Weak and strong solutions of general stochastic models

Thomas Kurtz

doi:10.1214/ecp.v19-2833

Abstract

Typically, a stochastic model relates stochastic “inputs” and, perhaps, controls tostochastic “outputs”. A general version of the Yamada-Watanabe and Engelbert the-orems relating existence and uniqueness of weak and strong solutions of stochasticequations is given in this context. A notion of compatibility between inputs and out-puts is critical in relating the general result to its classical forebears. The usualformulation of stochastic differential equations driven by semimartingales does notrequire compatibility, so a notion of partial compatibility is introduced which doeshold. Since compatibility implies partial compatibility, classical strong uniquenessresults imply strong uniqueness for compatible solutions. Weak existence argumentstypically give existence of compatible solutions (not just partially compatible solu-tions), and as in the original Yamada-Watanabe theorem, existence of strong solutionsfollows.

Highlights

Introduction and main theoremThis paper is essentially a rewrite of Kurtz (2007) following a realization that the general, abstract theorem in that paper was neither as abstract as it could be nor as general as it should be
As with the results of the earlier paper, the main theorem given here generalizes the famous theorem of Yamada and Watanabe (1971) giving the relationship between weak and strong solutions of an Ito equation for a diffusion and their existence and uniqueness
A second reason for this rewrite is that the main observation ensuring that the main theorem gives the Yamada-Watanabe result is buried in a proof in the earlier paper

Summary

Introduction and main theorem

This paper is essentially a rewrite of Kurtz (2007) following a realization that the general, abstract theorem in that paper was neither as abstract as it could be nor as general as it should be. Pointwise uniqueness for jointly C-compatible solutions holds if for every triple of processes (X1, X2, Y ) defined on the same probability space such that μX1,Y , μX2,Y ∈ SΓ,C,ν and (X1, X2) is jointly compatible with Y , X1 = X2 a.s. With reference to Lemma 2.4, uniqueness for jointly temporally compatible solutions is the usual kind of uniqueness considered for stochastic differential equations driven by Brownian motion, Levy processes, and/or Poisson random measures. The following lemma ensures that pointwise uniqueness of jointly compatible solutions is equivalent to the notion of pointwise uniqueness used in Theorem 1.5 and for example, Theorem 1.5 implies the classical Yamada-Watanabe theorem. (In particular, in the temporal compatibility setting, X is adapted to the filtration {FtY }.) if FαX ⊂ FαY for each α ∈ A and σ(X) ⊂ ∨α∈AFαX, X is a strong, compatible solution.

Xn of

Assume that

Xn converges to

Note that since