Optimal stopping of BSDEs with constrained jumps and related double obstacle PDEs

Mean-field backward stochastic differential equations and related partial differential equations

: In this paper we first review the classical Feynman-Kac formula and then introduce its generalization obtained by Pardoux-Peng via backward stochastic differential equations. It is because of the usefulness of the Feynman-Kac formula in the study of parabolic partial differential equations we see clearly how worthy to study the backward stochastic differential equations in more detail. We hence further review the work of Pardoux and Peng on backward stochastic differential equations and establish a new theorem on the existence and uniqueness of the adapted solution to a backward stochastic differential equation under a weaker condition than Lipschitz one. Key Words : Backward stochastic differential equation, parabolic partial differential equation, adapted solution, Bihari's inequality. Introduction In 1990, Pardoux & Peng initiated the study of backward stochastic differential equations motivated by optimal stochastic control (see Bensoussan, Bismut, Haussmann and Kushner). It is even more important that Pardoux & Peng, Peng recently gave the probabilistic representation for the given solution of a quasilinear parabolic partial differential equation in term of the solution of the corresponding backward stochastic differential equation. In other words, they obtained a generalization of the well-known Feynman-Kac formula (cf. Freidlin or Gikhman & Skorokhod). In view of the powerfulness of the Feynman-Kac formula in the study of partial differential equations e.g. K.P.P. equation (cf. Freidlin), one may expect that the Pardoux-Peng generalized formula will play an important role in the study of quasilinear parabolic partial differential equations. Hence from both viewpoints of the optimal stochastic control and partial differential equations, we see clearly how worthy to study the backward stochastic differential equations in more detail.

This thesis constitutes a research work on deriving viscosity solutions to optimal stopping problems for Feller processes. We present conditions on the process under which the value function is the unique viscosity solution to a Hamilton-Jacobi-Bellman equation associated with a particular operator. More speci cally, assuming that the underlying controlled process is a Feller process, we prove the uniqueness of the viscosity solution. We also apply our results to study several examples of Feller processes. On the other hand, we try to extend our results by iterative optimal stopping methods in the rest of the work. This approach gives a numerical method to approximate the value function and suggest a way of nding the unique viscosity solution associated to the optimal stopping problem. We use it to study several relevant control problems which can reduce to corresponding optimal stopping problems. e.g., an impulse control problem as well as an optimal stopping problem for jump di usions and regime switching processes. In the end, as a complementary, we are trying to construct optimal stopping problems with multiplicative functionals related to a non-conservative Feller semigroup. As a consequence, viscosity solutions were obtained for such kind of constructions.

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in \cite{weinan2017deep} when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minima as it can be the case with the algorithm designed in \cite{weinan2017deep}, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

Since 1990 Pardoux and Peng, proposed the theory of backward stochastic differential equation Backward stochastic differential equation and is backward stochastic differential equations (short for FBSDE) theory has been widely research (see El Karoui, Peng and Cauenez, Ma and Yong, etc.) Generally, a backward stochastic differential equation is a type Ito stochastic differential equation and a coupling Pardoux - Peng and backward stochastic differential equation. Antonelli, Ma, Protter and Yong is backward stochastic differential equation for a series of research, and apply to the financial. One of the research direction is put forward by Hu and Peng first. Peng and Wu Peng and Shi made a further research, and Yong to a more detailed discussion of this method, by introducing the concept of the bridge, systematically studied the FBSDE continuity method. Because such a system can be applied to random Feynman - Kac of partial differential equations of research, And a double optimal control problem of stochastic control systems, we will be working in Peng and Shi further in-depth study on the basis of this category are backward stochastic differential equation. In this paper, we are considering various constraint conditions with backward stochastic differential equation.

In this article we study a decoupled forward backward stochastic differential equation (FBSDE) and the associated system of partial integro-differential obstacle problems, in a flexible Markovian set-up made of a jump-diffusion with regimes. These equations are motivated by numerous applications in financial modeling, whence the title of the paper. This financial motivation is developed in the first part of the paper, which provides a synthetic view of the theory of pricing and hedging financial derivatives, using backward stochastic differential equations (BSDEs) as main tool. In the second part of the paper, we establish the well-posedness of reflected BSDEs with jumps coming out of the pricing and hedging problems exposed in the first part. We first provide a construction of a Markovian model made of a jump-diffusion – like component X interacting with a continuous-time Markov chain – like component N. The jump process N defines the so-called regime of the coefficients of X, whence the name of jump-diffusion with regimes for this model. Motivated by optimal stopping and optimal stopping game problems (pricing equations of American or game contingent claims), we introduce the related reflected and doubly reflected Markovian BSDEs, showing that they are well-posed in the sense that they have unique solutions, which depend continuously on their input data. As an aside, we establish the Markov property of the model. In the third part of the paper we derive the related variational inequality approach. We first introduce the systems of partial integro-differential variational inequalities formally associated to the reflected BSDEs, and we state suitable definitions of viscosity solutions for these problems, accounting for jumps and/or systems of equations. We then show that the state-processes (first components Y ) of the solutions to the reflected BSDEs can be characterized in terms of the value functions of related optimal stopping or game problems, given as viscosity solutions with polynomial growth to related integro-differential obstacle problems. We further establish a comparison principle for semi-continuous viscosity solutions to these problems, which implies in particular the uniqueness of the viscosity solutions. This comparison principle is subsequently used for proving the convergence of stable, monotone and consistent approximation schemes to the value functions. Finally in the last part of the paper we provide various extensions of the results needed for applications in finance to pricing problems involving discrete dividends on a financial derivative or on the underlying asset, as well as various forms of discrete path-dependence.KeywordsStochastic Differential EquationViscosity SolutionPrice ProcessObstacle ProblemPrice EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem.

