Electronic Communications in Probability

Weak and strong solutions of general stochastic models

Thomas Kurtz

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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.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 58, 16 pp.

Accepted: 25 August 2014
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60G05: Foundations of stochastic processes

weak solution strong solution stochastic models pointwise uniqueness pathwise uniqueness compatible solutions stochastic differential equations stochastic partial differential equations backward stochastic differential equations Meyer-Zheng condi

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Kurtz, Thomas. Weak and strong solutions of general stochastic models. Electron. Commun. Probab. 19 (2014), paper no. 58, 16 pp. doi:10.1214/ECP.v19-2833. https://projecteuclid.org/euclid.ecp/1465316760

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