Existence and uniqueness of solutions of stochastic functional differential equations

By Max-K. von Renesse, M. Scheutzow. 2008

78 citations recorded by the source corpus.

Abstract Using a variant of the Euler–Maruyama scheme for stochastic functional differential equations with bounded memory driven by Brownian motion we show that only weak one-sided local Lipschitz (or “monotonicity”) conditions are sufficient for local existence and uniqueness of strong solutions. In case of explosion the method yields the maximal solution up to the explosion time. We also provide a weak growth condition which prevents explosions to occur. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general stochastic versions of Gronwall's Lemma, the final one provides tail bounds for Hölder norms of stochastic integrals.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/acec24d4de0b9881daf00a9869106c803d769ef7

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In Computational Theory and Mathematics

1. M. Scheutzow
2. Oleg Butkovsky
3. Martin Hairer
4. Jonathan C. Mattingly
5. A. Kulik
6. Massimiliano Gubinelli
7. Jeremy Quastel
8. Bixiang Wang

In Control and Systems Engineering

1. Martin Hairer
2. M. Scheutzow
3. Oleg Butkovsky
4. A. Kulik
5. Jonathan C. Mattingly
6. Arnaud Debussche
7. C. Odasso
8. Michael Röckner

In Finance

1. Martin Hairer
2. M. Scheutzow
3. A. Kulik
4. Oleg Butkovsky
5. Jonathan C. Mattingly
6. Andreas Eberle
7. Arnaud Debussche
8. Arnaud Guillin

Citations in the corpus

• Generalized couplings and ergodic rates for SPDEs and other Markov models (2018, background): “The existence and uniqueness of Y x,y,λ follows again from [36].”
• Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations (2009, background): “ERIA TO SDDES 13 where jjjMjjj2 = Tr(MM ), then the equation (3.1) has at most one invariant measure. Remark 3.2 This assumption is sufficient to ensure the existence of (unique) global solutions (see [RS08] for even weaker hypotheses). Remark 3.3 Notice that Theorem 3.1 does not ensure that there ex”
• Subgeometric rates of convergence of Markov processes in the Wasserstein metric (2012, background): “4) imply [27] the existence and uniqueness of the strong solution of SDDE (3.”