On the quasi-linear reflected backward stochastic partial differential equations

We study an optimal control problem on infinite horizon for a controlled stochastic differential equation driven by Brownian motion, with a discounted reward functional. The equation may have memory or delay effects in the coefficients, both with respect to state and control, and the noise can be degenerate. We prove that the value, i.e. the supremum of the reward functional over all admissible controls, can be represented by the solution of an associated backward stochastic differential equation (BSDE) driven by the Brownian motion and an auxiliary independent Poisson process and having a sign constraint on jumps. In the Markovian case when the coefficients depend only on the present values of the state and the control, we prove that the BSDE can be used to construct the solution, in the sense of viscosity theory, to the corresponding Hamilton-Jacobi-Bellman partial differential equation of elliptic type on the whole space, so that it provides us with a Feynman-Kac representation in this fully nonlinear context. The method of proof consists in showing that the value of the original problem is the same as the value of an auxiliary optimal control problem (called randomized), where the control process is replaced by a fixed pure jump process and maximization is taken over a class of absolutely continuous changes of measures which affect the stochastic intensity of the jump process but leave the law of the driving Brownian motion unchanged.

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett. 14 (1990) 55–61; Pardoux and Peng, Lecture Notes in Control and Information Sciences 176 (1992) 200–217]. We have at our disposal stochastic processes which solve the so-called backward stochastic differential equations . These processes provide us with a Feynman-Kac representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear in many applications in the field of Mathematical Finance. Therefore there is a great interest among both practitioners and theoreticians to develop reliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods for approximating solutions of semilinear PDEs all based on the corresponding Feynman-Kac representation. We also include a general introduction to backward stochastic differential equations and their connection with PDEs and provide a generic framework that accommodates existing probabilistic algorithms and facilitates the construction of new ones.

This work considers the infinite horizon discounted risk-sensitive optimal control problem for the switching diffusions with a compact control space and controlled through the drift; thus, the the generator of the switching diffusions also depends on the controls. Note that the running cost of interest can be unbounded, so a decent estimation on the value function is obtained, under suitable conditions. To solve such a risk-sensitive optimal control problem, we adopt the viscosity solution methods and propose a numerical approximation scheme. We can verify that the value function of the optimal control problem solves the optimality equation as the unique viscosity solution. The optimality equation is also called the Hamilton–Jacobi–Bellman (HJB) equation, which is a second-order partial differential equation (PDE). Since, the explicit solutions to such PDEs are usually difficult to obtain, the finite difference approximation scheme is derived to approximate the value function. As a byproduct, the ϵ-optimal control of finite difference type is also obtained.

This paper provides a comprehensive examination of the pivotal roles played by Partial Differential Equations (PDE), Stochastic Differential Equations (SDE), and Backward Stochastic Differential Equations (BSDE) in financial asset pricing. It traces their theoretical evolution from the foundational Black-Scholes PDE model, through flexible SDE capturing market randomness, to advanced BSDE adept at handling complex, nonlinear problems with terminal conditions. The study details their diverse applications in pricing interest rate derivatives, exotic options, XVAs, and portfolio optimization. It also analyzes key challenges: the curse of dimensionality for PDE, unrealistic assumptions in SDE, and computational intensity for BSDE. Future research directions are explored, focusing on integration with fractional calculus, artificial intelligence, and high-performance computing to enhance efficiency and adaptability. This work concludes that PDE, SDE, and BSDE form a synergistic, indispensable toolkit for modern quantitative finance.

We employ the viscosity solution technique to analyze optimal stopping problems with regime switching. To be specific, we show the viscosity property of value functions with the help of dynamic programming, and more importantly, provide a mild and verifiable condition and an available bound that both can guarantee the uniqueness of viscosity solutions.

In this paper we show the existence and uniqueness of a solution for backward stochastic differential equations driven by a Levy process with moments of all orders. The results are important from a pure mathematical point of view as well as in the world of finance: an application to Clark-Ocone and Feynman-Kac formulas for Levy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equation provide an analogue of the famous Black-Scholes partial differential equation and thus can be used for the purpose of option pricing in a Levy market.

We present the main concepts of the theory of Markov processes: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov's backward and forward equations, and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward stochastic differential equations (BSDEs) and describe the links between Markov processes and parabolic partial differential equations (PDEs). In particular, we state the Feynman–Kac formula for linear PDEs and BSDEs, and we give some examples of the correspondence between stochastic control problems and Hamilton–Jacobi–Bellman (HJB) equations and between optimal stopping problems and variational inequalities. Several examples of financial applications are given to illustrate each of these results, including European options, Asian options, and American put options.