Nonlinear Techniques for Stochastic Systems of Differential Equations

Transcription

1 University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School January 2013 Nonlinear Techniques for Stochastic Systems of Differential Equations Tadesse G. Zerihun University of South Florida, Follow this and additional works at: Part of the Applied Mathematics Commons, and the Mathematics Commons Scholar Commons Citation Zerihun, Tadesse G., "Nonlinear Techniques for Stochastic Systems of Differential Equations" (2013. Graduate Theses and Dissertations. This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact

2 Nonlinear Techniques for Stochastic Systems of Differential Equations by Tadesse G. Zerihun A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics and Statistics College of Arts and Sciences University of South Florida Major Professor: Gangaram S. Ladde, Ph.D. Marcus McWaters, Ph.D. Manoug Manougian, Ph.D. Kandethody M. Ramachandran, Ph.D. Yuncheng You, Ph.D. Date of Approval: November 22, 2013 Keywords: Generalized Variation of Constants Formula, Nonlinear Systems of Stochastic Differential Equations, Energy Function Method, Dynamics of Photosynthetic Process. Copyright c 2013, Tadesse G. Zerihun

3 Acknowledgments My greatest thanks goes to my major professor Gangaram S. Ladde, Ph.D, for the unlimited guidance and follow up he has making both in the course work and in doing this dissertation. Without his guidance and persistence help, this dissertation is impossible. I also like to thankful to my dissertation committee members, Marcus McWaters, Manoug Manougian, Kandethody M. Ramachandran and Yuncheng You. My thanks also goes to Getachew Dagne, chairperson of defense. Moreover, I would like to thank my families, friends; specially Engineer Alemayehu Yemanebrhan, and my sweet wife for their unfolding moral support and help in any aspect till the end. Finally, I would like to thank Department of Mathematics and Statistics, University of South Florida and Mathematical Sciences Division, US Army Office, Grant Number W911NF for their financial support.

4 Table of Contents List of Figures Abstract iii iv 1 Preliminary Concepts and Results Basic Properties of Stochastic Differential Equations Nominal System and Error Estimate Comparison System and Qualitative Properties 5 2 Fundamental Properties of Solutions of Nonlinear Stochastic Differential Equations and Method of Variation of Parameters Introduction Problem Formulation Auxiliary Results Method of Variation of Constants Formula Examples 38 3 Method of Generalized Variation of Constants Formula: Relative Stability Introduction Energy Function Method and Generalized Variation of Constants Formula Robustness of Solution Process Stability Analysis 67 i

5 3.5 Relative Stability 74 4 Variational Comparison Method: Relative Stability Introduction Energy Function Method and Variational Comparison Theorems Comparison Theorems Robustness of Solution Process Stability Analysis Error Estimate and Relative Stability Stochastic Dynamic Model for Photosynthesis Introduction A Dynamic Model for Photosynthesis Light Dependent Reactions Dark (Light Independent Reactions Enzyme Reactions Normalizing and Equilibrium States of Deterministic Model Stochastic Dynamical model for photosynthesis Numerical Illustration 131 References 135 ii

6 List of Figures 5.1 NADPH concentration as a function of time(min PGA concentration as a function of time(min Carbon-dioxide concentration as a function of time(min The product p as a function of time(min Concentration of the complex s e as a function of time(min Substrate concentration s as a function of time(min 134 iii

7 Abstract Two of the most well-known nonlinear methods for investigating nonlinear dynamic processes in sciences and engineering are nonlinear variation of constants parameters and comparison method. Knowing the existence of solution process, these methods provide a very powerful tools for investigating variety of problems, for example, qualitative and quantitative properties of solutions, finding error estimates between solution processes of stochastic system and the corresponding nominal system, and inputs for the designing engineering and industrial problems. The aim of this work is to systematically develop mathematical tools to undertake the mathematical frame-work to investigate a complex nonlinear nonstationary stochastic systems of differential equations. A complex nonlinear nonstationary stochastic system of differential equations are decomposed into nonlinear systems of stochastic perturbed and unperturbed differential equations. Using this type of decomposition, the fundamental properties of solutions of nonlinear stochastic unperturbed systems of differential equations are investigated(1. The fundamental properties are used to find the representation of solution process of nonlinear stochastic complex perturbed system in terms of solution process of nonlinear stochastic unperturbed system(2. Employing energy function method and the fundamental properties of Itô-Doob type stochastic auxiliary system of differential equations, we establish generalized variation of constants formula for solution process of perturbed stochastic system of differential equations(3. Results regarding deviation of solution of perturbed system with respect to solution of nominal system of stochastic differential equations are developed(4. The obtained results are used to study the qualitative properties of perturbed stochasiv

8 tic system of differential equations(5. Examples are given to illustrate the usefulness of the results. Employing energy function method and the fundamental properties of Itô-Doob type stochastic auxiliary system of differential equations, we establish generalized variational comparison theorems in the context of stochastic and deterministic differential for solution processes of perturbed stochastic system of differential equations(6. Results regarding deviation of solutions with respect to nominal stochastic system are also developed(7. The obtained results are used to study the qualitative properties of perturbed stochastic system(8. Examples are given to illustrate the usefulness of the results. A simple dynamical model of the effect of radiant flux density and CO 2 concentration on the rate of photosynthesis in light, dark and enzyme reactions are analyzed(9. The coupled system of dynamic equations are solved numerically for some values of rate constant and radiant flux density. We used Matlab to solve the system numerically. Moreover, with the assumption that dynamic model of CO 2 concentration is studied. v

9 1 Preliminary Concepts and Results This chapter deals with a basic existing preliminary concepts and tools needed to undertake the study of nonlinear techniques for stochastic systems of differential equations. Two of the most well known nonlinear methods for investigating nonlinear dynamic processes in engineering and sciences, are nonlinear variation of constant parameters and comparison methods[1-4,6-27,30]. Moreover, knowing the existence of solution process, these methods provide very powerful tools for investigating qualitative properties of solution process[14, 15, 16, 19, 36]. Moreover, the qualitative properties are also used for designing plants. In this chapter, definitions and some results are outlined. 1.1 Basic Properties of Stochastic Differential Equations Let us consider a mathematical description of a nonlinear phenomenon under a random environmental perturbation described by a complex system of nonlinear nonstationary Itô-Doob-type stochastic differential equations: dx = f(t, xdt + σ(t, xdw(t, x( = x 0, (1.1.1 where x R n ; f and column vectors of σ C[J R n, R n ]; C[J R n, R n ] stands for the class of continuous functions defined on J R n into R n for a positive integer n, and J = [, + a for some positive real number a > 0; σ = [σ 1, σ 2,...σ j..., σ m ] is n m matrix; x 0 is an n-dimensional random variable defined on a complete prob- 1

10 ability space (Ω, F, P ; F t is an increasing family of sub-σ-algebras of F; w(t = (w 1 (t, w 2 (t,..., w m (t T is an m-dimensional normalized Wiener process with independent increments; x 0 and w(t are mutually independent for each t. Definition The random process x(t is said to be a solution of (1.1.1 on J if it satisfies the following conditions: 1. denoting by F t, t J the minimal σ-algebra with respect to which the variables x(s for s t and w(s for s t are measurable, the process w t (s = w(t + s w(t does not depend on F t ; 2. denoting by H 2 [J] the space of measurable random functions ϕ(t which, for each t J are F t -measurable and for which the integral +a ϕ 2 (tdt is w.p.1 finite, f(t, x(t 1/2 and σ(t, x(t belong to H 2 [J]; 3. the process x(t has on J the stochastic differential dx(t = f(tdt + σ(tdw(t, also, for all t J we have w.p.1 f = f(t, x(t, σ(t = σ(t, x(t. Let x(t = x(t,, x 0 be the solution process of (1.1.1 existing for t. Let us modify the stochastic differential equation (1.1.1 as: dx = f(t, x, λdt + σ(t, x, λdw(t, x(, λ = x 0 (λ, (1.1.2 and the corresponding nominal system, dx = f(t, x, λ 0 dt + σ(t, x, λ 0 dw(t, x( = x 0, (1.1.3 where (, x 0, λ 0 J R n Λ, with Λ being an open system parameter λ set in R m. In the following, we present a very simple result that exhibits the continuous dependence of solution process of (1.1.2 with respect to (, x 0, λ 0 J R n Λ. The proof is given in [15, 17, 19, 20, 21]. 2

