Solutions Of Backward Stochastic Differential Equations Research Articles

A linear-quadratic mean-field game of backward stochastic differential equation with partial information and common noise

This paper studies a linear-quadratic (LQ) mean-field game of stochastic large-population system with partial information and common noise, where the large-population system satisfies a class of backward stochastic differential equations (BSDEs), and both coupling structure and ki (a part of solution of BSDE) enter state equations and cost functionals. By virtue of stochastic maximum principle and optimal filter technique, we obtain a Hamiltonian system first, which is a fully-coupled forward-backward stochastic differential equation (FBSDE). Decoupling the Hamiltonian system, we obtain three ordinary differential equations (ODEs), a forward and a backward optimal filtering equations, which enable us to derive an optimal control of an auxiliary limiting control problem. Next, we verify that a decentralized control strategy is an ϵ-Nash equilibrium of the LQ game. Finally, we solve a product pricing problem and a network security control problem in applications. We give a near-optimal price strategy and a near-optimal control strategy with numerical simulations, respectively.

Backward stochastic differential equations with Markov chains and associated PDEs

In this paper we study the regularity of the solutions for backward stochastic differential equations (BSDEs) with finite state Markov chains and establish its link with associated partial differential equations (PDEs) in classical sense. Moreover, we study the existence and uniqueness of solutions for such BSDEs under Lipschitz conditions on f in the space Lρ2(Rd;Rk)⊗Lρ2(Rd;Rd×k)⊗Lρ2(Rd×I;Rk). In this way, we establish a new connection between Lρ2(Rd;Rk)⊗Lρ2(Rd;Rd×k)⊗Lρ2(Rd×I;Rk) valued solutions of BSDEs and the solutions of PDEs in a Sobolev space.

Backward stochastic differential equations with two barriers and generalized reflection

We prove the existence and uniqueness of solutions of backward stochastic differential equations (BSDEs) with generalized reflection at time dependent càdlàg barriers. The reflection model we consider includes, as special cases, the standard reflection as well as the mirror reflection studied earlier in the theory of forward stochastic differential equations. We also show that the solution of BSDEs with generalized reflection corresponds to the value of an optimal stopping problem.

Lp Solutions of BSDEs with a New Kind of Non-Lipschitz Coefficients

In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) with a new kind of non-Lipschitz coefficients. We establish an existence and uniqueness result of the Lp (p > 1) solutions, which includes some known results as its particular cases.

Optimal investment-consumption and life insurance selection problem under inflation. A BSDE approach

We discuss an optimal investment, consumption and insurance problem of a wage earner under inflation. Assume a wage earner investing in a real money account and three asset prices, namely: a real zero-coupon bond, the inflation-linked real money account and a risky share described by jump-diffusion processes. Using the theory of quadratic-exponential backward stochastic differential equation (BSDE) with jumps approach, we derive the optimal strategy for the two typical utilities (exponential and power) and the value function is characterized as a solution of BSDE with jumps. Finally, we derive the explicit solutions for the optimal investment in both cases of exponential and power utility functions for a diffusion case.

An existence theorem for multidimensional BSDEs with mixed reflections

In this note, we consider the pricing problem for a type of real option, which gives the right to switch investment modes and abandon the investment project before its maturity. The value of this option can be characterized by solutions to multidimensional backward stochastic differential equations (BSDEs) with both normal and oblique reflections, whose coefficients are of linear growth and are left-Lipschitz with respect to (w.r.t) y and Lipschitz w.r.t. z. We provide an existence theorem of minimal solutions for BSDEs in this framework.

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

We design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares Monte-Carlo (LSMC) algorithms. The Radon–Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization; secondly, we develop refined concentration-of-measure techniques to capture the variance reduction. Our theoretical results are supported by numerical experiments.

A Random Parameter Model for Continuous-Time Mean-Variance Asset-Liability Management

We consider a continuous-time mean-variance asset-liability management problem in a market with random market parameters; that is, interest rate, appreciation rates, and volatility rates are considered to be stochastic processes. By using the theories of stochastic linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), we tackle this problem and derive optimal investment strategies as well as the mean-variance efficient frontier analytically in terms of the solution of BSDEs. We find that the efficient frontier is still a parabola in a market with random parameters. Comparing with the existing results, we also find that the liability does not affect the feasibility of the mean-variance portfolio selection problem. However, in an incomplete market with random parameters, the liability can not be fully hedged.

BSDE driven by Poisson point processes with discontinuous coefficient

In this paper, we deal with the one-dimensional backward stochastic differential equation (BSDE) driven by Poisson processes. By means of the comparison theorem, we first prove the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies an improved linear growth assumption. Then we extend the result to the case where the coefficient is left or right continuous.

Reducing variance in the numerical solution of BSDEs

Numerical methods based on time discretization and estimation of conditional expectations for solving backward stochastic differential equations (BSDEs) have been the object of considerable research, particularly in view of the applications to finance. We introduce and implement a simple control variate technique to reduce the simulation error of the conditional expectation estimates in BSDE methods. These modifications increase the accuracy of the existing algorithms without additional computational cost.

