Logarithmic estimates for the density of an anticipating stochastic differential equation
Let \(\{X^\varepsilon _t\), \(t\in [0,1]\}\) be a family of \(R^d\)-valued stochastic processes indexed by the parameter \(\varepsilon \in (0,1)\), which satisfy the anticipating stochastic differential equation \[ X^\varepsilon _t= x_0^\varepsilon +\sqrt \varepsilon \int _0^t\sum _{j=1}^k \sigma _j(X^\varepsilon _s)\circ dW^j_s +\int _0^t \sigma _0 (X^\varepsilon _s) ds \] where \(W=\{W^j_t\), \(1\leq j\leq k\), \(t\in [0,1]\}\) is an \(R^k\)-valued standard Brownian motion and \(\sigma _j:R^d\to R^d\), \(j=0,\dots{},k\), and \(\{X^\varepsilon _0\}\) are random vectors not necessarily adapted to the filtration associated with the Wiener process. The stochastic integral is defined in the Stratonovich sense. Varadhan's type estimates for the density of the probability law of \(X^\varepsilon _t\) as \(\varepsilon \downarrow 0\) are proved.
scientific article; zbMATH DE number 1554287