E. Khmaladze

Victorian University of Wellington, New Zealand

STOCHASTIC DIFFERENTIAL EQUATION FOR BROWNIAN MOTION UNDER BOUNDARY CONDITIONS

In ℝ^{d}, consider an arbitrary star-shaped set C, containing O and Wiener random measure b on the class of Borel subsets B(C) of C. The values of this measure, b(A), A ∈ B(C), are Gaussian random variables with expectation O and covariance

Eb(A)b(A′) = μ_{(d)}(A ∩ A′),  (1)

where μ_{(d)} is Lebesgue measure in ℝ^{d}. Given the boundary values of b this random measure is not Wiener random measure any more and has more complicated covariance structure than (1).

Using 1-dimensional innovation argument on each ray it is shown that certain filtration with 1-dimensional time parameter allows construction of unique innovation random measure w on B(C), which is again Wiener random measure and is in one-to-one correspondence with b, given values of b on the boundary.