Second order backward stochastic differential equations under a monotonicity condition

In a recent paper, Soner, Touzi and Zhang (2012) [19] have introduced a notion of second order backward stochastic differential equations (2BSDEs), which are naturally linked to a class of fully non-linear PDEs. They proved existence and uniqueness for a generator which is uniformly Lipschitz in the variables y$y$ and z$z$. The aim of this paper is to extend these results to the case of a generator satisfying a monotonicity condition in y$y$. More precisely, we prove existence and uniqueness for 2BSDEs with a generator which is Lipschitz in z$z$ and uniformly continuous with linear growth in y$y$. Moreover, we emphasize throughout the paper the major difficulties and differences due to the 2BSDE framework.     

Some results on general quadratic reflected BSDEs driven by a continuous martingale

We study the well-posedness of general reflected BSDEs driven by a continuous martingale, when the coefficient f$f$ of the driver has at most quadratic growth in the control variable Z$Z$, with a bounded terminal condition and a lower obstacle which is bounded above. We obtain the basic results in this setting: comparison and uniqueness, existence, stability. For the comparison theorem and the special comparison theorem for reflected BSDEs (which allows one to compare the increasing processes of two solutions), we give intrinsic proofs which do not rely on the comparison theorem for standard BSDEs. This allows to obtain the special comparison theorem under minimal assumptions. We obtain existence by using the fixed point theorem and then a series of perturbations, first in the case where f$f$ is Lipschitz in the primary variable Y$Y$, and then in the case where f$f$ can have slightly-superlinear growth and the case where f$f$ is monotonous in Y$Y$ with arbitrary growth. We also obtain a local Lipschitz estimate in BMO$BMO$ for the martingale part of the solution.     

Quadratic reflected BSDEs with unbounded obstacles

In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator f$f$ has quadratic growth in the z$z$-variable. In particular, we obtain existence, uniqueness, and stability results, and consider the optimal stopping for quadratic g$g$-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient. Finally, we prove a comparison theorem for these obstacle problems when the generator is concave in the z$z$-variable.     

BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to real valued backward stochastic differential equations (BSDEs for short) with generators which satisfy a stochastic Lipschitz condition involving BMO martingales. This framework arises naturally when looking at the BSDE satisfied by the gradient of the solution to a BSDE with quadratic growth in Z$Z$. We first prove an existence and uniqueness result from which we deduce the differentiability with respect to parameters of solutions to quadratic BSDEs. Finally, we apply these results to prove the existence and uniqueness of a mild solution to a parabolic partial differential equation in Hilbert space with nonlinearity having quadratic growth in the gradient of the solution.     

Backward SDEs driven by Gaussian processes

In this paper we discuss existence and uniqueness results for BSDEs driven by centered Gaussian processes. Compared to the existing literature on Gaussian BSDEs, which mainly treats fractional Brownian motion with Hurst parameter H>1/2$H>1/2$, our main contributions are: (i) Our results cover a wide class of Gaussian processes as driving processes including fractional Brownian motion with arbitrary Hurst parameter H∈(0,1)$H\in (0,1)$; (ii) the assumptions on the generator f$f$ are mild and include e.g. the case when f$f$ has (super-)quadratic growth in z$z$; (iii) the proofs are based on transferring the problem to an auxiliary BSDE driven by a Brownian motion.     

Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition

This article deals with the existence and the uniqueness of solutions to quadratic and superquadratic Markovian backward stochastic differential equations (BSDEs) with an unbounded terminal condition. Our results are deeply linked with a strong a priori estimate on Z$Z$ that takes advantage of the Markovian framework. This estimate allows us to prove the existence of a viscosity solution to a semilinear parabolic partial differential equation with nonlinearity having quadratic or superquadratic growth in the gradient of the solution. This estimate also allows us to give explicit convergence rates for time approximation of quadratic or superquadratic Markovian BSDEs.     

Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation–G$G$-expectation–generated by a nonlinear heat equation with infinitesimal generator G$G$. We first study multi-dimensional G$G$-normal distributions. With this nonlinear distribution we can introduce our G$G$-expectation under which the canonical process is a multi-dimensional G$G$-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our G$G$-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G$G$-expectation.     

General existence results for reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. We deal with the case of a bounded terminal condition ξ and a bounded barrier L as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in Essaky and Hassani (submitted for publication) [14]. The work is suggested by the interest the results might have in finance, control and game theory.     

