = e p (•, t 0 )y 0 .The generalized polynomials g k , h k : T × T → R, k ∈ N 0 are defined as follows.The functions g 0 and h 0 are g 0 (t, s) = h 0 (t, s) ≡ 1 for all s, t ∈ T, and for k for t ∈ T κ with h k+1 (s, s) = 0 where h Δ k (t, s) denotes for each fixed s the Δ-derivative of h k (t, s) w.r.t.t.Similarly g Δ k+1 (t, s) = g k (σ(t), s) for t ∈ T κ with g k+1 (s, s) = 0. Thence, it can be shown that for constant k ∈ R and t 0 , t ∈ T. To verify (4) we need the following lemma.
Lemma 1 Let s, t ∈ T with s ≤ t and h defined as in (3).Then for n ∈ N 0 .
To begin the verification of (4), let y N (t) From Lemma 1, it follows that N=1 converges uniformly on T T and the limit function Clearly k n h n (t, t 0 ) is a solution to the IVP y Δ = ky, y(t 0 ) = 1 on T. By uniqueness of solutions to this IVP, y(t) = e k (t, t 0 ) on T, i.e. (4) holds for all t ∈ T. Finally we present the following lemma (an analogue of Gronwall's inequality) without proof.
for all t ∈ T T .
In the next section we present the construction of the stochastic dynamic integral.

Stochastic Dynamic Integral
Let inf T > −∞ and sup T = ∞ and let ψ be a real, increasing, and right continuous function on T. Lebesgue-Stieltjes delta integrals w.r.t.ψ on intervals in T can be defined in terms of Lebesgue-Stieljes intervals on intervals in R in a manner analogous to the Lebesgue delta integral w.r.t. the forward shift function of T (Bohner & Peterson, 2003;Guseinov, 2003).Explicitly, define Ψ on R by with +∞ as a possible value of this integral.
For the construction of the stochastic dynamic integral T t X u ΔW u , we parallel the construction of the stochastic integral w.r.t. a continuous square integrable martingale on [0, ∞) as presented in (Karatzas & Shreve, 1998).Let W = {W t , F t : t ∈ T} be a Brownian motion indexed by T defined on (Ω, F , P).In order to avoid possible measure theoretic pathologies, we can and do assume that F t 0 contains all subsets of P-measure zero in F and the filtration for all t ∈ T. By a partition of the interval T t we mean a finite subset P : Definition 2 X be a stochastic process indexed by T with inf T > −∞.Then the quadratic variation of X over the partition P of the time scale T t is defined as If X 2 t 0 ,t (P) converges in probability sufficiently rapidly as mesh(P) → 0 then the limit, X t , is called the quadratic variation of X on T t .
In (Grow & Sanyal, 2012), it has been shown that where λ denotes the Lebesgue measure and ∪∞ Then P-a.e. ω ∈ Ω, the function ω → W t (ω) is real, increasing, and right continuous on T. Consequently, we may define a measure μ W on (T × Ω, B(T) ⊗ F ) by We will say that two measurable, adapted processes for t ∈ T defines an L 2 -norm for X regarded as a function of (s, ω) restricted to the space T T × Ω under the measure μ W , and [X − Y] T = 0 for all T ∈ T if and only if X and Y are equivalent.
Definition 3 A stochastic process X = {X t : t ∈ T} on (Ω, F ) is called progressively measurable w.r.t. the filtration Let L * denote the set of equivalent classes of stochastic processes X which are progressively measurable w.r.t.{F t : t ∈ T} and for which [X] T < ∞ for all T ∈ T. Following the standard convention in measure and integration, we will not distinguish between an equivalence class in L * and a representative process X from that class.Choose and fix a strictly increasing sequence {t n } ∞ n=1 of points from T tending to infinity and define a metric on L * by Let L * T denote the set of processes X in L * for which X t (ω) = 0 for all t ∈ [T, ∞) ∩ T and ω ∈ Ω. Define L * ∞ as the class of processes X ∈ L * for which . Definition 4 A process X indexed by T is called simple if there exists a strictly increasing sequence {s n } ∞ n=0 of points from T, tending to infinity with s 0 = t 0 , as well as a sequence of random variables {ξ n } ∞ n=0 and a constant for all t ∈ T and ω ∈ Ω.
Denote by L 0 the class of all simple processes and observe that L 0 ⊆ L * since the processes in L 0 are progressively measurable and bounded.A standard construction shows that the set L 0 of simple processes is dense in Definition 5 The Brownian motion transform of (10) in L 0 is a progressively measurable stochastic process given by Here n ≥ 0 is the unique integer for which s n ≤ t < s n+1 .
It is apparent from Definition 5 that the Brownian motion transform is linear, i.e.

I(αX
for all α, β ∈ R and all X, Y ∈ L 0 .Furthermore, if X ∈ L 0 then I(X) is a right continuous, square integrable martingale indexed by T with quadratic variation for all t ∈ T.

