Properties of solutions of stochastic differential equations
The paper considers the problem of differentiability of solutions \(X_ t=X_ t(x)\) of stochastic differential equations \[ dX_ t=\int g_ t(X_{t-},z)\mu (dt,dz)+G_ t(X_{t-})dM_ t+\int \tilde G_ t(X_{t-},z)q(dt,dz) \] (\(\mu\) is a random measure, q is a martingale measure, M is a continuous local martingale) with respect to the initial condition x and also the property of diffeomorphity and the possibility of decomposition of \(X_ t\) in the form \(X_ t=X^ 1_ t\circ X^ 2_ t\), where \(X^ 1_ t=X^ 1_ t(x)\), \(X^ 2_ t=X^ 2_ t(x)\) are solutions of analogous equations. Those results are then applied to establish the existence and smoothness of densities of the measures \(\mu_ t(\Gamma)=P(1_{\Gamma}(X_ t)Z_ t| \bar {\mathcal F}_ t)\) where \(\bar {\mathcal F}_ t\subset {\mathcal F}_ t\), \(t\geq 0\), Z is some martingale, with respect to the Lebesgue measure dx under the condition \(d\mu_ 0\ll dx\).
scientific article; zbMATH DE number 3909460
scientific article