11 Theorem Assume that 1. f and the m column vectors of σ C[J R n Λ, R n ]; 2. There exist positive number M and N such that f(t, x, λ + σ(t, x, λ N + M X for (t, x, λ J R n Λ; 3. There exist a positive number L such that f(t, x, λ f(t, y, λ + σ(t, x, λ σ(t, y, λ L x y (t, x, λ, (t, y, λ J R n Λ; 4. x(, λ = x 0 (λ is independent of w(t and lim E[ x(, λ x(, λ 0 2 ] = 0; λ λ 0 5. E[ x(, λ 4 c 1 for some constant c 1 > 0; 6. ɛ > 0, p > 0, lim P [sup x <p ( f(t, x, λ f(t, x, λ 0 + σ(t, x, λ σ(t, x, λ 0 > ɛ] = 0 λ λ 0 Then the IVP (1.1.2 admit unique solution processes x(t, t 1, x 0 (λ, λ and x(t,, x 0, λ 0 through (t 1, x 0 (λ and (, x 0, respectively. Moreover, for given ɛ > 0, there exists a δ(ɛ > 0 such that (E[ x(t, t 1, x 0 (λ, λ x(t,, x 0, λ 0 2 ] 1/2 < ɛ, t J, 3

12 whenever t 1 + (E[ x 0 (λ x 0 2 ] 1/2 + λ λ 0 < δ(ɛ. Theorem Assume that σ, f, x 0 in (1.1.1 satisfy the hypotheses (1,(2,(4, and (5 of Theorem Furthermore, σ and f are continuously differentiable with respect to x for fixed t. Let x(t,, x 0 be the solution process of (1.1.1 existing for t. Then, Φ(t,, x 0 = x x(t,, x 0 exists and is the solution of dy = H(t,, x 0 ydt + Γ(t,, x 0 ydw(t where Φ(,, x 0 is the n n identify matrix, n n matrices f x (t, x and σx(t, l x are continuous in (t, x for l = 1, 2,..., m; σ x (t, x is the n nm matrix σ x (t, x=[σx(t, 1 xσx(t, 2 x...σx(t, j x...σx m (t, x]; H(t,, x 0 = f x (t, x(t,, x 0 and Γ(t,, x 0 =σ x (t, x(t,, x 0. The proof is given in [15, 17, 19, 20, 21]. Definition The trivial solution process of (1.1.1 is said to be i (SM 1 stable in the p-th moment, if for each ɛ > 0, R + and p 1 there exist a positive function δ = δ(, ɛ such that the inequality x 0 p δ implies x(t p < ɛ, t, where x(t p = (E[ x(t p ] 1/p ; ii (SM 2 asymptotically stable in the p-th moment, if it is stable in the p-th moment and if for any ɛ > 0, R +, there exists δ 0 = δ 0 ( and T = T (, ɛ such that 4

13 the inequality x 0 p δ 0 implies x(t p < ɛ, t + T. 1.2 Nominal System and Error Estimate Let us consider a nominal system of Itô-Doob type stochastic differential equations dy = G(t, ydt + H(t, ydw(t, y( = y 0. (1.2.1 where G C[J R n, R n ] and H C[J R n, R n m ]. Definition The two differential systems (1.1.1 and (1.2.1 are said to be i (RM 1 relatively stable in p-th moment, if for each ɛ > 0, R +, and p 1, there exists a positive function δ = δ(, ɛ such that the inequality x 0 y 0 p δ implies x(t y(t p < ɛ, t ; ii (RM 2 relatively asymptotically stable in the p-th moment, if it is relatively stable in the p-th moment and if for any ɛ > 0, R +, there exist δ 0 = δ 0 ( and T = T (, ɛ such that the inequality x 0 y 0 δ 0 implies x(t y(t p < ɛ, t + T. 1.3 Comparison System and Qualitative Properties Let us consider the following Itô-Doob type stochastic comparison and auxiliary systems differential equations du = g(t, udt, u( = u 0, (

14 and dz = α(t, zdt, z( = x 0, (1.3.2 where g C[R + R N, R N ] and α C[J R n, R n ] Definition The function g(t, u is said to possess a quasi-monotone nondecreasing property if for u, v R N such that u v and u i = v i, then g i (t, u g i (t, v for any i = 1, 2,..., N and fixed t. Let V C[R + R n, R N ] and its partial derivatives V t, V x and V xx exists and are continuous on R + R n. Definition The trivial solution processes z 0 and u 0 of (1.2.1 and (1.3.1 are said to be i (JM 1 jointly stable in the mean, if for ɛ > 0, R +, there exists δ 1 = δ 1 (, ɛ > 0 such that N i=1 E[V i(, x 0 ] δ 1 implies N E[u i (t,, V (, z(t,, x 0 ] < ɛ, t ; i=1 ii (JM 2 jointly asymptotically stable in the mean, if it is jointly stable in the mean and if for any ɛ > 0, R +, there exist δ 0 = δ 0 ( > 0 and T = T (, ɛ > 0 such that the inequality N i=1 E[V i(, x 0 ] δ 0 implies N E[u i (t,, V (, z(t,, x 0 ] < ɛ, t + T. i=1 Definition The systems (1.2.1 and (1.3.1 are said to be i (JR 1 jointly relatively stable in the mean, if for each ɛ > 0, R +, there exists δ 1 = δ 1 (, ɛ > 0 such that the inequality N i=1 E[V i(, x 0 y 0 ] δ 1 implies N E[u i (t,, V (, z(t,, x 0 y 0 ] < ɛ, t ; i=1 6

15 whenever y 0 is small enough. ii (JR 2 jointly relatively asymptotically stable in the mean, if it is jointly relatively stable in the mean and if for each ɛ > 0, R +, there exists δ 0 = δ 0 ( > 0 and T = T (, ɛ > 0 such that N i=1 E[V i(, x 0 y 0 ] < δ 0 implies N E[u i (t,, V (, z(t,, x 0 y 0 ] < ɛ, t. i=1 Definition The differential system (1.1.1 has asymptotic equilibrium if every solution of the system (1.1.1 tends to a finite limit vector ξ as t and to every constant vector ξ there is a solution x(t of (1.1.1 on t < such that lim t x(t = ξ. Definition The differential systems (1.1.1 and (1.2.1 are said to be asymptotically equivalent if, for every solution y(t of (1.2.1, there is a solution x(t of (1.1.1 such that x(t y(t 0 as t. Theorem If φ is a real-valued continuous and concave function defined on a convex domain D R n, then E[φ(x] φ(e(x. The proof is given in [17, 19, 20]. 7

16 2 Fundamental Properties of Solutions of Nonlinear Stochastic Differential Equations and Method of Variation of Parameters 2.1 Introduction One of the most well known methods for investigating the nonlinear dynamic processes in sciences and engineering is the method of nonlinear variation of constant parameters [14, 15, 16, 19, 36]. Knowing the existence of solution process, the method of variation of parameters provides a very powerful tool for finding the solution representation of system of differential equations [14, 15, 16, 19, 36]. The idea is to decompose a complex system of differential equations in to two parts in such a way that a system of differential equations corresponding to the simpler part is either easily solvable in a closed form or analytically analyzable. However, the over all complex system of differential equations are neither easily solvable in a closed form nor analytically analyzable [14, 15]. The method of variation of parameters provides a formula for a solution to the complex system in terms of the solution process of simpler system of differential equations. In this chapter, an attempt is made to find a representation of solutions of nonlinear and nonstationary Itô-Doob type stochastic system of differential equations in terms of solutions processes of smoother system of Itô-Doob type stochastic differentials. The organization is as follows: In section 2.2, the problem is formulated. In section 2.3, several auxiliary results are established for unperturbed system of nonlinear Itô- Doob type stochastic differential equations. In section 2.4, a variation of constants 8

17 formula is established. In section 2.5, examples are given to illustrate the usefulness of the methods. The Developed results are a convenient tool in discussing the properties of solutions of the perturbed system. 2.2 Problem Formulation Let us formulate a problem. We consider a mathematical description of a nonlinear dynamic phenomenon under randomly varying environmental perturbations described by a complex system of nonlinear nonstationary Itô-Doob type system of stochastic differential equations: dy = c(t, ydt + Σ(t, ydw(t, y( = x 0, (2.2.1 where y R n, c C[J R n, R n ], Σ C[J R n, R n m ]; C[J R n, R n ] (C[J R n, R n m ] stands for a class of continuous functions defined on J R n into R n (R n m ; n and m are positive integers ; J = [, + a for some positive real number a; x 0 is an n-dimensional random variable defined on a complete probability space (Ω, F, P ; w(t = (w 1 (t, w 2 (t,..., w m (t T is an m-dimensional normalized Wiener process with independent increments; x 0 and w(t are mutually independent for each t. We decompose complex system of stochastic differential equations (2.2.1 into two parts. The decomposition of its drift and diffusion rate functions are as follows: c(t, y = f(t, y + F (t, y and Σ(t, y = σ(t, y + Υ(t, y where the rate functions f(t, y and σ(t, y are considered to be smooth and simpler form in the sense of better structure and conceptually smooth. Thus, (2.2.1 can be 9

18 rewritten as dy = [f(t, y + F (t, y]dt + [σ(t, y + Υ(t, y]dw(t = [f(t, y + F (t, y]dt + m [σl (t, y + Υ l (t, y]dw l (t, y( = x 0. (2.2.2 The smoother and simpler form of mathematical model of dynamic process corresponding to (2.2.2 is described by dx = f(t, xdt + σ(t, xdw(t = f(t, xdt + m σl (t, xdw l (t, x( = x 0. (2.2.3 Moreover, systems (2.2.2 and (2.2.3 are considered to be perturbed and unperturbed systems of stochastic differential equations, respectively. Remark In the absence of any reasonable decomposition of the type (2.2.2, it is always possible to consider the above decompositions with F (t, y = c(t, y f(t, y and Υ(t, y = Σ(t, y σ(t, y for any suitable choice of smoother and simpler rate functions f(t, y and σ(t, y. 2.3 Auxiliary Results Our main objective is to develop the variation of constants formula with respect to (2.2.3 and its perturbed system ( For this purpose, first we investigate the Itô-Doob stochastic partial differentials of solution process x(t,, x 0 of unperturbed system (2.2.3 with respect to initial conditions (, x 0. In the following, under certain smoothness assumption on the rate functions of unperturbed stochastic system of differential equations (2.2.3, we establish the second order Itô-Doob type of differentials of the solution process of (2.2.3 with respect to (, x 0. In this section, by recalling the existence of Itô-Doob type differential of solution process of unperturbed system of stochastic differential equations with 10

19 respect to initial state, we first establish the existence of second order differential with respect to x 0. Moreover, as a byproduct, we show that the differentials satisfy Itô-Doob type of stochastic non homogeneous matrix differential equation. In the following Lemma, we assume that i σ is B F measurable, where B denotes the Borel σ -algebra on [0, and F is a σ -algebra such that for t 1 < t 2 F t1 F t2 such that w t is a martingale with respect to F t ii f t and σ t are F t -adapted; iii P [ 0 σ2 ij(s, w < for all t 0] = 1; iv P [ 0 f i(s, w < for all t 0] = 1; v w(t is F t measurable and x is jointly measurable in (t, w. Lemma Assume that σ and f in (2.2.3 are twice continuously differentiable with respect to x for fixed t, and f xx, σ xx are bounded with respect to x for fixed t. Further, assume that the initial value problem (2.2.3 has a unique solution process x(t,, x 0 existing for t. Then Φ(t,, x 0 = 2 x(t, t x 0 x 2 0, x 0 ( exists, and is the solution process of the following Itô-Doob type nonhomogeneous stochastic matrix differential equation: dy = [H(t,, x 0 Y +P (t]dt+ [Γ l (t,, x 0 Y +Q(t]dw l (t, Y ( = 0; (2.3.2 where the n n matrices H(t,, x 0 = f x (t, x(t,, x 0 and Γ l (t,, x 0 = σ l x(t, x(t,, x 0 are continuous; P (t = ( 2 x 2 f(t, x(t n k=1 Φ(t,, x 0 e k Φ(t,, x 0 ; Q(t = ( 2 x 2 σ l (t, x(t n k=1 Φ(t,, x 0 e k Φ(t,, x 0, Φ(,, x 0 is the n n identity matrix and is the tensor product of two matrices. 11

20 Proof. From the assumptions of the lemma, we conclude that Φ(t,, x 0 = x 0 x(t,, x 0 exists and is the solution of the Itô-Doob type stochastic matrix differential equations along the solution process x(t,, x 0 of (2.2.3[15, 27]: dy = H(t,, x 0 Y dt + Γ(t,, x 0 Y dw(t, Y ( = I n n. (2.3.3 In the following, we show that 2 x 2 0 x(t,, x 0 exists and it satisfies the stochastic differential equation ( For this purpose, we consider the following: For small λ > 0, let x 0 = n k=1 λe k ; where e k = (0, 0,..., 1,..., 0 T whose k-th component is 1. Moreover, let Φ(t, λ = Φ(t,, x 0 + x 0 and Φ(t = Φ(t,, x 0 be solutions of (2.3.3 through (, x 0 + x 0 and (, x 0, respectively, and x(t, λ = x(t,, x 0 + x 0 and x(t = x(t,, x 0 be solutions of (2.2.3 through (, x 0 + x 0 and (, x 0 respectively. Under the assumptions of Lemma and applying Lemma 6.1[15], we conclude that We set lim Φ(t, λ = Φ(t uniformly on J. (2.3.4 λ 0 Φ(t, λ = Φ(t, λ Φ(t, Φ(, λ = x 0. (2.3.5 Let R(θ = x f(t, x(t,, x 0 + θ x 0 with 0 θ 1. From the assumptions, we note that R is continuously differentiable with respect to θ, and hence d dθ R(θ = f xx(t, x(t,, x 0 + θ x 0 (Φ(t,, x 0 + θ x 0 x 0. (2.3.6 By Integrating both sides of (2.3.6 with respect to θ over an interval [0,1], we obtain R(1 R(0 = 1 0 f xx (t, x(t,, x 0 + θ x 0 (Φ(t,, x 0 + θ x 0 x 0 dθ. This, together with the fact that R(1 = f(t, x(t, λ and R(0 = f(t, x(t, yields x x 12

21 f(t, x(t, λ f(t, x(t = J(t, x(t, λ, Φ(t, λ, x x where J(t, x(t, λ, Φ(t, λ = Similarly, by setting 1 0 f xx (t, x(t,, x 0 + θ x 0 (Φ(t,, x 0 + θ x 0 x 0 dθ (2.3.7 G(θ = σx(t, l x(t,, x 0 + θ x 0 and using the continuous differentiability of G with respect to θ and chain rule, we have m d dθ G(θ = σxx(t, l x(t,, x 0 + θ x 0 (Φ(t,, x 0 + θ x 0 x 0. (2.3.8 By Integrating both sides of (2.3.8 with respect to θ over an interval [0,1], we get G(1 G(0 = 1 0 σ l xx(t, x(t,, x 0 + θ x 0 (Φ(t,, x 0 + θ x 0 x 0 dθ. This, together with the fact that G(1 = m σl x(t, x(t, λ and G(0 = m σl x(t, x(t, yields where [σx(t, l x(t, λ σx(t, l x(t] = Λ l (t, x(t, Φ(t, λ = 1 0 Λ l (t, x(t, λ, Φ(t, λ, σ l xx(t, x(t,, x 0 + θ x 0 (Φ(t,, x 0 + θ x 0 x 0 dθ. (2.3.9 Note that the integrals in (2.3.7 and (2.3.9 are cauchy-riemann integrals. Using the hypotheses of the Lemma, n n matrices J(t, x(t, λ, Φ(t, λ and Λ l (t, x(t, λ, Φ(t, λ 13

