Strong Solutions of Stochastic Differential Equations with Boundary Conditions

Abstract

The class B#;$fl of strong solutions of the boundary problem for a stochastic differential equation with given drift and diffusion coefficients implies here a class of all continuous processes having the given stochastic differential within the interval [y,, y,] and not leaving it. It is shown that the class BR.;i;] can be characterized as a class of all solutions of some stochastic integral equation. The instantly reflecting process (IRP) in A. Skorokhod's sense is proved to be the extremal (with respect to some ordering) element of this class and in the case when the drift and diffusion coefficients satisfy the Lipschitz condition it can be constructed by the method of successive approximations. The convergence of discrete approximations, which are constructed by the natural analogy of Euler's formula to the IRP, is studied.

Stochastic differential equations with time-dependent reflecting barriers

“…[2] (Theorem 2.6) an explicit formula for the solution of the extended Skorokhod problem is given. Related results in the case of the classical Skorokhod problem and constant barriers were obtained earlier in Refs [5,10].…”

A note on the structure of processes the measure of which is absolutely continuous with respect to the wiener process modulus measure

“…inoertihle image of' X in X , i.e.an image 4 -' : X + X exists such thatx = 4-( ( x for Q.a.a. x E CT(6)Then the measure GQnduced by the image 4 is absolutely continuous with respect to the measure Q G n dThe proof of the Lemma follows from the following chain of equalitiesProof of Theorem 1 According to Theorem 1 from[2], conditions (2),(3), (4) from Definition 1 are equivalent to the requirement of 5 satisfying the following equation Downloaded by [University of California, San Diego] at 13:05 29 June 2016 and, besides, if ( is a solution of Eq. (8) then the following representation helds where i.e.…”

“…The aim of the given note is to show that weak solutions of such 2,=a(x,:sSt). Let Pq be a measure on (C, , 9,) induced by a continuous process q.…”

Euler Scheme for One-Dimensional SDEs with Time Dependent Reflecting Barriers

“…From (2) we deduce that in both cases of Bollinger and trading bands for every q ∈ R + , δ > 0 there exists C > 0 such that…”

“…In the case G = 0, F = +∞ and G = l, F = d the rate of mean-square convergence was examined earlier by many authors (see, e.g., [2,4,5,7,9,10]). …”