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Unformatted text preview: Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE’s without going first through the machinery of stochastic integrals. 5.1 Itˆo Integrals and Itˆo Differential Equations Let us start with a review of the invariance principle. Let { ξ n } n ∈ N be a sequence of i.i.d. random variables such that E ξ n = 0, E ξ 2 n = 1. Define X N t by X N t n = ∑ n i =1 ξ i √ N , t n = n N , ≤ n ≤ N (5.1) and piecewise linear interpolation, then the invariance principle asserts that X N d → W (5.2) in distribution. Alternatively, we can define { X N t n } N n =1 by the recursion relation X N t n +1 = X N t n + √ Δ tξ n +1 , X N = 0 , (5.3) where Δ t = N 1 . We can think of (5.3) as a forward Euler scheme for solving the differential equation dX t = dW t (5.4) Obviously both (5.3) and (5.4) are a bit unusual. In (5.3), the multiplier in front of ξ n +1 is √ Δ t instead of the usual Δ t . In (5.4) we write dX t = dW t instead of ˙ X t = ˙ W t . This is because ˙ W t is not a standard stochastic process, but rather a generalized stochastic process as in generalized functions. In fact one can define ˙ W t as a Gaussian process with mean zero and covariance K ( s,t ) = δ ( t s ) . 41 42 CHAPTER 5. STOCHASTIC DIFFERENTIAL EQUATIONS This is a very important process called the Gaussian white noise. But its sample path are not the standard functions, but rather distributions, see [5]. We can now combine (5.4) with standard ordinary differential equation and study dX t = b ( X t ,t ) dt + dW t . (5.5) One can think of this as the distributional limit of the forward Euler scheme X N t n +1 = X N t n + Δ tb ( X t n ,t n ) + √ Δ tξ n +1 (5.6) where as before { ξ n } ∞ n =1 is a sequence of i.i.d. random variables such that E ξ n = 0, E ξ 2 n = 1. In fact if we denote by X N t the process obtained using (5.6) and the piecewise linear interpolation, then Theorem 5.1.1. There exists a stochastic process X t such that E  X N t X t  ≤ C √ Δ t (5.7) for t ∈ [0 , 1] , where C is independent of Δ t = 1 /N , and vextendsingle vextendsingle vextendsingle E g [ X N [0 , 1] ] E g [ X [0 , 1] ] vextendsingle vextendsingle vextendsingle ≤ C Δ t (5.8) for any continuous functional g on C [0 , 1] , where C may depend on g but not on Δ t . More generally, we can consider SDE’s of the type dX t = b ( X t ,W [0 ,t ] ,t ) dt + σ ( X t ,W [0 ,t ] ,t ) dW t (5.9) where B and sigma are functions of X t and t , and functional of W [0 ,t ] . Notice that they are nonanticipative functional, i.e. the Wiener process up to time t only enters the right handsidde of (5.9). (5.9) is defined it as the limit of the forward Euler scheme X n +1 = X n + Δ tb ( X n ,W m ≤ n ,t n ) + √ Δ tσ ( X n ,W m ≤ n ,t n ) ξ n +1 . (5.10) where, for simplicity we have used the slightly abusive notation X N t n = X n . We can also write this as X n +1 = X n + Δ tb ( X n ,W m ≤ n ,t n ) + σ ( X n ,W m ≤ n ,t n )(...
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 Fall '06
 gallivan
 Wj, Xt, Wiener process, STOCHASTIC DIFFERENTIAL EQUATIONS

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