22 are continuous in (t, x, λ for l = 1, 2, 3,..., m. Furthermore, from (2.3.7, (2.3.9 and applying the bounded convergence theorem[34], we obtain and J(t, x(t, λ, Φ(t, λ lim λ 0 λ Λ l (t, x(t, λ, Φ(t, λ lim λ 0 λ = f xx (t, x(t,, x 0 ( = σ l xx(t, x(t,, x 0 ( n Φ(t,, x 0 e k ( k=1 n Φ(t,, x 0 e k. ( From (2.3.5, using the fact that Φ(t, λ and Φ(t are solutions of (2.3.3, we obtain k=1 d(φ(t, λ Φ(t = dφ(t, λ dφ(t = f x (t, x(t, λφ(t, λdt + σx(t, l x(t, λφ(t, λdw l (t [f x (t, x(tφ(tdt + σx(t, l x(tφ(tdw l (t] = [f x (t, x(t, λφ(t, λ f x (t, x(tφ(t]dt + [σx(t, l x(t, λφ(t, λ σx(t, l x(tφ(t]dw l (t. ( By adding and subtracting f x (t, x(tφ(t, λdt and m σl x(t, x(tφ(t, λdw l (t in (2.3.12, we obtain d(φ(t, λ Φ(t = [f x (t, x(t, λφ(t, λ f x (t, x(tφ(t, λ +f x (t, x(tφ(t, λ f x (t, x(tφ(t]dt +[ [σx(t, l x(t, λφ(t, λ σx(t, l x(tφ(t, λ]dw l (t + [σx(t, l x(tφ(t, λ σx(t, l x(tφ(t]dw l (t] 14

23 = [f x (t, x(t(φ(t, λ Φ(t +(f x (t, x(t, λ f x (t, x(tφ(t, λ]dt + [σx(t, l x(t[φ(t, λ Φ(t] +[σ l x(t, x(t, λ σ l x(t, x(t]φ(t, λ]dw l (t. ( This, together with (2.3.7, (2.3.9 and the definitions of Φ(t, λ in (2.3.5, yields d( Φ(t, λ Φ(t, λ = [f x (t, x(t + λ λ + [σx(t, l Φ(t, λ x(t λ J(t, x(t, λ, Φ(t, λ Φ(t, λ]dt λ + Λl (t, x(t, λ, Φ(t, λ Φ(t, λ]dw l (t. λ ( From ( and (2.3.11, system (2.3.2 can be considered as the nominal system corresponding to ( with initial data Y ( = 0. It is obvious that the initial value problem ( satisfies all the hypothesis of Lemma 6.1 [15], and hence by its application, we have Φ(t, λ lim λ 0 λ = Y (t uniformly on J, ( where Y(t is the solution process of ( Because of (2.3.4 and (2.3.5, we note that the limit of Φ(t,λ λ in ( is equivalent to x 0 Φ(t,, x 0. Thus is the solution process of ( Moreover, x 0 Φ(t,, x 0 = 2 x 2 0 x(t,, x 0. x 0 Φ(t,, x 0 Example Let us consider a scalar nonlinear unperturbed stochastic differential equation: dx = αx(ρ xdt + βxdw(t, x( = x 0. (

24 where α, β and ρ are any constant. Find x 0 x(t,, x 0 and 2 x 2 0 x(t,, x 0, if it exists. Solution: We note that f(t, x = αx(ρ x and σ(t, x = βx are twice continuously differentiable with respect to x. In fact, x f(t, x = α(ρ 2x, 2 x 2 f(t, x = 2α, 2 σ(t, x = β, and σ(t, x = 0. The closed form solution of ( is x x 2 x(t,, x 0 = [ ] 1, Φ(t, x α Φ(t, sds where Φ(t, = exp[ (αρ 1 2 β2 (t β(w(t w( ]. The partial derivative of solution process x(t,, x 0 with respect to x 0 is x 0 x(t,, x 0 = Φ(t, (Φ(t, + αx 0 Φ(t, sds 2 ( and 2 x 2 0 x(t,, x 0 = 2αΦ(t, Φ(t, sds (Φ(t, + αx 0 Φ(t, sds 3. ( Moreover, 2 x 2 0 x(t,, x 0 satisfies the following differential equation: dy = [[ ( [ ] 1 ] ] α ρ 2 Φ(t, x α Φ(t, sds y 2αΦ 2 (t, dt +βydw(t, y( = 0. ( Example Consider a scalar nonlinear autonomous differential equation: [ dx = a(t 1 ] 2 x3 + b(tx dt + σxdw(t, x( = x 0. ( where a, b, and σ are continuous functions defined on R + into R. Find and 2 x 2 0 x(t,, x 0, if it exists. x 0 x(t,, x 0 Solution: We note that f(t, x = a(t 1 2 x3 + b(tx and σ(t, x = σx are continuously differentiable with respect to x. In fact x f(t, x = 3 2 a(tx2 + b(t, 2 x 2 f(t, x = 16

25 2 3a(tx, σ(t, x = σ, and σ(t, x = 0. The closed form solution of ( is x x 2 x(t,, x 0 = Φ(t, x x 2 t 0 a(sφ 2 (s, ds [ ] t where Φ(t, = exp [b(s 1 2 σ2 (s ds + ] t σ(sdw(s. The solution to the IVP is differentiable with respect to x 0 except at x 0 = 0. In this case, by the uniqueness of the solution process of (2.3.20, x(t,, x 0 0. This process is always differentiable with respect to x 0. The partial derivative of solution processes x(t,, x 0 with respect to x 0 is and x 0 x(t,, x 0 = 2 x 2 0 Φ(t, sgn(x 0 [1 + x 2 t ( a(sφ 2 (s, ds] 3/2 x(t,, x 0 = 3 x 0 Φ(t, a(sφ 2 (s, ds [1 + x 2 t. ( a(sφ 2 (s, ds] 5/2 Moreover, 2 x 2 0 x(t,, x 0 satisfies the following differential equation: dy = [[ 3a(tΦ 2 (t, x 2 ] 0 2(1 + x 2 t 0 a(sφ 2 (s, ds + b(t y 1 + x 2 0 3a(tΦ 3 (t, x 0 a(sφ 2 (s, ds +σydw(t, Y ( = 0. ( ] dt The following result shows the existence of partial differential of solution process of (2.2.3 with respect to. Lemma Let us assume that all the hypothesis of Lemma be satisfied. Let x(t,, x 0 be the solution process of (2.2.3 existing for t. Then t0 x(t,, x 0 17

26 exists and: with t0 x(t,, x 0 = 1 ( n 2 [ Φ ij (t,, x 0 σ l (, x 0 σ x j(t l 0, x 0 j=1 0 n 1 +Φ(t,, x 0 [ σx(t l 0, x 0 σ l (, x 0 f(, x 0 ]]d Φ(t,, x 0 σ l (, x 0 dw l ( ( t0 x(,, x 0 = [ σx(t l 0, x 0 σ l (, x 0 f(, x 0 ]d σ l (, x 0 dw l ( ( Proof. Let = λ > 0 be a positive increment to, and define x(t, λ = x(t, + λ, x 0 x(t,, x 0 ( where x(t, +λ, x 0 and x(t,, x 0 are solution processes of (2.2.3 through ( +λ, x 0 and (, x 0, respectively. Let x( = x( + λ,, x 0 x(,, x 0. Set R(θ = x(t, +λ, x 0 +θ x(. It is obvious that R is continuously differentiable with respect to θ, and hence d dθ R(θ = x 0 x(t, + λ, x 0 + θ x( x( = Φ(t, + λ, x 0 + θ x( x(. (

27 By Integrating both sides of ( with respect to θ over an interval [0,1], we obtain R(1 R(0 = 1 0 Φ(t, + λ, x 0 + θ x( x( dθ. This, together with the fact that R(1 = x(t, + λ, x( + λ,, x 0 and R(0 = x(t, + λ, x 0, yields x(t, + λ, x( + λ,, x 0 x(t, + λ, x 0 = 1 0 Φ(t, + λ, x 0 + θ x( x( dθ. ( Because of the uniqueness of solution of (2.2.3 we have x(t,, x 0 = x(t, +λ, x( + λ,, x 0 and equation ( can be written as x(t, + λ, x 0 x(t,, x 0 = 1 0 Φ(t, + λ, x 0 + θ x( x( dθ. ( By adding and subtracting Φ(t, + λ, x 0 x(, Φ(t, + λ, x( + λ,, x 0 x( and Φ(t,, x 0 x( in ( and using the fact that Φ(t,, x 0 = Φ(t, + λ, x( + λ,, x 0 Φ( + λ,, x 0, ( we have x(t, λ = = 1 0 [Φ(t, + λ, x 0 + θ x( Φ(t, + λ, x 0 ] x( dθ + [Φ(t, + λ, x( + λ,, x 0 Φ(t, + λ, x 0 ] x( + [Φ(t,, x 0 Φ(t, + λ, x( + λ,, x 0 ] x( Φ(t,, x 0 x( 1 0 [Φ(t, + λ, x 0 + θ x( Φ(t, + λ, x 0 ] x( dθ + [Φ(t, + λ, x( + λ,, x 0 Φ(t, + λ, x 0 ] x( + [Φ(t, + λ, x( + λ,, x 0 Φ( + λ,, x 0 19

