Moderate deviations-based importance sampling for stochastic recursive equations

Subsolutions to the Hamilton–Jacobi–Bellman equation associated with a moderate deviations approximation are used to design importance sampling changes of measure for stochastic recursive equations. Analogous to what has been done for large deviations subsolution-based importance sampling, these schemes are shown to be asymptotically optimal under the moderate deviations scaling. We present various implementations and numerical results to contrast their performance, and also discuss the circumstances under which a moderate deviation scaling might be appropriate.

On stochastic recursive equations of sum and max type

In this paper we consider stochastic recursive equations of sum type,, and of max type,, whereAi,bi, andbare random, (Xi) are independent, identically distributed copies ofX, anddenotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimalLs-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

On stochastic recursive equations of sum and max type

In this paper we consider stochastic recursive equations of sum type, , and of max type, , where Ai, bi, and b are random, (Xi) are independent, identically distributed copies of X, and denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal Ls-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.