Variational representations for continuous time processes
The authors prove a variational formula for functionals of a Poisson random measure and of a Brownian motion. Let \(\mathbb X\) be a locally compact metric space and \(T\in (0,\infty)\). For \(\theta>0\), let \(N^{\theta}\) be a Poisson random measure on \(\mathbb X_T=[0,T]\times \mathbb X\) with intensity \(\theta\nu_T\), where \(\nu_T=\lambda_T\otimes \nu\), \(\lambda_T\) is the Lebesgue measure on \([0,T]\) and \(\nu\) is a Radon measure on \(\mathbb X\). Given a bounded measurable functional \(F\) defined on the space of Radon measures on \(\mathbb X_T\), the authors establish a variational formula for \( \log \mathbb E [\exp(-F(N^{\theta}))] \). A similar result is proved for functionals of an infinite-dimensional Brownian motion. As an application, a large deviation principle, as \(\epsilon\downarrow 0\), is established for the solution \(\{Z^{\epsilon}(t): t\in [0,T]\}\) of a small-noise stochastic differential equation \[ dZ^{\epsilon}(t) = b(t,Z^{\epsilon}(t))dt +\sqrt{\epsilon} \sigma(t, Z^{\epsilon}(t))d\beta(t) + \int_{\mathbb X} \gamma(t, Z^{\epsilon}(t-),x) (\epsilon N^{1/\epsilon}(dtdx)-\nu_T(dtdx)). \] Here, \(Z^{\epsilon}(t)\) takes values in \(\mathbb R^d\), \(\beta\) is a \(d\)-dimensional standard Brownian motion and an initial condition \(Z^{\epsilon}(0)=z_0\) is imposed.
scientific article; zbMATH DE number 5956525
scientific article