BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples. - Archive ouverte HAL Access content directly
Journal Articles Journal of Theoretical Probability Year : 2021

BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.

Abstract

Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish space,defined on the canonical probability space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator.
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Dates and versions

hal-01505974 , version 1 (12-04-2017)
hal-01505974 , version 2 (05-04-2020)
hal-01505974 , version 3 (26-11-2020)
hal-01505974 , version 4 (10-05-2021)

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Adrien Barrasso, Francesco Russo. BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.. Journal of Theoretical Probability, 2021, 34, pp.1110-1148. ⟨10.1007/s10959-021-01092-7⟩. ⟨hal-01505974v4⟩
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