Explicit Solutions of the Extended Skorokhod Problems in Affine Transformations of Time-Dependent Strata

The goal of this paper is to expand the explicit formula for the solutions of the Extended Skorokhod Problem developed earlier for a special class of constraining domains in ℝ n with orthogonal reflection fields. We examine how affine transformations convert solutions of the Extended Skorokhod Problem into solutions of the new problem for the transformed constraining system. We obtain an explicit formula for the solutions of the Extended Skorokhod Problem for any ℝ n - valued càdlàg function with the constraining set that changes in time and the reflection field naturally defined by any basis. The evolving constraining set is a region sandwiched between two graphs in the coordinate system generating the reflection field. We discuss the Lipschitz properties of the extended Skorokhod map and derive Lipschitz constants in special cases of constraining sets of this type.

Numerical methods for Stochastic differential equations: two examples

The goal of this paper is to present a series of recent contributions arising in numerical probability. First we present a contribution to a recently introduced problem: stochastic differential equations with constraints in law, investigated through various theoretical and numerical viewpoints. Such a problem may appear as an extension of the famous Skorokhod problem. Then a generic method to approximate in a weak way the invariant distribution of an ergodic Feller process by a Langevin Monte Carlo simulation. It is an extension of a method originally developed for diffusions and based on the weighted empirical measure of an Euler scheme with decreasing step. Finally, we mention without details a recent development of a multilevel Langevin Monte Carlo simulation method for this type of problem.

On the Generalized Drift Skorokhod Problem in One Dimension

We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime.

On the Generalized Drift Skorokhod Problem in One Dimension

We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that isnotreflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/Nqueue in a many-server heavy traffic regime.

An Explicit Representation of the Extended Skorokhod Map with Two Time-Dependent Boundaries

We consider the extended Skorokhod problem for real-valued càdlàg functions with the constraining interval , where and change in time as values of two càdlàg functions. We find an explicit form of the solution and discuss its continuity properties with respect to the uniform, and , metrics on the space of càdlàg functions. We develop a useful technique of extending known results for the Skorokhod maps onto the larger class of extended Skorokhod maps.