One-dimensional stochastic differential equations with generalized and singular drift

Introducing certain singularities, we generalize the class of one-dimensional stochastic differential equations with so-called generalized drift. Equations with generalized drift, well-known in the literature, possess a drift that is described by the semimartingale local time of the unknown process integrated with respect to a locally finite signed measure ν$\nu$. The generalization which we deal with can be interpreted as allowing more general set functions ν$\nu$, for example signed measures which are only σ$\sigma$-finite. However, we use a different approach to describe the singular drift. For the considered class of one-dimensional stochastic differential equations, we derive necessary and sufficient conditions for existence and uniqueness in law of solutions.     

Backward stochastic differential equations associated to jump Markov processes and applications

In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X$X$ on a general state space K$K$. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K$K$, that generalize the Kolmogorov equation of X$X$. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton–Jacobi–Bellman equation and to identify it with the value function.     

A simple constructive approach to quadratic BSDEs with or without delay

This paper provides a simple approach for the consideration of quadratic BSDEs with bounded terminal conditions. Using solely probabilistic arguments, we retrieve the existence and uniqueness result derived via PDE-based methods by Kobylanski (2000) [14]. This approach is related to the study of quadratic BSDEs presented by Tevzadze (2008) [19]. Our argumentation, as in Tevzadze (2008) [19], highly relies on the theory of BMO martingales which was used for the first time for BSDEs by Hu et al. (2005) [12]. However, we avoid in our method any fixed point argument and use Malliavin calculus to overcome the difficulty. Our new scheme of proof allows also to extend the class of quadratic BSDEs, for which there exists a unique solution: we incorporate delayed quadratic BSDEs, whose driver depends on the recent past of the Y$Y$ component of the solution. When the delay vanishes, we verify that the solution of a delayed quadratic BSDE converges to the solution of the corresponding classical non-delayed quadratic BSDE.     

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h$h$–ε$\epsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.     

Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.     

On the Wiener–Hopf factorization for Lévy processes with bounded positive jumps

We study the Wiener–Hopf factorization for Lévy processes with bounded positive jumps and arbitrary negative jumps. We prove that the positive Wiener–Hopf factor can be expressed as an infinite product involving solutions to the equation ψ(z)=q$\psi \left(z\right)=q$, where ψ$\psi$ is the Laplace exponent. Under additional regularity assumptions on the Lévy measure we obtain an asymptotic expression for these solutions. When the process is spectrally negative with bounded jumps, we derive a series representation for the scale function. In order to illustrate possible applications, we discuss the implementation of numerical algorithms and present the results of several numerical experiments.     

The stochastic wave equation with fractional noise: A random field approach

We consider the linear stochastic wave equation with spatially homogeneous Gaussian noise, which is fractional in time with index H>1/2$H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in Dalang (1999) [10], where the noise is white in time. Under this condition, we show that the solution is L²(Ω)$L^2\left(\Omega \right)$-continuous. Similar results are obtained for the heat equation. Unlike in the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.     

An Itô–Stratonovich formula for Gaussian processes: A Riemann sums approach

The aim of this paper is to establish a change of variable formula for general Gaussian processes whose covariance function satisfies some technical conditions. The stochastic integral is defined in the Stratonovich sense using an approximation by middle point Riemann sums. The change of variable formula is proved by means of a Taylor expansion up to the sixth order, and applying the techniques of Malliavin calculus to show the convergence to zero of the residual terms. The conditions on the covariance function are weak enough to include processes with infinite quadratic variation, and we show that they are satisfied by the bifractional Brownian motion with parameters (H,K)$(H,K)$ such that 1/6<HK<1$1/6<HK<1$, and, in particular, by the fractional Brownian motion with Hurst parameter H∈(1/6,1)$H\in (1/6,1)$.     

Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum

In this article, we consider an m$m$-dimensional stochastic differential equation with coefficients which depend on the maximum of the solution. First, we prove the absolute continuity of the law of the solution. Then we prove that the joint law of the maximum of the i$i$th component of the solution and the i^′$i^{\prime }$th component of the solution is absolutely continuous with respect to the Lebesgue measure in a particular case. The main tool to prove the absolute continuity of the laws is Malliavin calculus.     

Functions of bounded variation on the classical Wiener space and an extended Ocone–Karatzas formula

We prove an extension of the Ocone–Karatzas integral representation, valid for all BV$BV$ functions on the classical Wiener space. We also establish an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of BV$BV$ composite random variables.     

Some properties of the Itô–Wiener expansion of the solution of a stochastic differential equation and local times

In this paper, we use the formula for the Itô–Wiener expansion of the solution of the stochastic differential equation proven by Krylov and Veretennikov to obtain several results concerning some properties of this expansion. Our main goal is to study the Itô–Wiener expansion of the local time at the fixed point for the solution of the stochastic differential equation in the multidimensional case (when standard local time does not exist even for Brownian motion). We show that under some conditions the renormalized local time exists in the functional space defined by the _L2$L_2$-norm of the action of some smoothing operator.