Hyperbolic stochastic differential equations: Absolute continuity of the LKW of the solution at a fixed point |
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Authors: | M Farré |
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Institution: | 1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain
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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 ∈ ?. 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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