A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise

We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we prove a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality.

• Publication year

  2019

• Venue

  Latin American Journal of Probability and Mathematical Statistics

• Publication date

  2019-08-28

• Fields of study

  Mathematics

• Identifiers
  DOI 10.30757/alea.v18-09arXiv 1908.10646
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Singular Brownian Diffusion Processes

  2018cited by this paper
• Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions

  2018cited by this paper
• Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching

  2017cited by this paper
• A discrete stochastic Gronwall lemma

  2016cited by this paper
• Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay Under Local Lipschitz Condition

  2013cited by this paper
• A Stochastic Gronwall Lemma

  2013cited by this paper
• A note on the domination inequalities and their applications

  2012cited by this paper
• Invariance and Monotonicity for Stochastic Delay Differential Equations

  2012cited by this paper
• A Note on Euler Approximations for Stochastic Differential Equations with Delay

  2012cited by this paper
• Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay

  2008cited by this paper
• Existence and uniqueness of solutions of stochastic functional differential equations

  2008cited by this paper
• The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay

  2007cited by this paper
• A Concise Course on Stochastic Partial Differential Equations

  2007cited by this paper
• Martingale-Valued Measures, Ornstein-Uhlenbeck Processes with Jumps and Operator Self-Decomposability in Hilbert Space

  2006cited by this paper
• The stable manifold theorem for non-linear stochastic systems with memory: II. The local stable manifold theorem

  2004cited by this paper
• Stochastic Differential Equations and Applications

  1998cited by this paper
• Lyapunov exponents of linear stochastic functional-differential equations. II. Examples and case studies

  1997cited by this paper
• Martingale measures and stochastic calculus

  1990cited by this paper
• Continuous martingales and Brownian motion

  1990cited by this paper
• An introduction to stochastic partial differential equations

  1986cited by this paper
• École d'été de probabilités de Saint Flour XIV, 1984

  1986cited by this paper
• Relation de domination entre deux processus

  1977cited by this paper
• Capacités et processus stochastiques

  1972influential reference

• EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE IN FINITE-DIMENSIONAL SPACES

  2025cites this paper
• Quantitative boundary H\"{o}lder estimates for the inhomogeneous Poisson problem through a probabilistic approach

  2025cites this paper
• The critical variational setting for stochastic evolution equations

  2024cites this paper
• An efficient Wasserstein-distance approach for reconstructing jump-diffusion processes using parameterized neural networks

  2024influential citation
• A note on the stochastic version of the Gronwall lemma

  2022cites this paper
• Concave and other generalizations of stochastic Gronwall inequalities

  2022influential citation
• A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations

  2022influential citation
• Stabilization of Highly Nonlinear Hybrid Stochastic Differential Delay Equations with Lévy Noise by Delay Feedback Control

  2022cites this paper
• Sharp nonlinear generalizations of stochastic Gronwall inequalities

  2021influential citation
• Robust Portfolio Choice with Sticky Wages

  2021cites this paper
• Sharpness of Lenglart’s domination inequality and a sharp monotone version

  2021cites this paper
• Electron. Commun. Probab. 26 (2021), article no. 44, https://doi.org/10.1214/21-ECP413

  2021cites this paper
• Sharp convex generalizations of stochastic Gronwall inequalities

  2021cites this paper
• Mathematical analysis of large-scale biological neural networks with delay

  2019cites this paper