We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we prove a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality.
There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of general Itô processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz con…
We prove a stochastic Gronwall lemma of the following type: if $Z$ is an adapted nonnegative continuous process which satisfies a linear integral inequality with an added continuous local martingale $M$ and a process $H$ on the right hand side, then for any $p \in (0,1)$ the $p$-th moment of the supremum of $Z$ is bounded by a constant $κ_p$ (which does not depend on $M$) times the $p$-th moment of the supremum of $H$. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical …
We deal with reflected solutions of anticipated backward doubly stochastic differential equations (RABDSDEs) driven by Teugels martingales associated with Lévy process under a Lipschitz generator where the coefficients of these BDSDEs depend on the future and present value of the solution $\left( Y,Z\right) $. Also we study the existence of a solution for anticipated BDSDEs.
The primary goal of this paper is to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. We prove the existence and uniqueness of solutions using two approaches: (1) Ayed-Kuo differential formula using an ansatz, and (2) a novel braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin-Wentzell type large deviations result for solution of such equations.
We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward pr…
We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard's iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingales. We work with a new outer measure which assigns zero value exactly to those properties which are instantly blockable.
In this paper, we propose a new notion of Forward--Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward--backward stochastic differential equations (FBSDEs). The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of an FBSDE. We first prove a general sufficient condition for the existence of the solution to the FBMP. In the Markovian case with uniformly continuous coefficients, we show that the weak solution to the FBSDE (or equivalently, the solution t…
By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.
The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation.
We study a class of stochastic differential equations driven by a possibly tempered L{é}vy process, under mild conditions on the coefficients. We prove the well-posedness of the associated martingale problem as well as the existence of the density of the solution. Two sided heat kernel estimates are given as well. Our approach is based on the Parametrix series expansion
A general maximum principle (necessary and sufficient conditions) for an optimal control problem governed by a stochastic differential equation driven by an infinite dimensional martingale is established. The solution of this equation takes its values in a separable Hilbert space and the control domain need not be convex. The result is obtained by using the adjoint backward stochastic differential equation.
We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak H{ö}rmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.
We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.
In this paper, a class of reflected generalized backward doubly stochastic differential equations (reflected GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of reflected stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.
Let $A$ be a pseudo-differential operator with symbol $q(x,ξ)$. In this paper we derive sufficient conditions which ensure the existence of a solution to the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem. If the symbol $q$ depends continuously on the space variable $x$, then the existence of solutions is well understood, and therefore the focus lies on martingale problems for pseudo-differential operators with discontinuous coefficients. We prove an existence result which allows us, in particular, to obtain new insights on the existence of weak solutions to a class of Lévy-driven SDEs wi…
The Bessel process in low dimension (0 $\le$ $δ$ $\le$ 1) is not an It{ô} process and it is a semimartingale only in the cases $δ$ = 1 and $δ$ = 0. In this paper we first characterize it as the unique solution of an SDE with distributional drift or more precisely its related martingale problem. In a second part, we introduce a suitable notion of path-dependent Bessel processes and we characterize them as solutions of path-dependent SDEs with distributional drift.
We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.
In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with Lévy process (RGBDSDELs, in short) with one continuous barrier. Under uniformly Lipschitz coefficients, we prove existence and uniqueness result by means of the penalization method and the fixed point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs, in short) with a nonlinear Neumann boundary condition.
The paper estimates the rate of convergence of the weak Euler approximation for the solutions of SDEs with Hoelder continuous coefficients driven by point and martingale measures. The equation considered has a non-degenerate main part whose jump intensity measure is absolutely continuous with respect to the Levy measure of a spherically-symmetric stable process. It includes the nondegenerate diffusions and SDEs driven by Levy processes.
A Milstein-type scheme was proposed to improve the rate of convergence of its approximation of the solution to a stochastic differential equation driven by a vector of continuous semimartingales. A necessary and sufficient condition was provided for this rate to be $1/n$ when the SDE is driven by a vector of continuous local martingales, or continuous semimartingales under an additional assumption on their finite variation part. The asymptotic behavior (weak convergence) of the normalized error processes was also studied.
Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性期望前沿科学中心（教育部））；Shandong Province Key Laboratory of Financial Risk, Shandong University（山东大学山东省金融风险重点实验室）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）
This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $g$ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $y$, and a stochastic-Lipschitz condition in the state variable $z$. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in \citet{XiaoFan2017Stochastics} and \citet{LiuLiFan2019CAM}. In particular, we put forward and prove…
We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the Hörmander hypoellipticity condition. In the proof we use global $L^p$-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti-Cupini-Lanconelli-Priola (Math. Z. 266 (2010)) and adapt the localization procedure introduced by Stroock and Va…
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in \cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non dens…
By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker--Planck equations for probability measures $(μ_t)_{t \geq 0}$ on the path space $\mathcal C:=C([-r_0,0];\mathbb R^d),$ is analyzed: $$\partial_t μ(t)=L_{t,μ_t}^*μ_t,\ \ t\ge 0,$$ where $μ(t)$ is the image of $μ_t$ under the projection $\mathcal C\niξ\mapsto ξ(0)\in\mathbb R^d$, and $$L_{t,μ}(ξ):= \frac 1 2\sum_{i,j=1}^d a_{ij}(t,ξ,μ)\frac{\partial^2} {\partial_{ξ(0)_i} \partial_{ξ(0)_j}} +\sum_{i=1}^d b_i(t,ξ,μ)\frac{\partial}{\partial_{ξ(0)_i}},\ \ t\ge 0, ξ\in \mathcal C, μ\…
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. This condition is satisfied by the Brownian motion and the fractional Brownian motion with Hurst parameter greater than 1/4. As application, we obtain rather flexible results such as support theorems, large deviation princ…
In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is componentwise nondeacreasing. Our approach consists of showing the result for bounded and componentwise nondecreasing drift using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by product, we obtain the existence of a unique strong solution of mult…
This paper focuses on the time-changed Q-Wiener process, a Hilbert space-valued sub-diffusion. It is a martingale with respect to an appropriate filtration, hence a stochastic integral with respect to it is definable. For the resulting integral, two change of variables formulas are derived. Via a duality theorem for integrals, existence and uniqueness theorems for stochastic differential equations (SDEs) driven by the time-changed Q-Wiener process are discussed. Associated fractional Fokker-Planck-Kolmogorov equations are derived using either a time-changed Itô formula or duality. Connections …
We consider Mc Kean-Vlasov stochastic differential equations (MVSDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. This type of SDEs was studied in statistical physics and represents the natural setting for stochastic mean-field games. We will first discuss questions of existence and uniqueness of solutions under an Osgood type condition improving the well known Lipschitz case. Then we derive various stability properties with respect to initial data, coefficients and driving processes, g…
Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process
Graduate School of Science, Kyoto University（京都大学大学院理学研究科）；Department of Business Economics, School of Management, Tokyo University of Science（东京理科大学经营学部商业经济学科）
We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingal…
A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process -, strong Markov processes and Gaussian processes. Using the Burkholder-Davis-Gundy inequalities we als discuss some relation…
We show that solutions of free stochastic differential equations with regular drifts and diffusion coefficients, when considered backwards in time, still satisfy free SDEs for an explicit free Brownian motion and drift. We also study the stochastic integral part with respect to this reversed free Brownian motion of canonical closed martingales. We deduce that conjugate variables computed along a free Brownian motion, an example of such a reversed martingale appearing in the definition of non-microstates free entropy, are in the $L^2$ domain of corresponding free difference quotients for almost…
We investigate the (functional) convex order of for various continuous martingale processes, either with respect to their diffusions coefficients for Lévy-driven SDEs or their integrands for stochastic integrals. Main results are bordered by counterexamples. Various upper and lower bounds can be derived for path wise European option prices in local volatility models. In view of numerical applications, we adopt a systematic (and symmetric) methodology: (a) propagate the convexity in a {\em simulatable} dominating/dominated discrete time model through a backward induction (or linear dynamical pr…
