Weak and strong solutions of general stochastic models

By Thomas G. Kurtz. 2013

92 citations recorded by the source corpus.

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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/e86c3ada8eafa5d3f772999c4fa9fed08ea8fa7b

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Citations in the corpus

Causal transport plans and their Monge–Kantorovich problems (2013, background): “the final version of the manuscript, we were acquainted of [17] of stochastic calculus, and of [25] (see also [26]) which investigates a distance in a specific framework with a symmetric constraint; the general analytic optimal transport framework of the present paper enlightens those works and shou”
A weak solution theory for stochastic Volterra equations of convolution type (2019, background): “This in turn yields uniqueness in law via the abstract machinery of Kurtz (2014) and, as a byproduct, strong existence.”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “1, and is natural from the perspective of prior work on weak solutions of SDEs with random coefficients [41, 49].”
Yamada-Watanabe Results for Stochastic Differential Equations with Jumps (2013, background): “Kurtz [15], [16] continued the direction of Engelbert [8] and Jacod [11].”
Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations (2016, background): “The famous Yamada–Watanabe principle (see [12], [21], [32], [33]) concerning weak existence and pathwise uniqueness here has a remote analogue in terms of the equivalence of weak and strong uniqueness, yet, under additional assumptions.”
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.”
Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization (2016, methodology): “The origins of this concept can be found in the work of Ya-mada and Watanabe [YW71]; see also [Jac80, Kur14] for a generalization of the latter.”
Mean Field Games of Timing and Models for Bank Runs (2016, unknown): “T.G. Kurtz, Weak and strong solutions of general stochastic models, Electronic Communications in Probability 19 (2014), no. 58, 1–16.”