28 = Φ(t, + λ, x( + λ,, x 0 ] x( Φ(t,, x 0 x( 1 0 [Φ(t, + λ, x 0 + θ x( Φ(t, + λ, x 0 ] x( dθ + [Φ(t, + λ, x( + λ,, x 0 Φ(t, + λ, x 0 ] x( + [Φ(t, + λ, x( + λ,, x 0 (Φ( + λ,, x 0 Φ(,, x 0 ] x( Φ(t,, x 0 x(. ( We set G(ψ = Φ(t, + λ, x 0 + ψθ x( for 0 ψ 1. It is obvious that G is continuously differentiable with respect to ψ, and hence d dψ G(ψ = x 0 Φ(t, + λ, x 0 + ψθ x( (θ x( ( By Integrating both sides of ( with respect to ψ over an interval [0,1], we have G(1 G(0 = 1 0 x 0 Φ(t, + λ, x 0 + ψθ x( (θ x( dψ. This, together with G(1 = Φ(t, + λ, x 0 + θ x( and G(0 = Φ(t, + λ, x 0, yields Φ(t, +λ, x 0 +θ x( Φ(t, +λ, x 0 = 1 0 x 0 Φ(t, +λ, x 0 +ψθ x( (θ x( dψ ( Similarly, by setting g(β = Φ(t, + λ, x 0 + β x( for 0 β 1, and repeating the previous argument, we obtain d dβ g(β = x 0 Φ(t, + λ, x 0 + β x( x(. (

29 This, together with g(1 = Φ(t, + λ, x 0 + x( and g(0 = Φ(t, + λ, x 0, yields Φ(t, +λ, x 0 + x( Φ(t, +λ, x 0 = 1 Using ( and (2.3.35, ( reduces to x(t, λ = x 0 Φ(t, + λ, x 0 + ψθ x( (θ x( x( dψdθ x 0 Φ(t, +λ, x 0 +β x( x( dβ. x 0 Φ(t, + λ, x 0 + β x( x( x( dβ ( [Φ(t, + λ, x( + λ,, x 0 (Φ( + λ,, x 0 Φ(,, x 0 ] x( Φ(t,, x 0 x(. ( Adding and subtracting x 0 Φ(t, + λ, x 0 (θ x( x( and x 0 Φ(t, + λ, x 0 ( x( x( in ( yields, x(t, λ = x 0 [Φ(t, + λ, x 0 + ψθ x( Φ(t, + λ, x 0 ] (θ x( x( dψdθ x 0 [Φ(t, + λ, x 0 + β x( Φ(t, + λ, x 0 ] x( x( dβ +[Φ(t, + λ, x( + λ,, x 0 (Φ( + λ,, x 0 Φ(,, x 0 ] x( + 1 Φ(t, + λ, x 0 x( x( Φ(t,, x 0 x(. ( x 0 Using the bounded convergence theorem[34], the concept of Itô-Doob type differential and sufficiently small increment to, ( reduces to t0 x(t,, x 0 = [Φ(t,, x 0 dφ( ]dx( + 1 Φ(t,, x 0 dx( dx( 2 x 0 Φ(t,, x 0 dx( 21

30 = Φ(t,, x 0 = 1 2 σx(t l 0, x 0 dw l ( + 1 Φ(t,, x 0 2 x 0 σ l (, x 0 dw l ( σ l (, x 0 dw l ( Φ(t,, x 0 [f(, x 0 d + ( n j=1 +Φ(t,, x 0 [ σ l (, x 0 dw l ( σ l (, x 0 dw l ( ] Φ ij (t,, x 0 σ l (, x 0 σ l x j(, x 0 0 n 1 σx(t l 0, x 0 σ l (, x 0 f(, x 0 ]d Φ(t,, x 0 σ l (, x 0 dw l ( ( This shows that t0 x(t,, x 0 exists and it is represented as in ( This together with t = and (2.3.2 yields ( d Example Let us consider a scalar linear unperturbed stochastic differential equation: dx = f(txdt + σ(txdw(t, x( = x 0, ( where f and σ are any differentiable functions defined on J = [, +a into R, where a > 0. Find t0 x(t,, x 0, if it exists. Solution: Note that f(t, x = f(tx and σ(t, x = σ(tx are continuously differen- tiable with respect to x. Moreover, f(t, x = f(t and σ(t, x = σ(t. The closed x x form solution of ( is given by x(t,, x 0 = Φ(t, x 0. 22

31 Let us consider the following: x(t, + λ, x 0 x(t,, x 0 = x(t, + λ, x 0 x(t, + λ, x( + λ,, x 0 = Φ(t, + λx 0 Φ(t, + λx( + λ,, x 0 = Φ(t, + λ[x( + λ,, x 0 x 0 ] = Φ(t, + λ x( = [Φ(t, + λ Φ(t, + Φ(t, ] x( = [Φ(t, + λ Φ(t, ] x( Φ(t, x( = [Φ(t, Φ(t, + λ] x( Φ(t, x( = [Φ(t, + λφ( + λ, Φ(t, + λ] x( Φ(t, x( = Φ(t, + λ[φ( + λ, I] x( Φ(t, x( = Φ(t, + λ Φ(, x( Φ(t, x(. ( Using the bounded convergence theorem [34], the concept of Itô-Doob type differential and sufficiently small increment λ to, ( reduces to t0 x(t,, x 0 = Φ(t, dφ(, dx( Φ(t, dx( = Φ(t, [f( Φ(, d + σ( Φ(, dw( ][f( x 0 d + σ( x 0 dw( ] Φ(t, [f( x 0 d + σ( x 0 dw( ] = Φ(t, σ( σ( x 0 d Φ(t, [f( x 0 d + σ( x 0 dw( ] = Φ(t, [σ 2 ( f( ]x 0 d Φ(t, σ( x 0 dw(. ( Example Let us consider a scalar linear perturbed stochastic differential equation: dx = [f(tx + p(t]dt + [σ(tx + q(t]dw(t, x( = x 0, (

32 where f, σ, p and q are any differentiable functions defined on J = [, + a into R, where a > 0. Find t0 x(t,, x 0, if it exists. Solution: Note that f(t, x = f(tx+p(t and σ(t, x = σ(tx+q(t are continuously differentiable with respect to x. Moreover, f(t, x = f(t and σ(t, x = σ(t. x x Using the application of Lemma we obtain t0 x(t,, x 0 = Φ(t, [(σ 2 ( f( x 0 + σ( q( p( ]d Φ(t, [σ( x 0 + q( ]dw(. ( In the following, we state and prove the existence of Itô-Doob type mixed partial differentials of solution process of ( Lemma Assume that all the hypothesis of Lemma hold. Let x(t,, x 0 be the solution process of (2.2.3 existing for t. Then the mixed Itô-Doob type partial differentials x0 ( t0 x(t,, x 0 and t0 ( x0 x(t,, x 0 exists and they are equal. Moreover, [ ( n x0 ( t0 x(t,, x 0 = + j=1 Φ ij (t,, x 0 σ l (, x 0 σ l x j(, x 0 0 ] Φ(t,, x 0 σx(t l 0, x 0 σ l (, x 0 d n 1 ( With initial condition: x0 ( t0 x(,, x 0 = σx(t l 0, x 0 σ l (, x 0 d ( Proof. Let x( = x( + λ,, x 0 x(,, x 0. Using (2.3.30, Lemma and the continuous dependence of solution process of (2.3.1, we examine the following 24

33 differential: x0 x(t, + λ, x 0 x0 x(t,, x 0 = x(t, + λ, x 0 dx x 0 2 ( Φ(t, + λ, x 0 dx 0 dx 0 x 0 [ x(t,, x 0 dx x 0 2 ( Φ(t,, x 0 dx 0 dx 0 ] x 0 = [Φ(t, + λ, x 0 Φ(t,, x 0 ]dx [ x 0 (Φ(t, + λ, x 0 Φ(t,, x 0 dx 0 ]dx 0. ( Since Φ(t,, x 0 = Φ(t, +λ, x( +λ,, x 0 Φ( +λ,, x 0, by adding and subtracting Φ(t, + λ, x( + λ,, x 0 dx 0 in (2.3.46, using generalized mean value theorem and algebraic manipulations, we get x0 x(t, + λ, x 0 x0 x(t,, x 0 = [Φ(t, + λ, x 0 Φ(t, + λ, x( + λ,, x 0 Φ( + λ,, x 0 ]dx [ (Φ(t, + λ, x 0 Φ(t,, x 0 dx 0 ]dx 0 x 0 = [Φ(t, + λ, x 0 Φ(t, + λ, x( + λ,, x 0 + Φ(t, + λ, x( + λ,, x 0 Φ(t, + λ, x( + λ,, x 0 Φ( + λ,, x 0 ]dx [ (Φ(t, + λ, x 0 Φ(t,, x 0 dx 0 ]dx 0 x 0 1 = [ Φ(t, + λ, x 0 + θ x( x( dθ]dx 0 x 0 0 Φ(t, + λ, x( + λ,, x 0 (Φ( + λ,, x 0 I n n dx [ x 0 (Φ(t, + λ, x 0 Φ(t,, x 0 dx 0 ]dx 0 (

34 Again by adding and subtracting x 0 Φ(t, + λ, x 0 x( dx 0 in (2.3.47, we get x0 x(t, + λ, x 0 x0 x(t,, x 0 = [ 1 0 x 0 (Φ(t, + λ, x 0 + θ x( Φ(t, + λ, x 0 x( dθ]dx 0 Φ(t, + λ, x( + λ,, x 0 Φ( dx 0 x 0 Φ(t, + λ, x 0 x( dx [ x 0 (Φ(t, + λ, x 0 Φ(t,, x 0 dx 0 ]dx 0 ( For sufficiently small = λ > 0, uniform convergence theorem, solution process of Itô-Doob type stochastic differential equations (2.2.3 and (2.3.3, Itô-Doob calculus and continuous dependence of solutions with respect to initial conditions, we obtain [ ( n t0 ( x0 x(t,, x 0 = + j=1 Φ ij (t,, x 0 σ l (, x 0 σ l x j(, x 0 0 n 1 ] Φ(t,, x 0 σx(t l 0, x 0 σ l (, x 0 d. ( On the other hand, using ( we examine the following differential t0 x(t,, x 0 + x 0 t0 x(t,, x 0 ( n = [ 1 Φ ij (t,, x 0 + x 0 σ l (, x 0 + x 0 σ l 2 x j(, x 0 + x 0 0 j=1 +Φ(t,, x 0 + x 0 [ n 1 σx(t l 0, x 0 + x 0 σ l (, x 0 + x 0 f(, x 0 + x 0 ]]d Φ(t,, x 0 + x 0 σ l (, x 0 + x 0 dw l ( [[ 1 2 ( n j=1 +Φ(t,, x 0 [ Φ ij (t,, x 0 σ l (, x 0 σ l x j(, x 0 0 n 1 σx(t l 0, x 0 σ l (, x 0 f(, x 0 ]]d Φ(t,, x 0 σ l (, x 0 dw l ( ] 26

35 = [ ( n j=1 ( n j=1 +Φ(t,, x 0 + x 0 [ Φ(t,, x 0 [ Φ ij (t,, x 0 + x 0 σ l (, x 0 + x 0 σ l x j(, x 0 + x 0 0 x 0 Φ ij (t,, x 0 σ l (, x 0 σ l j(, x 0 n 1 ]d n 1 σx(t l 0, x 0 + x 0 σ l (, x 0 + x 0 f(, x 0 + x 0 ]d σx(t l 0, x 0 σ l (, x 0 f(, x 0 ]d Φ(t,, x 0 + x 0 σ l (, x 0 + x 0 dw l ( + Φ(t,, x 0 σ l (, x 0 dw l (. ( By adding and subtracting Φ(t,, x 0 m σl (, x 0 + x 0 dw l ( in ( yields t0 x(t,, x 0 + x 0 t0 x(t,, x 0 ( n = [ 1 Φ ij (t,, x 0 + x 0 σ l (, x 0 + x 0 σ 2 x j(t l 0, x 0 + x 0 j=1 0 ( n 1 Φ ij (t,, x 0 σ l (, x 0 σ l 2 x j(, x 0 ]d 0 j=1 +Φ(t,, x 0 + x 0 [ Φ(t,, x 0 [ n 1 n 1 σx(t l 0, x 0 + x 0 σ l (, x 0 + x 0 f(, x 0 + x 0 ]d σx(t l 0, x 0 σ l (, x 0 f(, x 0 ]d [Φ(t,, x 0 + x 0 Φ(t,, x 0 ] Φ(t,, x 0 σ l (, x 0 + x 0 dw l ( (σ l (, x 0 + x 0 σ l (, x 0 dw l ( ( From the continuity of rate coefficient matrices and the continuous dependence of 27

36 solution process, we have [ ( n x0 ( t0 x(t,, x 0 = + j=1 Φ ij (t,, x 0 σ l (, x 0 σ l x j(, x 0 0 ] Φ(t,, x 0 σx(t l 0, x 0 σ l (, x 0 d n 1 ( This establishes the proof of ( Since I n n at t =, we have x0 ( t0 x( = This completes the proof of the Lemma. x 0 Φ(,, x 0 = 0 and Φ(,, x 0 = σx(t l 0, x 0 σ l (, x 0 d. ( Example Let us consider Example Find x0 ( t0 x(t,, x 0. Solution: Using (2.3.30, Lemma and the continuous dependence of solution process of (2.3.1, we examine the following differential: x0 x(t, + λ, x 0 x0 x(t,, x 0 = [ x(t, + λ, x 0 x(t, + λ, x( + λ,, x 0 ]dx 0 x 0 x 0 = [Φ(t, + λ Φ(t, ]dx 0 = [Φ(t, + λ Φ(t, + λφ( + λ, ]dx 0 = Φ(t, + λ[φ( + λ, Φ(, ]dx 0 = Φ(t, + λ Φ(, dx 0. ( Using the bounded convergence theorem[34], the concept of Itô-Doob type differential 28

37 and sufficiently small increment λ to, ( reduces to t0 ( x0 x(t,, x 0 = Φ(t, dφ(, dx 0 = Φ(t, [f( Φ(, d + σ( Φ(, dw( ][f( x 0 d + σ( x 0 dw( ] = Φ(t, σ 2 ( x 0 d. ( Method of Variation of Constants Formula In this section we shall establish the method of variation of constants formula with respect to (2.2.3 and its perturbed system ( Theorem Let the assumption of Lemma be satisfied. Let y(t,, x 0 and x(t,, x 0 be solution processes of (2.2.2 and (2.2.3, respectively, through the same initial data (, x 0, for all t. Then [ 1 n y(t,, x 0 = x(t,, x 0 + Φ ij (t, s, y(s[υ 2( l (s, y(sσ x j(s, l y(s j=1 0 +Υ l (s, y(sυ l j(s, y(s σ l (s, y(sυ l j(s, y(s] +Φ(t, s, y(s[f (s, y(s + n 1 ] σx(s, l y(sυ l (s, y(s] ds Φ(t, s, y(sυ l (s, y(sdw l (s. (

38 Proof. From the application of Lemma 2.3.1, Lemma 2.3.4, Lemma and the Itô-Doob differential formula with respect to s for s t, we have d s x(t, s, y(s = t0 x(t, s, y(s + x0 x(t, s, y(s + x0 ( t0 x(t, s, y(s ( Φ(t, s, y(s dydy x ( 0 n = 1 Φ ij (t, s, y(sσ l (s, y(sσ l 2 x j(s, y(s 0 j=1 +Φ(t, s, y(s( Φ(t, s, y(s ( n j=1 Φ(t, s, y(s n 1 σx(s, l y(sσ l (s, y(s f(s, y(sds σ l (s, y(sdw l (s + Φ(t, s, y(sdy(s Φ ij (t, s, y(sσ l (s, y(s(σ l x j(s, y(s + Υ l j(s, y(s 0 σx(s, l y(s(σ l (s, y(s + Υ l (s, y(sds ( x 0 Φ(t, s, y(s ( x 0 Φ(t, s, y(s ( x 0 Φ(t, s, y(s ( x 0 Φ(t, s, y(s σ l (s, y(sdw l (s Υ l (s, y(sdw l (s σ l (s, y(sdw l (s Υ l (s, y(sdw l (s ds σ l (s, y(sdw l (s σ l (s, y(sdw l (s Υ l (s, y(sdw l (s Υ l (s, y(sdw l (s n 1 ds (

