Hyperbolic stochastic differential equations: Absolute continuity of the LKW of the solution at a fixed point

1432-0606

Hyperbolic stochastic partial differential equations ; Two-parameter representable semimartingales ; Malliavin calculus ; 60H15 ; 60H07

Springer Online Journal Archives 1860-2000

Mathematics

Abstract LetW be the Wiener process onT=[0, 1]2. Consider the stochastic integral equation $$\begin{gathered} X_\zeta = x_0 + \int_{R_\zeta } {a_1 (\zeta \prime )X(s\prime ,dt\prime )ds\prime + } \int_{R_\zeta } {a_2 (\zeta \prime )X(ds\prime ,t\prime )dt\prime } \hfill \\ + \int_{R_\zeta } {a_3 (X_{\zeta \prime , } \zeta \prime )W(ds\prime ,dt\prime ) + } \int_{R_\zeta } {a_4 (X_{\zeta \prime , } \zeta \prime )ds\prime ,dt\prime ,} \hfill \\ \end{gathered} $$ whereR ζ =(s, t) ∈ T, andx 0 ∈ ℝ. Under some assumptions on the coefficients ai, the existence and uniqueness of a solution for this stochastic integral equation is already known (see [6]). In this paper we present some sufficient conditions for the law ofX ζ to have a density.

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