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Stratonovich Integral. Let (2, F, P) be a probability space and let { W, : t 2 0} be a stan- dard Wiener process. Let the Stratonovich integral of W, . dW, be defined by the following limit S(1) = W . dW = lim I'Mi NI- 1=00 ( With + W. ) ( WITH - W ) where t; = it, 0 = to 0. Let the running maximum of the process X, up to time / be defined as MA = max XS. OSset Using Girsanov's theorem to find a measure Q under which X, is a standard Wiener pro- cess, show that the cumulative distribution function of the running maximum is x - V - ut 2#(x-V) P( M, S x) =Q -xtv- Ht 62 D x 2 V ovt ovt where (.) is the standard normal cumulative distribution function. Finally, deduce that the cumulative distribution function of the funning minimum me = min X OSSSt is P (m, S x) = Q x - v - ut 2#(x-V) 62 D x- v+ uit te x