39 By simplifying (2.4.2, we get d s x(t, s, y(s = ( [ n 1 2 ( n j=1 ( n j=1 j=1 +Φ(t, s, y(s[f (s, y(s + Φ ij (t, s, y(sσ l (s, y(sυ l x j(s, y(s 0 x 0 Φ ij (t, s, y(sυ l (s, y(sσ l j(s, y(s x 0 Φ ij (t, s, y(sυ l (s, y(sυ l j(s, y(s n 1 n 1 ] σx(s, l y(sυ l (s, y(s] ds n 1 Φ(t, s, y(sυ l (s, y(sdw l (s. (2.4.3 Since the right hand side of (2.4.3 is continuous with respect to s, we integrate from to t, and obtain the variation of constant formula ( Corollary Let the assumption of Lemma be satisfied, except that only c(t, y in (2.2.2 can be decomposed. Let y(t,, x 0 and x(t,, x 0 be solution processes of (2.2.2 and (2.2.3, respectively, through the same initial data (, x 0, for all t. Then y(t,, x 0 = x(t,, x ( n j=1 ( n j=1 [ 1 2 ( n j=1 Φ ij (t, s, y(sσ l (s, y(sσ l x j(s, y(s 0 x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s n 1 n 1 n 1 31

40 +Φ(t, s, y(s( [σx(s, l y(sσ l (s, y(s σx(s, l y(sσ l (s, y(s] ] +F (s, y(s ds + Φ(t, s, y(s[σ l (s, y(s σ l (s, y(s]dw l (s (2.4.4 Proof. From the application of Lemma 2.3.1, Lemma 2.3.4, Lemma and the Itô-Doob differential formula with respect to s for s t, we have d s x(t, s, y(s = t0 x(t, s, y(s + x0 x(t, s, y(s + x0 ( t0 x(t, s, y(s ( Φ(t, s, y(s dydy x ( 0 n = 1 Φ ij (t, s, y(sσ l (s, y(sσ l 2 x j(s, y(s 0 j=1 +Φ(t, s, y(s( Φ(t, s, y(s n 1 σx(s, l y(sσ l (s, y(s f(s, y(sds σ l (s, y(sdw l (s +Φ(t, s, y(s[(f(s, y(s + F (s, y(sds + ( n j=1 Φ(t, s, y(s ds Σ l (s, y(sdw l (s] Φ ij (t, s, y(sσ l (s, y(sσ l x j(s, y(s 0 σx(s, l y(sσ l (s, y(sds ( x 0 Φ(t, s, y(s Σ l (s, y(sdw l (s n 1 ds Σ l (s, y(sdw l (s (

41 By simplifying (2.4.5, we get d s x(t, s, y(s = [ 1 2 ( n j=1 ( n j=1 ( n j=1 Φ ij (t, s, y(sσ l (s, y(sσ l x j(s, y(s 0 x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s +Φ(t, s, y(s( x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s n 1 n 1 n 1 [σx(s, l y(sσ l (s, y(s σx(s, l y(sσ l (s, y(s] ] +F (s, y(s ds + Φ(t, s, y(s[σ l (s, y(s σ l (s, y(s]dw l (s (2.4.6 Since the right hand side of (2.4.6 is continuous with respect to s, we integrate from to t, and obtain the variation of constant formula ( Corollary Let the assumption of Lemma be satisfied, except that only Σ(t, y in (2.2.2 can be decomposed. Let y(t,, x 0 and x(t,, x 0 be solution processes of (2.2.2 and (2.2.2, respectively, through the same initial data (, x 0, for all t. Then y(t,, x 0 = x(t,, x j=1 ( n ( n j=1 [ 1 2 ( n j=1 Φ ij (t, s, y(sυ l (s, y(sσ l x j(s, y(s 0 x 0 Φ ij (t, s, y(sσ l (s, y(sυ l j(s, y(s x 0 Φ ij (t, s, y(sυ l (s, y(sυ l j(s, y(s n 1 n 1 n 1 33

42 +Φ(t, s, y(s[c(s, y(s f(s, y(s + ] σx(s, l y(sυ l (s, y(s] ds Φ(t, s, y(sυ l (s, y(sdw l (s. (2.4.7 Proof. From the application of Lemma 2.3.1, Lemma 2.3.4, Lemma and the Itô-Doob differential formula with respect to s for s t, we have d s x(t, s, y(s = t0 x(t, s, y(s + x0 x(t, s, y(s + x0 ( t0 x(t, s, y(s ( Φ(t, s, y(s dydy x ( 0 n = 1 Φ ij (t, s, y(sσ l (s, y(sσ l 2 x j(s, y(s 0 j=1 +Φ(t, s, y(s( Φ(t, s, y(s n 1 σx(s, l y(sσ l (s, y(s f(s, y(sds σ l (s, y(sdw l (s +Φ(t, s, y(s[c(s, y(sds + ( n j=1 Φ(t, s, y(s ds (σ l (s, y(s + Υ l (s, y(sdw l (s] Φ ij (t, s, y(sσ l (s, y(s(σ l x j(s, y(s + Υ l j(s, y(s 0 σx(s, l y(s(σ l (s, y(s + Υ l (s, y(sds ( x 0 Φ(t, s, y(s ( x 0 Φ(t, s, y(s ( x 0 Φ(t, s, y(s ( x 0 Φ(t, s, y(s σ l (s, y(sdw l (s Υ l (s, y(sdw l (s σ l (s, y(sdw l (s Υ l (s, y(sdw l (s σ l (s, y(sdw l (s σ l (s, y(sdw l (s Υ l (s, y(sdw l (s Υ l (s, y(sdw l (s n 1 ds (

43 By simplifying (2.4.8, we get d s x(t, s, y(s = [ 1 2 ( n j=1 ( n 1 2 j=1 ( n j=1 Φ ij (t, s, y(sυ l (s, y(sσ l x j(s, y(s 0 x 0 Φ ij (t, s, y(sσ l (s, y(sυ l j(s, y(s x 0 Φ ij (t, s, y(sυ l (s, y(sυ l j(s, y(s +Φ(t, s, y(s[c(s, y(s f(s, y(s + n 1 n 1 n 1 ] σx(s, l y(sυ l (s, y(s] ds Φ(t, s, y(sυ l (s, y(sdw l (s. (2.4.9 Since the right hand side of (2.4.9 is continuous with respect to s, we integrate from to t, and obtain the variation of constant formula ( Corollary Let the assumption of Lemma be satisfied, except that c(t, y in (2.2.2 and Σ(t, y in (2.2.3 cannot be decomposed. Let y(t,, x 0 and x(t,, x 0 be solution processes of (2.2.2 and (2.2.3, respectively, through the same initial data (, x 0, for all t. Then y(t,, x 0 = x(t,, x ( n ( n j=1 j=1 [ 1 2 ( n j=1 Φ ij (t, s, y(sσ l (s, y(sσ l x j(s, y(s 0 x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s n 1 n 1 n 1 35

44 +Φ(t, s, y(s[ σx(s, l y(s(σ l (s, y(s Σ l (s, y(s + c(s, y(s ] f(s, y(s] ds + Φ(t, s, y(s[σ l (s, y(s σ l (s, y(s]dw l (s. ( Proof. From the application of Lemma 2.3.1, Lemma 2.3.4, Lemma and the Itô-Doob differential formula with respect to s for s t, we have d s x(t, s, y(s = t0 x(t, s, y(s + x0 x(t, s, y(s + x0 ( t0 x(t, s, y(s ( Φ(t, s, y(s dydy x ( 0 n = 1 Φ ij (t, s, y(sσ l (s, y(sσ l 2 x j(s, y(s 0 j=1 +Φ(t, s, y(s[( n 1 σx(s, l y(sσ l (s, y(s f(s, y(sds σ l (s, y(sdw l (s] +Φ(t, s, y(s[c(s, y(sds + ( n j=1 Φ(t, s, y(s Σ l (s, y(sdw l (s] Φ ij (t, s, y(sσ l (s, y(sσ l x j(s, y(s 0 σx(s, l y(sσ l (s, y(sds ( x 0 Φ(t, s, y(s Σ l (s, y(sdw l (s n 1 ds ds Σ l (s, y(sdw l (s (

