Backward Stochastic Difference Equations with Finite States

Considerably much work has been done on Backward Stochastic Differential Equations (BSDEs) in continuous-time with deterministic terminal horizon or stopping times. Various new models have been introduced in these years in order to generalize BSDEs to solve new practical financial problems.
\nOne strand is focused on discrete-time models. Backward Stochastic Difference Equations (also called BSDEs if no ambiguity) on discrete-time finite-state space were introduced by Cohen and Elliott (2010a). The associated theory required only weak assumptions. In the first topic of this thesis, properties of non-linear expectations defined using the discrete-time finite-state BSDEs were studied. A converse comparison theorem was established. Properties of risk measures defined by non-linear expectations, especially the representation theorems, were given. Then the theory of BSDEs was applied to optimal design of dynamic risk measures. Another strand is about a general random terminal time, which is not necessarily a stopping time. The motivation of this model is a financial problem of hedging of defaultable contingent claims, where BSDEs with stopping times are not applicable. In the second topic of this thesis, discrete-time finite-state BSDEs under progressively enlarged filtration were considered. Martingale representation
\ntheorem, existence and uniqueness theorem and comparison theorem were established. Application to nonlinear expectations was also explored. Using the theory of BSDEs, the explicit solution for optimal design of dynamic default risk measures was obtained.
\nIn recent work on continuous-time BSDEs under progressively enlarged filtration, the reference filtration is generated by Brownian motions. In order to deal with cases with jumps, in the third topic of this thesis, a general reference filtration with predictable representation property and an initial time with immersion property were considered. The martingale representation theorem for square-integrable martingales under progressively enlarged filtration was established. Then the existence and uniqueness theorem of BSDEs under enlarged filtration using Lipschitz continuity of the driver was proved. Conditions for a comparison theorem were also presented. Finally applications to nonlinear expectations and hedging of defaultable contingent claims on Brownian-Poisson setting were explored.

Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$\Delta$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$\Delta$Es and BSDEs are governed by drivers $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BS$\Delta$Es are based on d-dimensional random walks $W^N$ approximating the d-dimensional Brownian motion W underlying the BSDE and that $f^N$ converges to f. Conditions are given under which for any bounded terminal condition $\xi$ for the BSDE, there exist bounded terminal conditions $\xi^N$ for the sequence of BS$\Delta$Es converging to $\xi$, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when $f^N$ and f are convex in z. We show that in this situation, the solutions of the BS$\Delta$Es converge to the solution of the BSDE for every uniformly bounded sequence $\xi^N$ converging to $\xi$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^N,\xi^N)$ is close to $(W,\xi)$ in distribution, then the solution of the Nth BS$\Delta$E is close to the solution of the BSDE in distribution too.

Bounded solutions, [formula omitted] solutions and [formula omitted] solutions for one dimensional BSDEs under general assumptions

The present work is devoted to a study of the solvability of a class of non-Lipschitz and noncanonical backward stochastic differential equations (BSDEs) that naturally arises from an intertemporal mutual fund management problem; to this end, we propose a method of combining the techniques of Malliavin calculus and a discussion on the Jacobian flow of the BSDE. Specifically, based on the intimate relationship between $Y_t$ and $Z_t$ of the BSDE via the Malliavin derivative of the former, namely, $D_tY_t=Z_t$, we construct an iterative Picard converging scheme for approximating the underlying solution pair by first obtaining $Z_t$ from the derived BSDE with respect to the Malliavin derivative and then recovering $Y_t$ from the underlying BSDE. A local unique existence result is first warranted over a short time horizon with carefully examined a priori estimates; indeed, each term in the iterative sequence is related to different Girsanov transforms for change of measure, and comparing them demands a delicate analysis. The use of Jacobian flow further enables us to properly control the lower and upper bounds for a certain product of the forward process and $Z_t$, which enables us to extend the solution globally by an inductive argument. Our proposed method is fundamentally different from other probabilistic methods that also involve estimating or bounding Malliavin traces such as [E. Pardoux and S. Peng, Some Backward SDEs with Non-Lipschitz Coefficients, Technical note, Université de Provence, Aix-en-Provence, France, 1996] and [M. C. Zedouri, Equations Différentielles Stochastiques Rétrogrades avec Générateurs Lipschitiziens Stochastiques, Master's Thesis, Université Mohammed Seddik Ben Yahia-Jijel, Jijel, Algeria, 2010]. We believe that our new approach proposed here can be potentially applied to resolve many other general non-Lipschitz forward-backward stochastic differential equations (FBSDEs) encountered in economics and finance, especially in the presence of generic utility functions.

In the present paper, the notion of a weak solution of a general backward stochastic differential equation (BSDE), which was introduced by the authors and A. Răşcanu in [Theory Probab. Appl., 49 (2005), pp. 16–50], will be discussed. The relationship between continuity of solutions, pathwise uniqueness, uniqueness in law, and existence of a pathwise unique strong solution is investigated. The main result asserts that if all weak solutions of a BSDE are continuous, then the solution is pathwise unique. One should notice that this is a specific result for BSDEs and there is of course no counterpart for (forward) stochastic differential equations (SDEs). As a consequence, if a weak solution exists and all solutions are continuous, then there exists a pathwise unique solution and this solution is strong. Moreover, if the driving process is a continuous local martingale satisfying the previsible representation property, then the converse is also true. In other words, the existence of discontinuous solutions to a BSDE is a natural phenomenon, whenever pathwise uniqueness or, in particular, uniqueness in law is not satisfied. Examples of discontinuous solutions of a certain BSDE were already given in [R. Buckdahn and H.-J. Engelbert, Proceedings of the Fourth Colloquium on Backward Stochastic Differential Equations and Their Applications, to appear]. This was the motivation for the present paper which is aimed at exploring the general situation.

In this paper we investigate a class of backward stochastic differential equations (BSDE) whose terminal values are allowed to depend on the history of a forward diffusion. We first establish a probabilistic representation for the spatial gradient of the viscosity solution to a quasilinear parabolic PDE in the spirit of the Feynman--Kac formula, without using the derivatives of the coefficients of the corresponding BSDE. Such a representation then leads to a closed-form representation of the martingale integrand of a BSDE, under only a standard Lipschitz condition on the coefficients. As a direct consequence we prove that the paths of the martingale integrand of such BSDEs are at least c\\`{a}dl\\`{a}g, which not only extends the existing path regularity results for solutions to BSDEs, but contains the cases where existing methods are not applicable. The BSDEs in this paper can be considered as the nonlinear wealth processes appearing in finance models; our results could lead to efficient Monte Carlo methods for computing both price and optimal hedging strategy for options with nonsmooth, path-dependent payoffs in the situation where the wealth is possibly nonlinear.

Motivated by structural, reduced-form and hybrid models of the third party and counterparty credit risk, we study a generalized backward stochastic differential equations (BSDE) up to a random time horizon ϑ, which is not a stopping time with respect to a reference filtration. In contrast to the existing literature in the area of credit risk modeling, we do not impose specific assumptions on the random time ϑ and we study the existence of solutions to BSDE and reflected BSDE with a random time horizon through the method of reduction. For this purpose, we also examine BSDE and reflected BSDE with a làdlàg driver where the driver is allowed to have a finite number of jumps overlapping with jumps of the martingale part. Theoretical results are illustrated by particular instances of a random time and explicit BSDEs in either the Brownian or Brownian-Poisson filtration.

The theory of backward stochastic differential equations (BSDEs) was pioneered by Pardoux and Peng [PaPe90]. It became now very popular, and is an important field of research due to its connections with stochastic control, mathematical finance, and partial differential equations. BSDEs provide a probabilistic representation of nonlinear PDEs, which extends the famous Feynman-Kac formula for linear PDEs. As a consequence, BSDEs can be used for designing numerical algorithms to nonlinear PDEs.

This article is interested in solutions of general time interval backward stochastic differential equations (BSDEs) with generator g satisfying a p-order weak stochastic-monotonicity condition and a general growth condition in y. An existence and uniqueness result of the Lp solution is proved for the multidimensional BSDEs where the generator g is stochastic-Lipschitz in z. And, a general comparison theorem and an existence and uniqueness theorem of the Lp solution is established for the one-dimensional BSDEs under a stochastic uniform-continuity condition of the generator g in z. These results strengthen some known works. The stochastic Gronwall-type inequality and the stochastic Bihari-type inequality play an important role in the whole article.

In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a backward stochastic differential equation (BSDE) that it is shown to converge towards a reduced BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation is required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space.

Backward stochastic differential equations with reflection and weak assumptions on the coefficients

Suppose that a road network model is given, together with some given demand for travel by (say) car and that the demand for travel varies with time of day but not from day to day. Suppose that this demand is given in the form of specified total outflow rates from each origin headed towards each destination, for each origin-destination pair and for each time of day, and that some initial time-dependent routeinflow rates, meeting the given demand, are given. Finally, suppose that within-day time is represented by a continuous variable. This paper specifies a natural smooth day-to-day route-swapping procedure wherein drivers swap toward less expensive routes as day succeeds day, and shows that under reasonable conditions there is an equilibrium state of this dynamical system. If such a collection of route-inflows has arisen today, say, then there is no incentive for any route-inflow to change tomorrow, in the sense that at each moment of today each of today's route-inflows isalready on a route which today yielded the smallest travel cost. Such a set of “no-incentive-to-change” route-inflows is called adynamic equilibrium, or adynamic user-equilibrium, and may be regarded as a solution of the dynamic equilibrium traffic assignment problem. Thus, the paper introduces a smooth day-to-day dynamic assignment model and, using this model, shows that there is a dynamic user-equilibrium in a continuous time setting. The paper briefly considers the day-to-day stability of the route-swapping process, also in a continuous setting. Finally, the paper gives a simple dynamical example illustrating the stability of the route-swapping process in a simple two-route network when there is deterministic queueing at bottlenecks.

In this paper, we consider the problem of numerical solution of the system of forward– backward stochastic differential equations and its Cauchy problem for a quasilinear parabolic equation. We propose a layer method of solving the Cauchy problem with simultaneous estimation of the solution gradient, which is based on the probabilistic representation of the solution. We consider the ways of constructing layer methods for nonlinear boundary value problems. The system of stochastic differential equations is solved by the Euler method using the preliminarily obtained numerical solution of a quasilinear equation and its gradient.

This paper is concerned with backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure. The motivation is a constrained stochastic Riccati equation derived from a stochastic linear quadratic optimal control problem with both Poisson and Markovian jumps. The existence and uniqueness of an adapted solution under global Lipschitz condition on the generator is obtained. The continuous dependence of the solution on parameters is proved. Two comparison theorems are also derived by a generalized Girsanov transformation theorem.