Related papers: A regularization approach to functional It\^o calc…

In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional It\^o calculus, we introduce a path-dependent PDE and prove that its solution is uniquely…

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process $\X$,…

In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of $L^p$…

We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical…

We consider high order approximations of the solution of the stochastic filtering problem, derive their pathwise representation in the spirit of the earlier work of Clark and Davis and prove their robustness property. In particular, we show…

We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber, Probab. Theory…

In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204-236], which satisfies a…

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking…

We prove a representation for the support of McKean Vlasov Equations. To do so, we construct functional quantizations for the law of Brownian motion as a measure over the (non-reflexive) Banach space of H\"older continuous paths. By solving…

In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,\omega)$, and generator Lipschitz continuous in $(y,z,\gamma)$. We prove…

We establish It\^o's formula along flows of probability measures associated with general semimartingales; this generalizes existing results for flows of measures on It\^o processes. Our approach is to first establish It\^o's formula for…

In this paper we analyze the optimal value function $v$ associated to a general parametric optimization problems via the theory of viscosity solutions. The novelty is that we obtain regularity properties of $v$ by showing that it is a…

In this article, a notion of viscosity solutions is introduced for fully nonlinear second order path-dependent partial differential equations in the spirit of [Zhou, Ann. Appl. Probab., 33 (2023), 5564-5612]. We prove the existence,…

In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration…

This paper is concerned with H\"older regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain,…

The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…

The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., \& R\"ockner, M., A mild Ito formula for SPDEs, arXiv:1009.3526 (2012), To appear in the Trans.\ Amer.\ Math.\ Soc.] has turned out to be a useful instrument to study…

We derive a functional change of variable formula for {\it non-anticipative} functionals defined on the space of right continuous paths with left limits. The functional is only required to possess certain directional derivatives, which may…

Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and L\^e in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise…

Semilinear, $N-$dimensional stochastic differential equations (SDEs) driven by additive L\'evy noise are investigated. Specifically, given $\alpha\in\left(\frac{1}{2},1\right)$, the interest is on SDEs driven by $2\alpha-$stable,…