arXiv subjects

Maurycy Rzymowski

Publications and source records attributed to Maurycy Rzymowski.

9 recordsLinked to original sources

arXiv2026

A Quantitative Characterization of the Mokobodzki Condition

We establish a quantitative characterization of the Mokobodzki condition for two adapted càdlàg barriers on an arbitrary filtered probability space. For every $p\geq1$, we introduce a mean co-variation functional $Var_p(L,U)$, which for $p=1$ and $L=U$ reduces to Rao's mean variation, and show that it is quantitatively equivalent to the minimal $\underline H^p$-norm among all semimartingales lying between the barriers. The result covers both the case $p>1$ and the endpoint $p=1$, where the natural scale is based on Doob's class $(D)$.

math.PR↗

arXiv2026

Mean-Field Doubly Reflected Forward-Backward SDEs with Optional Barriers and $L^p$-Data

We study mean-field doubly reflected forward-backward stochastic differential equations with two optional barriers satisfying a strong Mokobodzki condition. For $L^p$-data, $p\in(1,2]$, we prove existence and uniqueness on sufficiently short time horizons when the coefficients may depend on the joint law of $(X,Y,Z)$. Under an additional monotonicity condition and using an exponentially weighted norm, we also obtain a global-in-time result for $p=2$. The setting is motivated by recursive mean-field Dynkin games and game-option valuation with irregular payoff barriers.

math.PR↗

arXiv2024

Mokobodzki's intervals: an approach to Dynkin games when value process is not a semimartingale

We study Dynkin games governed by a nonlinear $\mathbb E^f$-expectation on a finite interval $[0,T]$, with payoff càdlàg processes $L,U$ of class (D) which are not imposed to satisfy (weak) Mokobodzki's condition - the existence of a càdlàg semimartingale between the barriers. For that purpose we introduce the notion of Mokobodzki's stochastic intervals $\mathscr M(θ)$ (roughly speaking, maximal stochastic interval on which Mokobodzki's condition is satisfied when starting from the stopping time $θ$) and the notion of reflected BSDEs without Mokobodzki's condition (this is a generalization and modification of the notion introduced by Hamadéne and Hassani (2005)). We prove an existence and uniqueness result for RBSDEs with driver $f$ that is non-increasing with respect to the value variable (no restrictions on the growth) and Lipschitz continuous with respect to the control variable, and with data in $L^1$ spaces. Next, by using RBSDEs, we show numerous results on Dynkin games: existence of the value process, saddle points, and convergence of the penalty scheme. We also show that the game is not played beyond $\mathscr M(θ)$, when starting from $θ$.

math.PR↗

arXiv2022

Nonlinear BSDEs with two optional Doob's class barriers satisfying weak Mokobodzki's condition and extended Dynkin games

We study reflected backward stochastic differential equation (RBSDEs) on the probability space equipped with a Brownian motion. The main novelty of the paper lies in fact that we consider the following weak assumptions on the data: barriers are optional of class (D) satisfying weak Mokobodzki's condition, generator is continuous and non-increasing with respect to the value-variable (no restriction on the growth) and Lipschitz continuous with respect to the control-variable, and the terminal condition and the generator at zero are supposed to be merely integrable. We prove that under these conditions on the data there exists a solution to corresponding RBSDE. In the second part of the paper, we apply the theory of RBSDEs to solve basic problems in Dynkin games driven by nonlinear expectation based on the generator mentioned above. We prove that the main component of a solution to RBSDE represents the value process in corresponding extended nonlinear Dynkin game. Moreover, we provide sufficient condition on the barriers guaranteeing the existence of the value for nonlinear Dynkin games and the existence of a saddle point.

math.PR↗

arXiv2022

A priori estimates for multidimensional BSDEs with integrable data

We study Backward Stochastic Differential Equations on a probability space equipped with a Brownian filtration. We assume that the terminal value and the generator at zero are merely integrable. Moreover, the generator is assumed to be non-increasing with respect to the value variable (with no restrictions on the growth) and Lipschitz continuous, with sublinear growth, with respect to the control variable. We provide a priori estimate and stability result for solutions to the aforementioned BSDEs.

math.PR↗

arXiv2021

Nonlinear BSDEs in general filtration with drivers depending on the martingale part of a solution

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part $M$ of a solution. We assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on "the future" of the process $M$ as well as a dependence of its law (McKean-Vlasov-type equations). For such driver, we prove the existence and uniqueness of a global solution (i.e. for any maturity $T>0$) to BSDE with data satisfying natural integrability conditions.

math.PR↗

arXiv2020

Reflected BSDEs with two optional barriers and monotone coefficient on general filtered space

We consider reflected backward stochastic differential equations with two optional barriers of class (D) satisfying Mokobodzki's separation condition and coefficient which is only continuous and non-increasing. We assume that data are merely integrable and the terminal time is an arbitrary (possibly infinite) stopping time. We study the problem of existence and uniqueness of solutions, and their connections with the value process in nonlinear Dynkin games.

math.PR↗

arXiv2018

Reflected backward stochastic differential equations with two optional barriers

We consider reflected backward stochastic differential equations with two general optional barriers. The solutions to these equations have the so-called regulated trajectories, i.e trajectories with left and right finite limits. We prove the existence and uniqueness of $\mathbb L^p$ solutions, $p\geq 1$, and show that the solutions may be approximated by a modified penalization method.

math.PR↗

arXiv2016

Reflected BSDEs with regulated trajectories

We consider reflected backward stochastic different equations with optional barrier and so-called regulated trajectories, i.e trajectories with left and right finite limits. We prove existence and uniqueness results. We also show that the solution may be approximated by a modified penalization method. Application to an optimal stopping problem is given.

math.PR↗