SDE Coefficients. March 4, 2008

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1 SDE Coefficients March 4, 2008 The following is a summary of GARD sections 3.3 and 6., mainly as an overview of the two main approaches to creating a SDE model. Stochastic Differential Equations (SDE) allow for taking into account random process when modeling real world phenomena. Deterministic problems can be extended to SDE. The basic form of an SDE is dx = f(x(t), t)dt + G(X(t), t)dw (t) where dw (t) is a Wiener process. The functions f(x(t), t) and G(X(t), t) are determined from the parameters and form of the problem. In general there are two approaches for determining f and G. The first method results from veiwing the problem as a diffusion process while the second approach involves determining the functions from the deterministic equations. Diffusion Process A diffusion process is a continuous time Markov process with continuous sample paths. Definition: A Markov process X(t), for t 0 t T, with state space R n, and continuous sample paths with probability, is called a diffusion process if its transition probability P (s, x, t, B) satisfies the following conditions for ever s [t 0, T ), x R n, and ɛ > 0: limit t s t s y z>ɛ P (s, x, t, B) = 0 ()

2 limit t s t s limit t s t s y z ɛ y z ɛ (y x)p (s, x, t, B) = a(s, x) (2) (y x)(y x) T P (s, x, t, B) = B(s, x) (3) The functions a and B are the coefficients of the diffusion process. They are referred to as the drift vector and diffusion matrix respectively. B(x, s) is symmetric and nonnegative definite. Writing equations (2) and (3) in terms of expectation yields: E s,x (X(t) X(s)) = a(s, x)(t s) + o(t s) (4) E s,x ((X(t) X(s))(X(t) X(s)) T ) = B(s, x)(t s) + o(t s) (5) Then when X(t) = x a(s, x) is the mean velocity vector of the random motion of X(t), and B(s, x) is the measure of the local magnitude of fluctuation of X(t) X(s) about the mean value. Solutions to SDE are diffusion processes. However we are concerned with moving from a given diffusion process to a SDE. We begin by stating the existence and uniqueness theorem for SDE. Theorem. Suppose that we have a stochastic differential equation dx(t) = f(t, X(t))dt + G(t, X(t))dW (t), X(t 0 ) = c, t 0 t T, (6) where W (t) is an R m valued Wiener process and c is a random variable independent of W (t) W (t 0 ) for t t 0. Suppose that the R n valued function f(t, x) and the (n m) valued function G(t, x) are defined and measurable on [t 0, T ] R n and have the following properties: There exists a constant K > 0 such that for all t [t 0, T ], x R n, y R n, f(t, x) f(t, y) + G(t, x) G(t, y) K x y (7) and for all t [t 0, T ], x R n, f(t, x) 2 + G(t, x) 2 K 2 ( x y 2 ). (8) Then equation (6) has on [t 0, T ] a unique R n valued solution X(t), continuous with probability, that satisfies the initial condition X(t 0 ) = c; that is if 2

3 X(t) and Y (t) are continuous solutions of (6) with the same initial value c, then P (sup t0 t T X(t) Y (t) 0) = 0. (9) To move from a diffusion process to a SDE requires one more theorem. Theorem 2. Assume that the functions f and g satisfy the conditions guaranteeing existence and uniqueness of solutions to the initial value problem (6). Then any solution X(t) of (6) is a diffusion process on the interval [t 0, T ], with drift coefficient f(t, x) and diffusion coefficient g(t, x). Now let Y(t) be a diffusion process defined on some interval with drift coefficient a(t, x) and diffusion coefficient b(t, x). Then we might guess that for a(t, x) and b(t, x) satisfying the conditions of theorem () and for W = W (t) any standard Wiener process the solution X(t) of the initial value problem dx = a(t, X)dt + b(t, X)dW, X(t 0 ) = Y (t 0 ) (0) shares the same probability law and is thus equivalent to the process Y (t). But since the sample paths of X(t) and Y (t) may not coincide with probability, we can not be sure and must make the connection rigorous. To begin let Z(t) = g(t, Y (t)) where g(t, y) = y dν ; define the 0 (b(t,ν) function ā(t, z) = [ g t + a g y + 2 b 2 g t 2 )(t, g (t, z)); () then it can be shown that Z diffusion process with drift ā(t, z) and diffusion coefficient one. If the process W y is defined by W y = Z(t) Z(0) t 0 ā(s, Z(s)ds, (2) one can show that W y is a Wiener process and further that we have dz = ā(t, Z(t))dt + dw y (t). (3) 3

