Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density, u ( t , x ), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation for u ( t , x ) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations for u ( t , x ). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained.
Applied Mathematics and Optimization – Springer Journals
Published: Jan 1, 1997
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