Weak Dirichlet Processes Research Articles

Characteristics and Itô's formula for weak Dirichlet processes: an equivalence result

The main objective consists in generalizing a well-known Itô formula of J. Jacod and A. Shiryaev: given a càdlàg process S, there is an equivalence between the fact that S is a semimartingale with given characteristics ( B k , C , ν ) and a Itô formula type expansion of F ( S ) , where F is a bounded function of class C 2 . This result connects weak solutions of path-dependent SDEs and related martingale problems. We extend this to the case when S is a weak Dirichlet process. A second aspect of the paper consists of discussing some untreated features of stochastic calculus for finite quadratic variation processes.

A C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-Itô’s Formula for Flows of Semimartingale Distributions

We provide an Itô’s formula for C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 2006) to this context. As the first application, we study a class of McKean–Vlasov optimal control problems, and establish a verification theorem which only requires C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result.

Stochastic Differential Equations with Singular Coefficients: The Martingale Problem View and the Stochastic Dynamics View

We consider stochastic differential equations (SDEs) with (distributional) drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, such as continuity with respect to the drift and the link with the Fokker–Planck equation. We also show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. In the second part we identify the dynamics of the solution of the martingale problem by describing the proper associated SDE. Under suitable assumptions we show equivalence with the solution to the martingale problem.

Weak Dirichlet processes and generalized martingale problems

In this paper we explain how the notion of weak Dirichlet process is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition: in particular we introduce characteristics for weak Dirichlet processes. We also introduce a weak concept (in law) of finite quadratic variation. We investigate a set of new useful chain rules and we discuss a general framework of (possibly path-dependent with jumps) martingale problems with a set of examples of SDEs with jumps driven by a distributional drift.

A [formula omitted]-functional Itô’s formula and its applications in mathematical finance

Using Dupire’s notion of vertical derivative, we provide a functional (path-dependent) extension of the Itô’s formula of Gozzi and Russo (2006) that applies to C0,1-functions of continuous weak Dirichlet processes. It is motivated and illustrated by its applications to the hedging or superhedging problems of path-dependent options in mathematical finance, in particular in the case of model uncertainty. In this context, we also prove a new regularity result for the vertical derivative of candidate solutions to a class of path-depend PDEs, using an approximation argument which seems to be original and of own interest.

Rough Paths and Regularization

Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on deterministic techniques but there is still a probability in the background. The goal of this paper is to establish a connection between stochastically controlled-type processes, a concept reminiscent from rough paths theory, and the so-called weak Dirichlet processes. As a by-product, we present the connection between rough and Stratonovich integrals for cadlag weak Dirichlet processes integrands and continuous semimartingales integrators.

A Feynman–Kac result via Markov BSDEs with generalised drivers

In this paper, we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman–Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE. Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process.

Weak Dirichlet processes with jumps

This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that [N,A]=0, for any continuous local martingale N. Given a function u:[0,T]×R→R, which is of class C0,1 (or sometimes less), we provide a chain rule type expansion for u(t,Xt) which stands in applications for a chain Itô type rule.

HJB Equations in Infinite Dimension and Optimal Control of Stochastic Evolution Equations Via Generalized Fukushima Decomposition

A stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space is considered. Thanks to the identification of the mild solution of the state equation as a $\nu$-weak Dirichlet process, the value process is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is used to prove a verification theorem. Through that technique several of the required assumptions are milder than those employed in previous contributions about nonregular solutions of Hamilton--Jacobi--Bellman equations.

Special weak Dirichlet processes and BSDEs driven by a random measure

This paper considers a forward BSDE driven by a random measure, when the underlying forward process X is special semimartingale, or even more generally, a special weak Dirichlet process. Given a solution (Y, Z, U), generally Y appears to be of the type u(t, X_t) where u is a deterministic function. In this paper we identify Z and U in terms of u applying stochastic calculus with respect to weak Dirichlet processes.

Infinite dimensional weak Dirichlet processes and convolution type processes

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 values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process.The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs).In particular the mentioned decomposition appears to be a substitute of an Itô’s type formula applied to f(t,X(t)) where f:[0,T]×H→R is a C0,1 function and X a convolution type process.

GENERALIZED COVARIATION AND EXTENDED FUKUSHIMA DECOMPOSITION FOR BANACH SPACE-VALUED PROCESSES: APPLICATIONS TO WINDOWS OF DIRICHLET PROCESSES

This paper is concerned with the notion of covariation for Banach space-valued processes. In particular, we introduce a notion of quadratic variation, which is a generalization of the classical restrictive formulation of Métivier and Pellaumail. Our approach is based on the notion of χ-covariation for processes with values in two Banach spaces B1 and B2, where χ is a suitable subspace of the dual of the projective tensor product of B1 and B2. We investigate some C1 type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If 𝕏1 and 𝕏2 admit a χ-covariation, Fi : Bi → ℝ, i = 1, 2 are of class C1 with some supplementary assumptions, then the covariation of the real processes F1(𝕏1) and F2(𝕏2) exist. A detailed analysis is provided on the so-called window processes. Let X be a real continuous process; the C([-τ, 0])-valued process X(⋅) defined by Xt(y) = Xt+y, where y ∈ [-τ, 0], is called window process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. Those will constitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As application, we provide a new technique for representing a path-dependent random variable as its expectation plus a stochastic integral with respect to the underlying process.

Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes

In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb F$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the structure $Q + R$ where $Q$ is a finite quadratic variation process and $R$ is {\it{strongly predictable}} in some technical sense: that condition implies in particular that $R$ is \textit{weak Dirichlet}, and it is fulfilled, for instance, when $R$ is independent of $M$. The method is based on a transformation which reduces the {{\it diffusion}} coefficient multiplying $\xi$ to 1. We use generalized Ito and Ito-Wentzell type formulae. A similar method allows to discuss existence and uniqueness theorem when $\xi$ is a Holder continuous process and $\sigma$ is only Holder in space. Using an Ito formula for {\it{reversible}} semimartingales we also show existence of a solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.

Weak Dirichlet processes with a stochastic control perspective

The motivation of this paper is to prove verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term, in the case where the value function is assumed to be continuous in time and once differentiable in the space variable ( C 0 , 1 ) instead of once differentiable in time and twice in space ( C 1 , 2 ), like in the classical results. For this purpose, the replacement tool of the Itô formula will be the Fukushima–Dirichlet decomposition for weak Dirichlet processes. Given a fixed filtration, a weak Dirichlet process is the sum of a local martingale M plus an adapted process A which is orthogonal, in the sense of covariation, to any continuous local martingale. The decomposition mentioned states that a C 0 , 1 function of a weak Dirichlet process with finite quadratic variation is again a weak Dirichlet process. That result is established in this paper and it is applied to the strong solution of a Cauchy problem with final condition. Applications to the proof of verification theorems will be addressed in a companion paper.

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions