Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Stochastic models in biology.

http://directory.umm.ac.id/Data%20Elmu/jurnal/S/Stochastic%20Processes%20And%20Their%20Applications/Vol87.Issue2.2000/911.pdf

Weak approximation of killed diffusion using Euler schemes

The radiation (reactive or Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary; however, the relation between the reaction constant in the Robin boundary condition and the reflection probability is not well defined. In this paper we define the partially reflected process as a limit of the Markovian jump process generated by the Euler scheme for the underlying Ito dynamics with partial boundary reflection. Trajectories that cross the boundary are terminated with probability $P\sqrt{\Delta t}$ and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the Fokker–Planck equation in a half-space, wi...

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Partially reflected diffusion

This paper is concerned with the problem of simulation of (Xt)0≤t≤T , the
 solution of a stochastic differential equation constrained by some boundary conditions in a smooth domain
 D: namely, we consider the case where the boundary ∂D is killing, or where it is instantaneously
 reflecting in an oblique direction. Given N discretization times equally spaced on the interval [0,T],
 we propose new discretization schemes: they are fully implementable and provide a weak error of order
 N -1 under some conditions. The construction of these schemes is based on a natural principle of local
 approximation of the domain into a half space, for which efficient simulations are available.

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Euler schemes and half-space approximation for the simulation of diffusion in a domain

The aim of this paper is to approximate the expectation of a large class of functionals of the solution (X,xi) of a stochastic differential equation with normal reflection in a piecewise smooth domain of Rd . This also yields a Monte Carlo method for solving partial differential problems of parabolic type with mixed boundary conditions. The approximation is based on a modified Euler scheme for the stochastic differential equation. The scheme can be driven by a sequence of bounded independently and identically distributed (i.i.d.) random variables, or, when the domain is convex, by a sequence of Gaussian i.i.d. random variables. The order of (weak) convergence for both cases is given. In the former case the order of convergence is 1/2, and it is shown to be exact by an example. In the last section numerical tests are presented. The behavior of the error as a function of the final time T, for fixed values of the discretization step, and as a function of the discretization step, for fixed values of the final...

Numerical approximation for functionals of reflecting diffusion processes

The Skorohod oblique reflection problem for (D, Γ, w) (D a general domain in ℝ d , Γ(x),x∈∂D, a convex cone of directions of reflection,w a function inD(ℝ+,ℝ d )) is considered. It is first proved, under a condition on (D, Γ), corresponding to Γ(x) not being simultaneously too large and too much skewed with respect to ∂D, that given a sequence {wn} of functions converging in the Skorohod topology tow, any sequence {(xn, ϕn)} of solutions to the Skorohod problem for (D, Γ, wn) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, Γ, w). Next it is shown that if (D, Γ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, Γ, w) exist for everyw∈D(ℝ+,ℝ d ) with small enough jump size. The requirement is met in the case when ∂D is piecewiseC b 1 , Γ is generated by continuous vector fields on the faces ofD and Γ(x) makes and angle (in a suitable sense) of less than π/2 with the cone of inward normals atD, for everyx∈∂D. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

We study the sensitivity, with respect to a time dependent domain ${\cal D}_s,$ of expectations of functionals of a diffusion process stopped at the exit from ${\cal D}_s$ or normally reflected at the boundary of ${\cal D}_s.$ We establish a differentiability result and give an explicit expression for the gradient that allows the gradient to be computed by Monte Carlo methods. Applications to optimal stopping problems and pricing of American options, to singular stochastic control and others are discussed.

/pdf/boundary-sensitivities-for-diffusion-processes-in-time-35mji97122.pdf

Boundary Sensitivities for Diffusion Processes in Time Dependent Domains

Let E be a complete, separable metric space and A be an operator on Cb(E). We give an abstract denition of viscosity sub/supersolution of the resolvent equation u Au = h and show that, if the comparison principle holds, then the martingale problem for A has a unique solution. Our proofs work also under two alternative denitions

/pdf/viscosity-methods-giving-uniqueness-for-martingale-problems-3vcjbywc0a.pdf

Viscosity methods giving uniqueness for martingale problems

Consider a stochastic process consisting of the pair of a position and a velocity, in a piecewise $\mathscr{L}^1 d$-dimensional domain. In the interior of the domain the dynamics are assigned by a potential and by random changes of the velocity occurring at exponentially distributed times, according to a probability distribution which may depend on the current position and velocity. On the boundary the process reflects physically (the angle of reflection equals the angle of incidence). First it is shown that the process is well defined for all times. Then, when the coefficients depend on a diverging parameter $N$, in particular such that the speed and the jump rate of the velocity go to $\infty$ with order $\sqrt{N}$ and at least $N$ respectively, a diffusion approximation is sought. The position process is represented as a solution of a Skorohod reflection equation: A skewing effect on the boundary results from the interaction between the dynamics and the reflection law, so that the direction of reflection is in general oblique. The assumption that the mean change of the velocity in the interior is linear in the current velocity, up to order at least $1/2$ in $1/N$, ensures that the cone of directions of reflection is independent of $N$. The continuity properties of the Skorohod oblique reflection problem enable one to show tightness of the position processes without having to estimate explicitly local times (boundary layers) and, together with a suitable law of large numbers for the velocity, allow one to identify the limit stochastic differential equation with oblique reflection. The theory is illustrated by several applications, in particular one to a mechanical model of Brownian motion.

/pdf/diffusion-approximation-for-a-class-of-transport-processes-13tod6wt9d.pdf

Cristina Costantini

Papers

Numerical approximation for functionals of reflecting diffusion processes

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Boundary Sensitivities for Diffusion Processes in Time Dependent Domains

Viscosity methods giving uniqueness for martingale problems

Diffusion Approximation for a Class of Transport Processes with Physical Reflection Boundary Conditions