The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities

By Thomas G. Kurtz. 2007

128 citations recorded by the source corpus.

A general version of the Yamada-Watanabe and Engelbert results relating existence and uniqueness of strong and weak solutions for stochastic equations is given. The results apply to a wide variety of stochastic equations including classical stochastic differential equations, stochastic partial differential equations, and equations involving multiple time transformations.

Subfields: Control and Systems Engineering, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/76f3f8238ad94b2571b6fc0edb1e0da3981a6433

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In Control and Systems Engineering

In Finance

In Mathematical Physics

Citations in the corpus

Restricted Markov uniqueness for the stochastic quantization of $P(\Phi)_2$ and its applications (2015, unknown): “Zon [0,T] P− a.s.. Then the pathwise uniqueness holds for the stationary solution to (3.3). Now by the existence of the stationary martingale solution ( cf. [MR99]) and the Yamada-Watanabe Theorem in [Kur07] we obtain: Theorem 3.13 For any initial condition X(0) ∈ Cα with distribution ν and α<0, ”
Stochastic equations, flows and measure-valued processes (2010, background): “See El Karoui and Méléard (1990) and Kurtz (2007, 2010) for the general theory of stochastic equations driven by white noises and Poisson random measures.”
Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case (2008, methodology): “The conclusions of an F W t -adapted (strong) solution and uniqueness in law of (X, W) follow from the conclusions in Theorem 3.14 (b) (of ( Kur07 )) of a strong compatible solution and joint uniqueness in law, respectively.”
All adapted topologies are equal (2019, methodology): “Causal couplings have appeared in the work by Yamada and Watanabe [62], Jacod and Mémin [38] as well as Kurtz [42, 43], concerning weak solutions of stochastic differential equations, and by Rüschendof [58] concerning approximation theorems in probability theory.”
Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness (2021, result): “Furthermore, our construction directly implies the following result, essentially proving part of the classical Yamada–Watanabe–Engelbert’s theorem (see Kurtz [Kur07], Cherny [Che01] for a version applicable to finite dimensional stochastic differential equations).”
Continuous Time Markov Chain Models for Chemical Reaction Networks (2011, background): “For a general discussion of compatibility, see [37], in particular, Example 3.”
The Wasserstein space of stochastic processes (2021, methodology): “Imposing a ‘causality’ constraint on a transport plan between laws of processes seems to go back to the Yamada–Watanabe criterion for stochastic differential equations [99] and is used under the name ‘compatibility’ by Kurtz [71].”
Yamada-Watanabe Results for Stochastic Differential Equations with Jumps (2013, background): “Kurtz [15], [16] continued the direction of Engelbert [8] and Jacod [11].”
Causal transport plans and their Monge–Kantorovich problems (2013, background): “165 of [12]) of the celebrated Yamada–Watanabe criterion (see [12], and [14] or [16]).”