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**The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities.**
*(English)*
Zbl 1157.60068

The author considers stochastic equations and studies the relationship between weak and strong existence and weak and strong uniqueness of solutions. Earlier results for special types of stochastic equations were obtained by T. Yamada and S. Watanabe [J. Math. Kyoto Univ. 11, 155–167 (1971; Zbl 0236.60037)] and H. J. Engelbert [Stochastics Stochastics Rep. 36, No. 3–4, 205–216 (1991; Zbl 0739.60046)]. In particular, Yamada and Watanabe proved for Itô equations that weak existence and strong uniqueness imply strong existence and weak uniqueness, whereas Engelbert considered a slightly more general type of equations and showed a converse: weak joint uniqueness and strong existence imply strong uniqueness. Further results for other classes of equations, e.g., backward stochastic differential equations and two-parameter stochastic differential equations, have been obtained subsequently by other researchers and in the paper under review the author generalises and unifies these results.

Thus, given two Polish spaces \(S_1\) and \(S_2\) and \(\nu\) a probability law on \(S_2\) the author studies stochastic equations driven by a process \(Y\), with \(P_Y=\nu\), and describes the joint law \(P_{(X,Y)}\) of a weak solution \((X,Y)\) of that equation by convex constraints \(\Gamma\) on the set of probability measures \(\mu\) over \(S_1\times S_2\) with \(\mu(S_1\times \cdot)=\nu\), and by a compatibility restriction \(\mathcal{C}\). Based on these notions, the author defines unique and strong solutions for the triplet \((\Gamma,\mathcal{C},\nu)\) and gives a number of results concerning their relationship with different notions of uniqueness, e.g., pointwise uniqueness, \(\mu\)-pointwise uniqueness and joint uniqueness in law. Applications to several classes of equations are also provided.

Thus, given two Polish spaces \(S_1\) and \(S_2\) and \(\nu\) a probability law on \(S_2\) the author studies stochastic equations driven by a process \(Y\), with \(P_Y=\nu\), and describes the joint law \(P_{(X,Y)}\) of a weak solution \((X,Y)\) of that equation by convex constraints \(\Gamma\) on the set of probability measures \(\mu\) over \(S_1\times S_2\) with \(\mu(S_1\times \cdot)=\nu\), and by a compatibility restriction \(\mathcal{C}\). Based on these notions, the author defines unique and strong solutions for the triplet \((\Gamma,\mathcal{C},\nu)\) and gives a number of results concerning their relationship with different notions of uniqueness, e.g., pointwise uniqueness, \(\mu\)-pointwise uniqueness and joint uniqueness in law. Applications to several classes of equations are also provided.

Reviewer: Evelyn Buckwar (Edinburgh)

### MSC:

60H99 | Stochastic analysis |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60H20 | Stochastic integral equations |

60H25 | Random operators and equations (aspects of stochastic analysis) |