Uniqueness of solutions of stochastic differential equations

By A. Davie. 2007

564 citations recorded by the source corpus.

It follows from a theorem of Veretennikov [4] that (1) has a unique strong solution, i.e. there is a unique process x(t), adapted to the filtration of the Brownian motion, satisfying (1). Veretennikov in fact proved this for a more general equation. Here we consider a different question, posed by N. V. Krylov [2]: we choose a Brownian path W and ask whether (1) has a unique solution for that particular path. The main result of this paper is the following affirmative answer:

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/017463e95954279ed46177ad2496165e04cdb40e

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Citations in the corpus

On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients (2015, methodology): “3 Yamada and Watanabe approximation technique To deal with the Hölder continuity of the diffusion coefficient σ, we use Yamada and Watanabe approximation technique (see [28] or [13]).”
Averaging along irregular curves and regularisation of ODEs (2012, unknown): “Introduction In [4] A. M. Davie showed that the integral equation xt = x0 + ∫ t 0 b(s, xs)ds+ wt, t ∈ [0, 1] (1) with x,w ∈ C([0, 1];Rd) and b : R × Rd → Rd bounded and measurable has a unique continuous solution for almost every path w sampled from the law of the d-dimensional Brownian motion.”
Stochastic equations of non-negative processes with jumps (2008, background): “In view of the result of Yamada and Watanabe (1971), weaker conditions would be sufficient for the existence and uniqueness of strong solutions 1Supported by NSFC (10525103 and 10721091). for one-dimensional equations.”
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 [78] and is used under the name ‘compatibility’ by Kurtz [52].”
On the connection between symmetric $N$-player games and mean field games (2014, methodology): “A construction analogous to the one used in the proof of Proposition 1 in Yamada and Watanabe (1971) [also see Section 5.3.”
Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, methodology): “To our knowledge, the only two results on path-by-path uniqueness are [26] and [18].”
Hölder Flow and Differentiability for SDEs with Nonregular Drift (2013, unknown): “8), so that strong existence for the solutions of the two SDEs follows by the classical Yamada-Watanabe principle [7].”
On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients (2010, background): “Under this condition, the existence and uniqueness results for classical finite dimensional stochastic differential equations can be found in WatanabeYamada [11] and Yamada [14], while the infinite dimensional case can be found in Hu-Lerner [4].”
High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations (2019, background): “The logic behind these attempts is the known fact that noise sometimes improves the theory of differential equations, fact certainly true in finite dimensions [40, 32, 14] and also true for some infinite dimensional systems, like [28, 19, 20, 15, 12, 16, 4, 13, 27, 7, 26], but not for all examples o”
Affine Diffusion Processes: Theory and Applications (2009, result): “This proof builds on the seminal result by Yamada and Watanabe [35].”
Skew convolution semigroups and affine Markov processes (2005, result): “In particular, Watanabe [28] gave a complete description of regular two-dimensional continuous state branching processes.”
Mean field games with common noise (2014, background): “In this paper, we develop a similar notion of pathwise uniqueness for mean field games and provide an analog of the theorem of Yamada and Watanabe in this context.”
Noiseless regularisation by noise (2020, background): “In the case H = 1/2, we can allow b ∈ Lt C x for any α > 0; this result is comparable to the one from [15], in which sharper estimates allow to reach b ∈ Lt,x.”