A coupling by reflection of a time-inhomogeneous diffusion process on a manifold are studied. The condition we assume is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow. As in time-homogeneous cases, our coupling provides a gradient estimate of the diffusion semigroup which yields the strong Feller property. To construct the coupling via discrete approximation, we establish the convergence in law of geodesic random walks as well as a uniform non-explosion type estimate.
In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained via a penalization method.
The present paper is devoted to the study of the well-posedness of mean field BSDEs with mean reflection and nonlinear resistance. By the contraction mapping argument, we first prove that the mean-field BSDE with mean reflection and nonlinear resistance admits a unique deterministic flat local solution on a small time interval. Moreover, we build the global solution by introducing a two-step approach, which is a combination of stitching method and fixed point method. We further provide an application to the super-hedging problem with risk constraint.
This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.
The aim of this short note is to fill in a gap in our earlier paper [16] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [15]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [13, 23] in the so--called $G-$framework.
Let $\{g_t\}_{t\in [0,T)}$ be a family of complete time-depending Riemannian matrics on a manifold which evolves under backwards Ricci flow. The Itô formula is established for the $\mathcal{L}$-distance of the $g_t$-Brownian motion to a fixed reference point ($\mathcal{L}$-base). Furthermore, as an application, we construct a coupling by parallel displacement which yields a new proof of some results of Topping.
We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as $Y$ by convention, but in terms of its conditional expectation $\mathbb{E}[Y_t|\mathcal{G}_{t}]$ on a general sub-filtration $\{\mathcal{G}_{t}\}.$ We thus term such equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: …
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our main result yields the equivalence of (i) dynamic convexity of the Boltzmann entropy on the (time-dependent) $L^2$-Wasserstein space; (ii) monotonicity of $L^2$-Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time); (iii) gradi…
We study solutions of a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We prove existence and uniqueness by a fixed point argument and we derive a comparison result. Moreover, we show how the solution of our problem is related to a time-inconsistent optimal stopping problem and derive an optimal strategy.
This paper studies a system of multi-dimensional reflected backward stochastic differential equations with oblique reflections (RBSDEs for short) in infinite horizon associated to switching problems. The existence and uniqueness of the adapted solution is obtained by using a method based oa combination of penalization, verification method and contraction property.
We consider systems of backward stochastic differential equations with càdlàg upper barrier $U$ and oblique reflection from below driven by an increasing continuous function $H$. Our equations are defined on general probability spaces with a filtration satisfying merely the usual assumptions of right continuity and completeness. We assume that the pair $(H(U),U)$ satisfies a Mokobodzki--type condition. We prove the existence of a solution for integrable terminal conditions and integrable quasi--monotone generators. Applications to the optimal switching problem are given.
Mean-field BSDEs with non-Lipschitz coefficients and double mean reflections
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究院）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性预期前沿科学中心）
Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretat…
We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pathwise contraction result established by Arnaudon, Coulibaly and Thalmaier (arXiv:0904.2762, to appear in Sem. Prob.). As an important tool which is of independent interest we derive an Ito formula for the distance from a fixed reference point, generalising a result of Kendall (An…
We prove existence and uniqueness of solutions of reflected backward stochastic differential equations in time-dependent adapted and càdlàg convex regions $\mathcal{D}=\{D_t;t\in[0,T]\}$. We also show that the solution may be approximated by solutions of backward equations with reflection in appropriately defined discretizations of $\mathcal{D}$ and by a modified penalization method. The approximation results are new even in the one-dimensional case.
We study an approximation by time-discretized geodesic random walks of a diffusion process associated with a family of time-dependent metrics on manifolds. The condition we assume on the metrics is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow, and no further a priori curvature bound is required. As an application, we construct a coupling by reflection which yields a nice estimate of coupling time, and hence a gradient estimate for the associated semigroups.
In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equation in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining \emph{a priori} estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both $Y$ and $Z$. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and $Y$. W…
In this paper we study multi-dimensional reflected backward stochastic differential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.
This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the components of the solution. We study the existence and uniqueness of a minimal solution for these so-called constrained BSDEs with jumps via a penalization procedure. This new type of BSDE offers a nice and practical unifying framework to the notions of constrained BSDEs presented in [19] and BSDEs with constrained jumps introduced in [14]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [11…
We consider a backward stochastic differential equation with a generator that can be subjected to delay, in the sense that its current value depends on the weighted past values of the solutions, for instance a distorted recent average. Existence and uniqueness results are provided in the case of possibly infinite time horizon for equations with, and without reflection. Furthermore, we show that when the delay vanishes, the solutions of the delayed equations converge to the solution of the equation without delay. We argue that these equations are naturally linked to forward backward systems, an…
In this paper, we study the solvability of a class of multi-dimensional forward backward stochastic differential equations (FBSDEs) with oblique reflection and unbounded stopping time. Under some mild assumptions on the coefficients in such FBSDE, the existence result of adapted solutions is done via a penalization method. The uniqueness is obtained by a verification theorem similarly to the one used by Hu and Tang \cite{HT10}. Finally, we establish the connection with the corresponding optimal switching problem. This latter is solved by using the previous results on FBSDEs.
We study four systems and their interactions. First, we formulate a unified system of coupled forward-backward stochastic partial differential equations (FB-SPDEs) with Levy jumps, whose drift, diffusion, and jump coefficients may involve partial differential operators. A solution to the FB-SPDEs is defined by a 4-tuple general dimensional random vector-field process evolving in time together with position parameters over a domain (e.g., a hyperbox or a manifold). Under an infinite sequence of generalized local linear growth and Lipschitz conditions, the well-posedness of an adapted 4-tuple st…
