Weak and strong solutions of general stochastic models

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.

[1]  A. Davie.  Uniqueness of solutions of stochastic differential equations , 2007, 0709.4147.

[2]  Weian Zheng,et al.  Tightness criteria for laws of semimartingales , 1984 .

[3]  D. Blackwell,et al.  An extension of Skorohod’s almost sure representation theorem , 1983 .

[4]  Thomas G. Kurtz,et al.  The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities , 2007 .

[5]  V. Lebedev,et al.  On the existence of weak solutions for stochastic differential equations with driving martingales and random measures , 1983 .

[6]  Jacod Jean,et al.  Weak and strong solutions of stochastic differential equations , 1980 .

[7]  Thomas G. Kurtz,et al.  Random Time Changes and Convergence in Distribution Under the Meyer-Zheng Conditions , 1991 .

[8]  Nuaa Nanjing.  On the Theorem of Ошкин , 1993 .

[9]  Adam Jakubowski,et al.  A Non-Skorohod Topology on the Skorohod Space , 1997 .

[10]  Jean Jacod,et al.  Existence of weak solutions for stochastic differential equations with driving semimartingales , 1981 .

[11]  P. Protter,et al.  Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations , 1991 .

[12]  A. Cherny.  On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations , 2002 .

[13]  P. Donnelly.  MARKOV PROCESSES Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics) , 1987 .

[14]  R. Buckdahn,et al.  On Weak Solutions of Backward Stochastic Differential Equations , 2005 .

[15]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .