By H. Jerome Keisler

ISBN-10: 0821822977

ISBN-13: 9780821822975

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**Extra resources for An Infinitesimal Approach to Stochastic Analysis**

**Example text**

A equals this naive approach we next look for a more (GWN} based on acceptable definition of Gaussian white noise is side Suppose X = (Xt} stationary Gaussian process with mean zero and covar- iance function R(T} == E(Xt+TXt}. Then by Bochner's theorem 00 R(T} = f eiTAdF(A} --00 where F is the spectral distribution F'(A} (if it function and f(A} == exists} is called the power spectral density and is then given by the inverse Fourier transform f(A} 1 2v 00 f e -iTA"R(T}dT. -00 Idealized GWN is that process X which has constant power spectral density.

Then for all s € [O,T]. is an ~t = ~+t-martingale. 1, let f € C ' ([O,T]xffi ). Then M~ is an ~t-local martingale. 2. Theorem 3~3: Let (Xt) be an ffid-valued diffusion process with diffusion and drift coefficients a,b respectively. Suppose that for some constants cl. c2. c3. c4. x) I ~ c1 • E exp(C3 1x0 12 ) ~ c4 . Then there exist constants c5. c6 depending only on cl. C3 . C4 , T and d such that c2. 7) The above theorem is an easy extension of the result proved in [34] where (Xt) is taken to be the solution of SDE with smooth coefficients an a,b.

7) The above theorem is an easy extension of the result proved in [34] where (Xt) is taken to be the solution of SDE with smooth coefficients an a,b. The part played by the smoothness assumption there was to ensure existence of a solution to the SDE. For later application we require two results due to Stroock and Varadhan on the existence, uniqueness vergence of and con- diffusion processes. It is beyond the scope of this monograph to include an outline of the proofs of results. T >0 To state them we need the following notation.

### An Infinitesimal Approach to Stochastic Analysis by H. Jerome Keisler

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