Approximation of Expectation of Diffusion Process and Mathematical Finance

Abstract

and Mathematical Finance Shigeo Kusuoka §1.Introduction Let (0, F, P) be a probability space and let {(B 1 (t), ... , Bd(t); t E [0, oo)} bead-dimensional Brownian motion.Let B 0 (t) = t, t E [0, oo).Let Vo, V1, ... , Vd E Cb'(RN; RN).Here Cb'(RN; Rn) denotes the space of Rn-valued smooth functions defined in RN whose devivatives of any order are bounded.We regard elements in Cb'(RN; RN) as vector fields on R N. Now let X(t, x), t E [0, oo), x ERN, be the solution to the Stratonovich stochastic integral equation (1Then there is a unique solution to this equation.Moreover we may assume that with probability one X(t, x) is continuous in t and smoothHere L = ~ 2:~1 Y; 2 + V0 .So to compute E[f(X(T, x))] is the same to compute the solution u(T, x) to PDE.However, in mathematical finance, if we think of the problem of pricing of Europian options, there are sometimes following difficulties.(1) L can be degenerate.Moreover, L may not satisfy even the Hi:irmander condition.

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