Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations

In this paper we study stochastic optimal control problems with jumps with the help of the theory of Backward Stochastic Differential Equations (BSDEs) with jumps. We generalize the results of Peng [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L. Wu, Topics in Stochastic Analysis, Science Press, Beijing, 1997 (Chapter 2) (in Chinese)] by considering cost functionals defined by controlled BSDEs with jumps. The application of BSDE methods, in particular, the use of the notion of stochastic backward semigroups introduced by Peng in the above-mentioned work allows a straightforward proof of a dynamic programming principle for value functions associated with stochastic optimal control problems with jumps. We prove that the value functions are the viscosity solutions of the associated generalized Hamilton–Jacobi–Bellman equations with integral-differential operators. For this proof, we adapt Peng’s BSDE approach, given in the above-mentioned reference, developed in the framework of stochastic control problems driven by Brownian motion to that of stochastic control problems driven by Brownian motion and Poisson random measure.     

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.     

Probabilistic approach to homogenization of a non-divergence form semilinear PDE with non-periodic coefficients

We consider a semilinear partial differential equation (PDE) of non-divergence form perturbed by a small parameter. We then study the asymptotic behavior of Sobolev solutions in the case where the coefficients admit limits in C?esaro sense. Neither periodicity nor ergodicity will be needed for the coefficients. In our situation, the limit (or averaged or effective) coefficients may have discontinuity. Our approach combines both probabilistic and PDEs arguments. The probabilistic one uses the weak convergence of solutions of backward stochastic differential equations (BSDE) in the Jakubowski S-topology, while the PDEs argument consists to built a solution, in a suitable Sobolev space, for the PDE limit. We finally show the existence and uniqueness for the associated averaged BSDE, then we deduce the uniqueness of the limit PDE from the uniqueness of the averaged BSDE.     

Mean-Field Backward Stochastic Differential Equations Driven by Fractional Brownian Motion

In this paper, we study a new class of equations called mean-field backward stochastic differential equations(BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H 1/2. First, the existence and uniqueness of this class of BSDEs are obtained. Second, a comparison theorem of the solutions is established. Third, as an application, we connect this class of BSDEs with a nonlocal partial differential equation(PDE, for short), and derive a relationship between the fractional mean-field BSDEs and PDEs.     

Linear forward-backward stochastic differential equations with random coefficients

Solvability of linear forward-backward stochastic differential equations (FBSDEs, for short) with random coefficients is studied. A decoupling reduction method is introduced via which a large class of linear FBSDEs with random or deterministic time-varying coefficients is proved to be solvable. On the other hand, by means of Four Step Scheme, a Riccati backward stochastic equation (BSDE, for short) for (m×n) matrix-valued processes is derived. Global solvability of such Riccati BSDEs is discussed for some special (but nontrivial) cases, which leads to the solvability of the corresponding linear FBSDEs. This work is supported in part by the NSFC, under grant 10131030, the Chinese Education Ministry Science Foundation under grant 2000024605, the Cheung Kong Scholars Programme, and Shanghai Commission of Science and Technology under grant 02DJ14063.     

Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs

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 interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs.     

Stationary backward stochastic differential equations and associated partial differential equations

By replacing the final condition for backward stochastic differential equations (in short: BSDEs) by a stationarity condition on the solution process we introduce a new class of BSDEs. In a natural manner we associate to such BSDEs the periodic solution of second order partial differential equations with periodic structure. Received: 11 October 1996 / Revised version: 15 February 1999     

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.     

Lp solutions of backward stochastic differential equations

In this paper, we are interested in solving backward stochastic differential equations (BSDEs for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic calculus related to BSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in L^p p>1, extending the results of El Karoui et al. (Math. Finance 7(1) (1997) 1) to the case where the monotonicity conditions of Pardoux (Nonlinear Analysis; Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht, pp. 503–549) are satisfied. We consider both a fixed and a random time interval. In the last section, we obtain, under an additional assumption, an existence and uniqueness result for BSDEs on a fixed time interval, when the data are only in L¹.     

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.     

BSDES IN GAMES,COUPLED WITH THE VALUE FUNCTIONS. ASSOCIATED NONLOCAL BELLMAN-ISAACS EQUATIONS

We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs' condition.     

