Uniqueness of strong solutions to stochastic differential equations in the plane with deterministic boundary process

Yeh, J

doi:10.2140/pjm.1987.128.391

Cited by 12 publications

(4 citation statements)

References 10 publications

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“…In [37], Yeh proves an existence theorem for weak solutions under the assumption that the drift and diffusion coefficients are continuous and satisfy a growth condition and that the boundary has finite sixth order moment. For the same equation, Nualart and Yeh established in [28] (see also [38]) a Yamada-Watanabe type result for stochastic differential equations driven by Brownian sheet. More precisely, they proved that existence and uniqueness of strong solutions is equivalent to existence of weak solutions and pathwise uniqueness.…”

Section: Introductionmentioning

confidence: 88%

Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

Bogso,

Dieye,

Menoukeu-Pamen

2021

Preprint

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In this paper with study a uniqueness result (in the sense of [13]) for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spacial linear growth condition and is componentwise nondeacreasing. We first show the result when the drift is bounded measurable and nondecreasing. Our proofs rely on a local time-space representation of Brownian sheet and a type of law of the iterated logarithm for the Brownian sheet. The result in the unbounded case then follows by using the Gronwall's lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by Brownian sheet when the drift is non-decreasing and satisfies a spacial linear growth condition.

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Section: Introductionmentioning

confidence: 88%

Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

Bogso,

Dieye,

Menoukeu-Pamen

2021

Preprint

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“…The integral equation in (1.2) has been extensively studied from a probabilistic point of view, mostly in the setting where w is a Brownian sheet, but also other processes have been considered. Specifically, in [35,36] strong existence and pathwise uniqueness of solutions to equations of the form…”

Section: Introductionmentioning

confidence: 99%

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Bechtold

Harang

Rana

2023

Stoch PDE: Anal Comp

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We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.

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“…The integral equation in (1.2) has been extensively studied from a probabilistic point of view, mostly in the setting where w is a Brownian sheet, but also other processes have been considered. Specifically, in [30,31] strong existence and pathwise uniqueness of solutions to equations of the form…”

Section: Introductionmentioning

confidence: 99%

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Bechtold¹,

Harang²,

Rana³

2022

Preprint

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We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli in [6]. To this end we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of regularity of the local time associated to the perturbing stochastic field. As an illustration we prove wellposedness of a 1D non-linear wave equation with a noisy boundary given by two independent fractional Brownian motions. A discussion of several open problems and further investigations is provided.

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Uniqueness of strong solutions to stochastic differential equations in the plane with deterministic boundary process

Cited by 12 publications

References 10 publications

Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

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