Definition 6
The space M 2 of all right continuous, square integrable martingales indexed by T consists of those adapted processes X = {X t , F t : t ∈ T} such that 1) X t 0 = 0 a.e.-P; 2) for P-a.e ω ∈ Ω, the function t → X t (ω) is right continuous on T; 4) the conditional expectations of X satisfy E (X t |F s ) = X s for all s and t in T such that t 0 ≤ s ≤ t.
For any X ∈ M 2 , if we define , almost all of whose sample paths t → X t (ω) and t → Y t (ω) are identical on T, then ||X − Y|| is a complete metric on the equivalence classes of M 2 .As usual, we will blur the distinction between an equivalence class in M 2 and a representative process X from that class.Observe that for X ∈ L 0 we have That is, the Brownian motion transform is a linear isometry from the space L 0 of simple processes into the space M 2 of right continuous, square integrable martingales.Moreover, L 0 is a dense subspace of the square integrable progressively measurable processes L * equipped with the metric d(X, Y) = [X − Y], and M 2 equipped with the metric D(U, V) = ||U − V|| is complete.Consequently, the Brownian motion transform extends in the usual manner to an isometry from L * into M 2 .
Definition 7 The stochastic dynamic integral of X ∈ L * w.r.t. a Brownian motion W indexed by T is the unique right continuous, square integrable martingale indexed by T, I(X) = {I t (X), F t : t ∈ T}, satisfying ||I(X (n) ) − I(X)|| → 0 for every sequence We denote the stochastic dynamic integral of X w.r.t.W by I t (X) = T t X u ΔW u for all t ∈ T.
Theorem 1 Let X and Y belong to L * , let s, t ∈ T with t 0 ≤ s < t, and let α, β ∈ R. Then the stochastic dynamic integral given in Definition 7 has the following properties.

1)
2) 3) Proof.Properties 1, 2, and 3 follow from Definition 7.For property 4, note that and from (8) we have Finally property 5 follows from ( 11) and property 4. ).By the definition of right-continuity of the filtration F t given in (7) note that F 1 F 2 .Let us take a simple process where a, b ∈ R.
Since by the definition of W given in Section 1, W 2 − W 1 is F 1 measurable and independent of F 1 , we observe that and hence In the next section we present the existence and uniqueness of stochastic dynamic equation.

Existence and Uniqueness
With the theory of stochastic dynamic integral established, the second integral in (2) has meaning.Consequently, we are able to rigorously cast the IVP (1) in the integral form (2). We now consider the problem of the existence and uniqueness of solutions of (1).
Theorem 2 Let T be a time scale and suppose a, b: for all t ∈ T T , all x, x 1 , x 2 ∈ R and for some constant L. Let X 0 be any real-valued random variable such that Then there exists a unique strong solution X of the stochastic dynamic IVP on T T .Moreover, there exists a constant k T ∈ R such that for t ∈ T T .
Proof.Let us suppose that X and X are solutions of ( 14).Then for t 0 , t, T ∈ T, Thus X t = Xt a.s.for all t ∈ T T .
To prove existence, let us define X 0 t := X 0 and for n ∈ N 0 and t ∈ T T .We claim that for all n ∈ N 0 , and t ∈ T T , where h n are the generalized polynomials defined by (3) and M a constant that depends on L, T , and X 0 .Indeed for n = 0, we have Under the assumption that the claim is valid for n − 1, we have ) 2 and this proves the claim.Now using ( 12) and( 16) we have Consequently the martingale inequality implies by claim (17), where C = 2L 2 (T − t 0 + 4).Since and by ( 4) with 4M ∈ R, the Borel-Cantelli lemma applies.Thus, P sup 16), we have That is ( 14) holds for all time t ∈ T T .Then we have for some constant k T ≥ max 2, 4L 2 (T − t 0 )(T − t 0 + 1), 4L 2 (T − t 0 + 1) .Note that k T > 1.By induction, therefore, for all t ∈ T T , which proves (15).
Definition 8 For a function f : T → R, let us define R W as a space of all stochastic regressive function (w.r.t.W) such that for all t ∈ T 1 + f (t) W σ(t) − W(t) 0 a.s.( 18) For T = R, since σ(t) = t, we have 1 0 which is trivially true.Likewise for T = Z, (18) reduces to 1 + f (t) (W t+1 − W t ) 0 a.s.for t ∈ Z.
Example 2 (Stochastic exponential) Consider the stochastic dynamic IVP (14) with a(X, t) ≡ 0 and b(X, t) = β(t)X for some β ∈ R W . Then according to Theorem 2, the solution of exists and is unique.For an isolated time scale T, i.e. μ(t) > 0 for all t ∈ T, it has been shown in (Bohner & Sanyal, 2010) that ( 19) with X 0 = 1 has a solution X = E β (•, t 0 ), with the following properties: E E β (t, t 0 ) = 1 and E E 2 β (t, t 0 ) = e β 2 (t, t 0 ).( 20) Now by virtue of Theorem 2 it is clear that for an arbitrary T the solution of (19) exists, is unique, and has properties given by (20).

Concluding Remarks
In this paper, we construct the stochastic dynamic integral and the existence and uniqueness of a certain class of stochastic dynamic equation (SΔE) on a general time scale T. We also present examples concerning generalization of stochastic exponential and geometric Brownian motion which are used in the model of stock price behavior and to model stock prices in the celebrated Black-Scholes model.
increasing, and right continuous of R such that ν(c, d] = Ψ(d) − Ψ(c) for all intervals in R of the form (c, d].If φ is a Borel measurable function on an interval I such that I |φ(x)|dν < ∞ then we say that φ is Lebesgue-Stieltjes integrable w.r.t.Ψ on I. Definition 1 Let a < b be points in T, let ψ be a real, increasing, and right continuous function on [a, b] ∩ T, and let φ be a Borel measurable function which is Lebesgue-Stieltjes integrable w.r.t.Ψ on [a, b].Then the Lebesgue-Stieltjes delta integral of φ w.r.t.ψ on the interval [a, b] ∩ T is defined by b