A Note on Time-Dependent Additive Functionals

2019

Stoch. Dyn.

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The paper introduces and investigates the natural extension to the path-dependent setup of the usual concept of canonical Markov class introduced by Dynkin and which is at the basis of the theory of Markov processes. That extension, indexed by starting paths rather than starting points will be called path-dependent canonical class. Associated with this is the generalization of the notions of semi-group and of additive functionals to the pathdependent framework. A typical example of such family is constituted by the laws (P s,η ) (s,η)∈R+×Ω , where for fixed time s and fixed path η defined on [0, s], P s,η being the (unique) solution of a path-dependent martingale problem or more specifically a weak solution of a path-dependent SDE with jumps, with initial path η. In a companion paper we apply those results to study pathdependent analysis problems associated with BSDEs. MSC 2010 Classification. 60H30; 60H10; 35S05; 60J35; 60J75. KEY WORDS AND PHRASES. Path-dependent martingale problems; path-dependent additive functionals. * ENSTA ParisTech, Unité de Mathématiques appliquées, 828, boulevard des Maréchaux,

mentioning

confidence: 62%

Section: A Proofs Of Sectionmentioning

confidence: 74%

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Path-dependent martingale problems and additive functionals

2019

Stoch. Dyn.

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“…For details concerning the exact mathematical framework for our Markov process, we refer to our previous paper [7] about canonical Markov classes and additive functionals.…”

Section: Martingale Problem and Canonical Markov Classesmentioning

confidence: 99%

“…From now on, E is a Polish space and Ω, F , (X t ) t∈[0,T ] , (F t ) t∈[0,T ] denotes the canonical space defined in Notation 3.1 of [7]. We also fix a canonical Markov class (È s,x ) (s,x)∈[0,T ]×E associated to a transition kernel P = (P s,t ) measurable in time as defined in Definitions 3.4, 3.5 and 3.7 in [7].…”

Section: Martingale Problem and Canonical Markov Classesmentioning

confidence: 99%

Martingale driven BSDEs, PDEs and other related deterministic problems

Stochastic Processes and their Applications

2021

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We focus on a class of BSDEs driven by a càdlàg martingale and the corresponding Markovian BSDEs which arise when the randomness of the driver appears through a Markov process. To those BSDEs we associate a deterministic equation which, when the Markov process is a Brownian diffusion, is nothing else but a parabolic semi-linear PDE. We prove existence and uniqueness of a decoupled mild solution of the deterministic problem, and give a probabilistic representation of this solution through the aforementioned BSDEs.

Decoupled Mild Solutions of Path-Dependent PDEs and Integro PDEs Represented by BSDEs Driven by Cadlag Martingales

2019

Potential Anal

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We focus on a class of path-dependent problems which include path-dependent (possibly Integro) PDEs, and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the afore mentioned BSDEs. This concept generalizes a similar notion introduced by the authors in previous papers in the framework of classical PDEs and IPDEs. For every initial condition (s, η), where s is an initial time and η an initial path, the solution of such BSDE produces a couple of processes (Y s,η , Z s,η ). In the classical (Markovian or not) literature the function u(s, η) := Y s,η s constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify u as (in our language) the unique decoupled mild solution, but also to solve quite generally the so called identification problem, i.e. to also characterize the (Z s,η ) s,η processes in term of a deterministic function v associated to the (above decoupled mild) solution u.MSC 2010 Classification. 60H30; 60H10; 35D99; 35S05; 60J35; 60J75.