Formula omitted] solutions to backward stochastic differential equations with discontinuous generators

In this paper, we deal with the Lp(p>1) solutions to one dimensional backward stochastic differential equations (BSDEs) with discontinuous generators. We obtain an existence theorem of Lp solutions for BSDEs whose generators satisfy a kind of discontinuous condition in y and are uniformly continuous in z.

Exponential Utility Maximization in an Incomplete Market with Defaults

In this paper, we study the exponential utility maximization problem in an incomplete market with a default time inducing a discontinuity in the price of stock. We consider the case of strategies valued in a closed set. Using dynamic programming and BSDEs techniques, we provide a characterization of the value function as the maximal subsolution of a backward stochastic differential equation (BSDE) and an optimality criterium. Moreover, in the case of bounded coefficients, the value function is shown to be the maximal solution of a BSDE. Moreover, the value function can be written as the limit of a sequence of processes which can be characterized as the solutions of Lipschitz BSDEs in the case of bounded coefficients. In the case of convex constraints and under some exponential integrability assumptions on the coefficients, some complementary properties are provided. These results can be generalized to the case of several default times or a Poisson process.

A Generalized Comparison Theorem for BSDEs and Its Applications

This paper establishes a generalized comparison theorem for one-dimensional backward stochastic differential equations (BSDEs) whose generators are uniformly continuous in z and satisfy a kind of weakly monotonic condition in y. As applications, two new existence and uniqueness theorems for solutions of BSDEs are obtained. In the one-dimensional setting, these results generalize some corresponding results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Mao (Stoch. Process. Their Appl. 58:281–292, 1995), El Karoui et al. (Math. Finance 7:1–72, 1997), Pardoux (Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, Kluwer Academic, Dordrecht, 1999), Cao and Yan (Adv. Math. 28(4):304–308, 1999), Briand and Hu (Probab. Theory Relat. Fields 136(4):604–618, 2006), and Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008).

Backward SDEs with superquadratic growth

In this paper, we discuss the solvability of backward stochastic differential equations (BSDEs) with superquadratic generators. We first prove that given a superquadratic generator, there exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. Finally, we prove the existence of a solution for Markovian BSDEs where the terminal value is a bounded continuous function of a forward stochastic differential equation.

On the Continuity of Weak Solutions of Backward Stochastic Differential Equations

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.

Properties of Solutions of BSDEs with Integrable Parameters

The main result of this study is to obtain, using the localization method in Briand et al.[3]. Levi, Fatou and Lebesgue type theorems for the solutions of certain one-dimensional backward stochastic differential equation (BSDEs) with integrable parameters with respect to the terminal condition.

A comonotonic theorem for BSDEs

Pardoux and Peng (Systems Control Lett. 14 (1990) 55) introduced a class of nonlinear backward stochastic differential equations (BSDEs). According to Pardoux and Peng's theorem, the solution of this type of BSDE consists of a pair of adapted processes, say ( y , z ) . Since then, many researchers have been exploring the properties of this pair solution ( y , z ) , especially the properties of the first part y. In this paper, we shall explore the properties of the second part z . A comonotonic theorem with respect to z is obtained. As an application of this theorem, we prove an integral representation theorem of the solution of BSDEs.

Existence and Uniqueness of Solutions for BSDEs with Locally Lipschitz Coefficient

We deal with multidimensional backward stochastic differential equations (BSDE) with locally Lipschitz coefficient in both variables $ y,z $ and an only square integrable terminal data. Let $ L_N $ be the Lipschitz constant of the coefficient on the ball $ B(0,N) $ of $ R^d\times R^{dr} $. We prove that if $ L_N = O (\sqrt {\log N }) $, then the corresponding BSDE has a unique solution. Moreover, the stability of the solution is established under the same assumptions. In the case where the terminal data is bounded, we establish the existence and uniqueness of the solution also when the coefficient has an arbitrary growth (in $ y $) and without restriction on the behaviour of the Lipschitz constant $ L_N $.

Backward stochastic differential equations and partial differential equations with quadratic growth

We provide existence, comparison and stability results for one- dimensional backward stochastic differential equations (BSDEs) when the coefficient (or generator) $F(t,Y, Z)$ is continuous and has a quadratic growth in $Z$ and the terminal condition is bounded.e also give, in this framework, the links between the solutions of BSDEs set on a diffusion and viscosity or Sobolev solutions of the corresponding semilinear partial differential equations.

On solutions of backward stochastic differential equations with jumps and applications

For backward stochastic differential equation (BSDE) with jumps and with non-Lipschitzian coefficient the existence and uniqueness of an adapted solution is obtained. By generalizing the existence result on partial differential and integral equations (PDIE) and Ito formula to the functions with only first and second Sobolev derivatives the probabilistic interpretation for solutions of PDIE (a new Feynman-Kac formula) by means of solutions of BSDE with jumps is got. With the help of this formula a new existence and uniqueness result for the solution of PDIE with non-Lipschitzian force is obtained. The convergence theorems of solutions to BSDE and PDIE are also derived.