45 By simplifying (2.4.11, we get d s x(t, s, y(s = [ 1 2 ( n j= ( n ( n j=1 j=1 +Φ(t, s, y(s[ Φ ij (t, s, y(sσ l (s, y(sσ l x j(s, y(s 0 x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s x 0 Φ ij (t, s, y(sσ l (s, y(sσ l j(s, y(s n 1 n 1 n 1 σx(s, l y(s(σ l (s, y(s Σ l (s, y(s + c(s, y(s ] f(s, y(s] ds + Φ(t, s, y(s[σ l (s, y(s σ l (s, y(s]dw ( l (s. Since the right hand side of ( is continuous with respect to s, we integrate from to t, and obtain the variation of constant formula ( Remark In Corollary (2.4.4, if 1. σ(t, x = 0 and c(t, x = f(t, x, then the variation of constant formula reduces to y(t,, x 0 = x(t,, x ( n j=1 Φ ij (t, s, y(sσ l (s, y(sσ l x j(s, y(s 0 Φ(t, s, y(sσ l (s, y(sdw l (s. ( f(t, x = 0 and Σ(t, x = σ(t, x, then the variation of constant formula reduces to y(t,, x 0 = x(t,, x 0 + n 1 Φ(t, s, y(sc(s, y(sd(s. ( ds 37

46 2.5 Examples Example Consider a scalar linear unperturbed and perturbed stochastic differential equations: dx = f(txdt + σ(txdw(t, x( = x 0, (2.5.1 and dy = [f(ty + p(t]dt + [σ(ty + q(t]dw(t, y( = x 0, (2.5.2 where f, σ, p and q are any differentiable functions defined on J = [, + a into R, where a > 0. Then y(t,, x 0 = x(t,, x 0 + Φ(t, s[p(s σ(sq(s]d(s+ Φ(t, sq(sdw(s. (2.5.3 Solution: Let x(t = x(t,, x 0 and y(t = y(t,, x 0 be solution processes of (2.5.1 and (2.5.2 through (, x 0, respectively. Since x(t = x(t,, x 0 = Φ(t, x 0 [11], the partial derivative of x(t,, x 0 with respect to x 0 will be x 0 x(t,, x 0 = Φ(t,. Moreover, 2 x 2 0 x(t,, x 0 = 0. From Lemma 2.3.4, the partial differential of x(t,, x 0 with respect to is given by t0 x(t,, x 0 = Φ(t, [σ 2 ( f( ]x 0 d Φ(t, σ( x 0 dw( (2.5.4 At t =, we have t0 x(,, x 0 = [σ 2 ( f( ]x 0 d σ( x 0 dw( (

47 Moreover, from Lemma 2.3.7, the corresponding Itô-Doob mixed partial differential of solution process x(,, x 0 of (2.5.1 is given by t0 x 0 x(t,, x 0 = Φ(t, σ 2 ( x 0 d (2.5.6 Using the method of variational constants, Theorem 2.4.1, the solution of (2.5.2 is given by ( Example Consider a scalar linear unperturbed and nonlinear perturbed stochastic differential equations: dx = f(txdt + σ(txdw(t, x( = x 0, (2.5.7 and dy = [f(ty p(t 1 2 y3 ]dt + σ(tydw(t, y( = x 0, (2.5.8 where f, σ and p are any differentiable functions defined on J into R. Then y(t,, x 0 = x(t,, x Φ(t, sp(ty3 (sds. (2.5.9 Solution: Let x(t = x(t,, x 0 and y(t = y(t,, x 0 be solution processes of (2.5.7 and (2.5.8 through (, x 0, respectively. The partial differential of x(t,, x 0 with respect to initial data is given in Example The closed form solution of (2.5.8 is y(t,, x 0 = Φ(t, x 0. ( x 2 t 0 p(sφ 2 (t, sds 39

48 The partial differential of y(t,, x 0 with respect to is t0 y(t,, x 0 = Φ(t, [(f( + σ 2 ( x p(tx 0 3 ]d Φ(t, σ( x 0 dw(. At t =, we have ( t0 y(,, x 0 = [(f( + σ 2 ( x p(tx 0 3 ]d σ( x 0 dw( ( Moreover, the corresponding Itô-Doob mixed partial differential of solution process y(,, x 0 of (2.5.7 is given by t0 x 0 y(,, x 0 = Φ(t, σ 2 ( x 0 d ( Using the method of variational constants, Theorem (2.4.1, the solution of (2.5.8 is given by ( Example Consider a nonlinear unperturbed and perturbed stochastic differential equation: dx = αx(n xdt + βxdw(t, x( = x 0, ( and dy = [αy(n y + g(t, y]dt + [βy + σ(t, y]dw(t, y( = x 0, ( where α, β and n are any constant, g and σ are differentiable functions. Then y(t,, x 0 = x(t,, x 0 + βσ(s, y(s]]ds + [ 1 Φ(t, s, y(sσ 2 (s, y(s + Φ(t, s, y(s[g(s, y(s 2 x 0 Φ(t, sσ(s, y(sdw(s. (

49 Solution: Let x(t = x(t,, x 0 and y(t = y(t,, x 0 be solution processes of ( and ( through (, x 0, respectively. The partial derivative of solution processes x(t,, x 0 with respect to x 0 is and x 0 x(t,, x 0 = 2 x 2 0 x(t,, x 0 = Φ(t, (Φ(t, + αx 0 Φ(t, sds 2 ( αΦ(t, Φ(t, sds (Φ(t, + αx 0 Φ(t, sds 3. ( Moreover, the partial differential of x(t,, x 0 with respect to is t0 x(t,, x 0 = [ αβ 2 x 2 0Φ(t, Φ(t, sds (Φ(t, + αx 0 Φ(t, sds 3 +β 2 Φ(t, x 0 Φ(t, αx 0 (n x 0 d ] Φ(t, βx 0 dw(. ( The corresponding Itô-Doob mixed partial differential of solution process x(t,, x 0 of ( is given by t0 x 0 x(t,, x 0 = [ 2αβ 2 x 2 0Φ(t, Φ(t, sds ] (Φ(t, + αx 0 Φ(t, sds + Φ(t, β 2 x 0 d. ( Using the method of variational constants method, Theorem (2.4.1, with f(t, x = αx(n x and σ(t, x = βx, the solution of ( is given by ( Example Consider a scalar linear unperturbed and perturbed stochastic differential equation as: dx = A(txdt + σ l (txdw l (t, x( = x 0, (

50 and dy = [A(ty + P (t]dt + [σ l (ty + Υ l (t]dw l (t, y( = x 0, ( where A and σ l are any differentiable functions defined on J into R n n and P, Υ l are any differentiable functions defined on J into R n.then y(t,, x 0 = x(t,, x 0 + Φ(t, s[p (s Solution: σ l (sυ l (s]ds+ Φ(t, sυ l (sdw l (s. ( Let x(t = x(t,, x 0 and y(t = y(t,, x 0 be solution processes of ( and ( through (, x 0, respectively. The partial differential of x(t,, x 0 with respect to initial data is given in Example Using the method of variational constants, Theorem (2.4.1 with f(t, x = A(tx, σ(t, x = m σl (tx the solution of ( is given by (

51 3 Method of Generalized Variation of Constants Formula: Relative Stability 3.1 Introduction A closed form representation of a dynamic process of a nonlinear nonstationary solution process is not always possible or we might not be interested in the closed form solution [15]. In the absence of this, qualitative or quantitative properties are investigated for both the ordinary and stochastic dynamic systems [15, 16, 17, 19, 20]. A well known nonlinear technique [2-30] is to measure a dynamic flow by means of a suitable auxiliary measurement device, and then to use this measured dynamic flow to determine the desired information about the original dynamic flow [15, 16, 17, 20]. Employing energy function method as a measurement device and the fundamental properties of Itô-Doob type stochastic auxiliary system of differential equations [40], we establish the relationship between the solution processes of stochastic perturbed, auxiliary and nominal systems of differential equations. In addition, several estimates are obtained with regard to the deviation of solution process of perturbed system with respect to the solution process of nominal system of differential equations. Moreover, stability and relative stability results are developed to illustrate the usefulness of the results. Presented results generalize the existing results in a systematic and unified way [2-4, 6-28, 31]. Let us consider the following Itô-Doob type stochastic perturbed and auxiliary 43