Second order backward stochastic differential equations with quadratic growth

We extend the well posedness results for second order backward stochastic differential equations introduced by Soner, Touzi and Zhang (2012)  [31] to the case of a bounded terminal condition and a generator with quadratic growth in the z$z$ variable. More precisely, we obtain uniqueness through a representation of the solution inspired by stochastic control theory, and we obtain two existence results using two different methods. In particular, we obtain the existence of the simplest purely quadratic 2BSDEs through the classical exponential change, which allows us to introduce a quasi-sure version of the entropic risk measure. As an application, we also study robust risk-sensitive control problems. Finally, we prove a Feynman–Kac formula and a probabilistic representation for fully non-linear PDEs in this setting.     

Reflected BSDEs with nonpositive jumps,and controller-and-stopper games

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution are proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman–Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affects both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.     

Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models

Over the past few years quadratic Backward Stochastic Differential Equations (BSDEs) have been a popular field of research. However there are only very few examples where explicit solutions for these equations are known. In this paper we consider a class of quadratic BSDEs involving affine processes and show that their solution can be reduced to solving a system of generalized Riccati ordinary differential equations. In other words we introduce a rich and flexible class of quadratic BSDEs which are analytically tractable, i.e. explicit up to the solution of an ODE. Our results also provide analytically tractable solutions to the problem of utility maximization and indifference pricing in multivariate affine stochastic volatility models. This generalizes univariate results of Kallsen and Muhle-Karbe (2010) and some results in the multivariate setting of Leippold and Trojani (2010) by establishing the full picture in the multivariate affine jump-diffusion setting. In particular we calculate the interesting quantity of the power utility indifference value of change of numeraire. Explicit examples in the Heston, Barndorff-Nielsen–Shephard and multivariate Heston setting are calculated.     

A Relationship Between the Conditional <Emphasis Type="Italic">g</Emphasis>-Evaluation System and the Generator <Emphasis Type="Italic">g</Emphasis> and Its Applications

In this paper, under the most elementary conditions on a backward stochastic differential equation （BSDE for short） introduced by Peng, a new relationship between the conditional g-evaluation system and the generator g of BSDE is obtained in the sense of ＂process＂, based on some recent results of Jiang. Moreover, as applications, two converse comparison theorems and two uniqueness theorems on the generators of BSDEs are proved.     

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.     

BSDEs under partial information and financial applications

In this paper we provide existence and uniqueness results for the solution of BSDEs driven by a general square-integrable martingale under partial information. We discuss some special cases where the solution to a BSDE under restricted information can be derived by that related to a problem of a BSDE under full information. In particular, we provide a suitable version of the Föllmer–Schweizer decomposition of a square-integrable random variable working under partial information and we use this achievement to investigate the local risk-minimization approach for a semimartingale financial market model.     

Continuous dependence properties on solutions of backward stochastic differential equation

The existence theorem and continuous dependence property in ”L²” sense for solutions of backward stochastic differential equation (shortly BSDE) with Lipschitz coefficients were respectively established by Pardoux-Peng and Peng in [1,2], Mao and Cao generalized the Pardoux-Peng’s existence and uniqueness theorem to BSDE with non-Lipschitz coefficients in [3,4]. The present paper generalizes the Peng’s continuous dependence property in ”L²” sense to BSDE with Mao and Cao’s conditions. Furthermore, this paper investigates the continuous dependence property in “almost surely” sense for BSDE with Mao and Cao’s conditions, based on the comparison with the classical mathematical expectation.     

Mean-field backward stochastic differential equations driven by G-Brownian motion and related partial differential equations

In this paper, we study mean-field backward stochastic differential equations driven by G-Brownian motion (G-BSDEs). We first obtain the existence and uniqueness theorem of these equations. In fact, we can obtain local solutions by constructing Picard contraction mapping for Y term on small interval, and the global solution can be obtained through backward iteration of local solutions. Then, a comparison theorem for this type of mean-field G-BSDE is derived. Furthermore, we establish the connection of this mean-field G-BSDE and a nonlocal partial differential equation. Finally, we give an application of mean-field G-BSDE in stochastic differential utility model.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a “point-wise” manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the ω-wise, “deterministic” viscosity solution to the PDE with random coefficients will be established.