A note on weak solutions to stochastic differential equations. (English) Zbl 1449.60103

Ondreját, Martin; Seidler, Jan

Kybernetika 54, No. 5, 888-907 (2018).

The paper presents an elegant and simple proof of a several classical results for weak solutions of stochastic differential equations, namely, their existence under weak assumptions on the coefficients and Yamada-Watanabe result of the form pathwise uniqueness and existence of weak solutions – existence of strong solutions. It is a continuation of previous works of M. Hofmanová and the second author [Stochastic Anal. Appl. 30, No. 1, 100–121 (2012; Zbl 1241.60025); ibid. 31, No. 4, 663–670 (2013; Zbl 1277.60106)], where another simplified proof of the existence result was presented. The present paper takes the method one step further and gives another proof of the identification of the stochastic integral in the equation. To be more precise, the construction of weak solutions relies on a suitable approximation procedure together with a compactness argument: the coefficients are approximated by, e.g., Lipschitz ones, so that existence of unique approximate solutions follows. Then uniform bounds are found to imply tightness of the induced probability laws, and the last step is the identification of the limit. In particular, the passage to the limit in the stochastic integrals is delicate as each approximate stochastic integral is with respect to a different Brownian motion. In the literature, one can find several methods to handle the last point mentioned above: the classical approach by the martingale representation theorem, the method from [2012, loc. cit.; 2013, loc. cit.], a convergence lemma from a paper by A. Debussche et al. [Physica D 240, No. 14–15, 1123–1144 (2011; Zbl 1230.60065)] and an auxiliary mollification argument from [A. Bensoussan, Acta Appl. Math. 38, No. 3, 267–304 (1995; Zbl 0836.35115)]. The idea in the present paper is similar to that of [Bensoussan, loc. cit.] but the mollification is different. The paper is very carefully written and presents the method in detail and in a nicely readable fashion. The method itself is indeed very elegant.

Reviewer: Martina Hofmanová (Bielefeld)

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

stochastic differential equations, continuous coefficients, weak solutions

Citations:

Zbl 1241.60025; Zbl 1277.60106; Zbl 1230.60065; Zbl 0836.35115