4 This implies that Y (t) is a solution to the stochastic equation dy (t) = a(t, Y (t)dt + b(t, Y (t))dw y. (4) The conditions on the coefficients are summed up in the following theorem. Theorem 3: Let Y (t) be a diffusion process on [0, T ] with coefficients a(t, y) and b(t, y) satisfying the following conditions for all y and 0 s t T :. a(t, y) is continuous in both variables and for some constant K, a(t, y) K( y ). 2. b(t, y) is continuous in both variables, is bounded, and the partial b derivatives b b and are continuous and bounded. t x 3. There is a function Ψ(y) > + y such that (a) sup [0,T ] EΨ(Y (t)) < (b) E s,y ( Y (t) Y (s) ) + E s,y ( Y (t) Y (s) ) 2 Ψ(y)(t s) (c) E s,y ( Y (t) ) + E s,y ( Y (t) ) 2 Ψ(y) Then there is a Wiener process W y (t) for which Y (t) solves the stochastic differential equation dy (t) = a(t, Y (t)dt + b(t, Y (t))dw y (t). (5) So given a diffusion process we can generate an SDE. 2 Deterministic Equation Alternatively we may start from a deterministic equation and move to a SDE. We begin with an ordinary differential equation dx dt = f(x) (6) 4

5 and assume that the addition of a noise perturbation given by g(x)ξ(t), where g(x) is the strength of the ξ(t) a normalized autocorrelated noise process, correctly models the system being investigated. We then have the equation dx = f(x) + g(x)ξ(t). (7) dt Note that ξ(t) is not in general a Gaussian noise process and that the above equation is often mathematically difficult to solve. Analysis of the model typically performed by replacing the model by a sequence of equations, dx n dt = f(x n ) + g(x n )ξ n (t). (8) The goal is to establish that X n X L where X L is similar to but more mathematically tractable than X. It is assumed that ξ n approaches Gaussian white noise in some sense. The following theorem states the convergence conditions. Theorem 4 (Papanicolau and Kohler, 974) Assume that, for all x R, the function f satisfies, for some constant C and integer q 0,. f(x) C( + x 2. f (x) C 3. f (x) + f (x) + f (4) (x) C( + x (q) and also that the function g satisfies 3. Suppose further that the ξ n processes satisfy ξ n (t) = nη(nt) where η is a continuous, strictly stationary, mean zero process with covariance function K and satisfying the mixing condition: there is a function φ such that for any event A and B, any t, and s > 0 P (η(t+s) A η(t) B) P (η(t+s) A) φ(s) and φ(s) 0 as s. Set γ = K(s)ds and 0 φ(s)ds <. Then let X n (t) be a solution of the sequence of equations on some interval [a, b] satisfying X n (a) = X a, a random variable. Then for t [a, b], X n (t) X γ (t) as n, in the distribution sense where X γ is the diffusion process 5

6 on [a, b] satisfying X γ (a) = X a and having drift coefficient f(x) + γg(x)g (x) and diffusion coefficient γg 2 (x). The problem resulting from the above theorem is that while our sequence of equations might be close to a white noise forced differential equation the appropriate diffusion equation may not be supplied by either the Ito or Stratonovich SDE. Thus the usefulness of the theorem depends on whether or not for large n the restriction on ξ n constitutes an acceptable model for the system of interest. Another method of finding an SDE from a deterministic equation is again to start with an equation dx = f dt 0 (x). Then letting λ 0 be a parameter involved in the functional form of f 0 that is affected by random environmental variations and can be written as λ = α + σ, where α is an average and σ is a fluctuation term. Assume the parameter appears linearly in f, f 0 (x) = f (x)+λf 2 (x). Then a choice for f and g in a stochastic counterpart of the deterministic equation is: f(x) = f (x) + αf 2 (x) (9) g(x) = σf 2 (x) (20) The problem with this approach is that a restricted deterministic parameter may be replaced by an analogous unrestricted behavior in the stochastic model. The situation can be dealt with by replacing f and g by smooth bounded truncations f n and g n. Theorem 5 (Turelli 977) Assume f and g are functions that are continuously differentiable on [0, ) with f(0) = g(0) = 0, and suppose that for any solution of dx = f(x)dt + g(x)dw on [0, ) the boundary at is unattainable. Let X N (t) be constructed as follows, for a t: X N (a+n+) X N (a+n) = f N (X N (a+n)) N +g N(X N (a+n)) η N(a + n + ) N /2 with initial value X N (a) = X a and where it is assumed that for each N, the functions f N and g N are bounded continuously differentiable functions on [0, ), and there are numbers C N, C N as N, such that f N (x) = f(x), g N (x) = g(x), x C N 6

7 and that each η N is a sequence of i.i.d. mean zero, variance one random variables with finite third moments. A solution X N is extended to a continuous time process by defining X N (t) = X N (a + n), a + n t < a + n +. Then if X(t) is the Ito solution of the SDE then, in the weak sense, X N (t) X(t) as N. 7

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