Existence of strong solutions for stochastic differential equations in the plane

Abstract

Let B be the 2-parameter Brownian motion on D = [0, oo] x [0, oo) and Z be a 2-parameter stochastic process defined on the boundary 3D of D. Consider the non-Markovian stochastic differential system in 2-parameter (dX(8, t) = a(s, t, X)dB(8, t) + β(s, t, X)dsdt for (β, t)eD, (X(s,t) = Z(s f t) for (s,t)edD. Under the assumption that the coefficients a and β satisfy a Lipschitz condition and a growth condition and the assumption that Z has continuous sample functions and locally bounded second moment on 3D, it is shown in this paper that the differential system has a strong solution. Pathwise uniqueness of solution is established under the assumption of the Lipschitz condition.

Existence of weak solutions to stochastic differential equations in the plane with continuous coefficients

“…In [18] we showed that under a Lipschitz condition on a and ß the solution of the stochastic differential equation, if it exists, is pathwise unique. We showed also that if a and ß satisfy, in addition to the Lipschitz condition, a certain order of growth condition (see condition (C2) in §1), then for every {gz}-adapted 2-parameter stochastic process Z defined on 9R2+ and having a locally bounded second moment on 9 R2+ the stochastic differential equation has a strong solution whose restriction to 9 R2+ is equal to Z.…”

“…Let us now replace the Lipschitz condition in [18] by a continuity condition (see condition (Cl) in §1) on a and ß. Under the assumption of (Cl) and (C2) on the coefficients a and ß and with an g0-measurable random variable ¿ on (ß, g, P), we consider the stochastic differential equation with the boundary condition X, = £ for z g 9R2+, i.e., we consider the stochastic differential system (0.1) X(z) = X(0) + f a($, X) dBt + [ ß($, X) d$ for z g R2+, JR.…”

“…will signify the subspace of square integrable members and the subscript c will signify the subspace of continuous ones. For the definition of the linear spaces C (gz), p g[1,qo), of equivalence classes of 2-parameter stochastic processes, see [18,Definition 2.4]. Similarly we define C0O(gz) to be the linear space of equivalence classes of measurable {g.}-adapted stochastic processes i> on (ß, g, P; gz) such that for every T > 0 there exists CT > 0 satisfying |$(z)|<Cr for all z < (T, T) a.s.…”

“…For the continuous, square integrable, {g^ ,}-strong martingale M*, consider (18) M*(z)2=f a(lX*)2dÇ forzGR2+.…”

Uniqueness of strong solutions to stochastic differential equations in the plane with deterministic boundary process

“…The coefficients a and β are real valued functions on R^X W satisfying certain measurability conditions that imply that for each ω e Ω, a (z, X(-,ω)) and β (z, X(-,ω)) depend only on that part of the sample function X (-,ω) which precedes z in the sense of the partial ordering of R 2 + . We refer to [8] or [10] for these measurability conditions. In this article, by an equipped probability space we mean a complete probability measure space (Ω, g, P) with an increasing and right continuous family {g z , z e R 2 + } of sub-σ-fields of g, each containing all the null 392 J. YEH sets in (Ω, g, P).…”