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This paper is concerned with optimal control of linear backward stochastic differential equations (BSDEs) with a quadratic cost criteria, or backward linear-quadratic (BLQ) control. The theory of BSDE, which was originally developed by Peng [17] for stochastic control theory, has been introduced to stochastic differential games by Hamad`ene and Lepeltier [ 13] and Hamadene et al. The BSDE approach is used to solve the game problem. Consider for example an electricity producer applications to stochastic control problems, dealing with examples in population monitoring. 1 Introduction This paper is dedicated to the obtention of a probabilistic representation for a general form of continuous optimal switching problems. 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 … Moreover, as has been shown in [40], arguably the most important open problem in stochastic financial mathematical interest, BSDE appear in numerous applications, including stochas-tic representations for partial differential equations, optimal stochastic control and stochastic games (see, e.g., [16, 26, 29]and[43]). Advances in Statistics, Probability and Actuarial Science Stochastic Processes, Finance and Control, pp. Huanjun Zhang, Zhiguo Yan, Backward stochastic optimal control with mixed deterministic controller and random controller and its applications in linear-quadratic control, Applied Mathematics and Computation, 10.1016/j.amc.2019.124842, 369, (124842), (2020). Springer, 2011. 19/44. There exists a unique optimal control for LQ stochastic optimal control problem -, and where the is the solution of the following BSDE driven by Lévy process. The optimal investment problem is then formulated as a zero-sum stochastic di erential game between the insurer and the market. • The process of estimating the values of the state variables is called optimal filtering . 2.BSDE Formulation of Parabolic PDE 3.Deep BSDE Method 4.Numerical Examples of High-Dimensional PDEs 5.Stochastic Control in Discrete Time 6.Convergence of the Deep BSDE Method 7.Summary 2/38. Backward Stochastic Di erential Equations 1 What is a BSDE? BSDE Approach to Non-Zero-Sum Stochastic Differential Games of Control and Stopping (I Karatzas and Q Li) Mathematical Finance: On Optimal Dividend Strategies in Insurance with a Random Time Horizon (H Albrecher and S Thonhauser) ... imum principle to study the stochastic optimal control problems is one kind of FBSDEs. The existence and the uniqueness of the solution $(Y^{t,x,P_\xi},Y^{t,\xi})$ of … SDEs - the di erential dynamics approach to BSDEs 2 Applications - Why do we need BSDEs? MSC Classi cation (2000): 93E20, 60H30, 60J75. For standard stochastic control in continuous time, we refer to the textbooks [13, 22, 14, 27, 26]. For this Peng’s BSDE method (Peng [14]) is translated from the framework of stochastic control theory into that of stochastic … BSDEs with convex standard ... Theorem 2.1 (Linear BSDE). process of our stochastic control problem as the unique solution of a generalized backward stochastic difierential equation with a quadratic driver. We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. associated to the stochastic control problem and can be computed approximatively by employing the policy Z(see (9) below for details). Some new BSDE results for an infinite–horizon stochastic control problem. The solution of this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique. In this paper we study the optimal stochastic control problem for a path-dependent stochastic system under a recursive path-dependent cost functional, whose associated Bellman equation from dynamic programming principle is a path-dependent fully nonlinear partial differential equation of second order. The connection between the PDE (PDE) and the stochastic control problem (1){(2) is based on the nonlinear Feynman-Kac formula which links PDEs and BSDEs (see (BSDE) and (3) below). The former results have been extended to cost functionals defined by For a typical Markovian singular stochastic control problems, it can be … Abstract: In this talk we consider a decoupled mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. A particular feature is that the problem is formulated and solved for an inflnite horizon and that we also obtain new results on a certain inflnite-horizon BSDE with quadratic generator. Assume , , , and are all deterministic; then Riccati equation ( 23 ) changes to Then from Theorem 6 we can get Corollary 9 . Backward stochastic differential equation, Infinite horizon, Reflected barriers, Stochastic optimal control, Stochastic differential game. Key words: robust control, model uncertainty, quadratic BSDE, stochastic control… In this context, the PDE is reformulated as a stochastic control problem through backward stochastic differential equation (BSDE) and the gradient of the unknown solution is approximated by neural networks. situation with control of a diffusion and a nonlinear cost functional defined as solution to a BSDE. This paper studies a stochastic control problem arising in the context of robust utility max-imisation, and proves new result via BSDE techniques. Secondly I will show, that Convex BSDEs, i.e. • A decision maker is faced with the problem of making good estimates of these state variables from noisy measurements on functions of them. The solution of such control problem is proved to identify with the solution of a Z-constrained BSDE, with dynamics associated to a … Approximate stochastic control based on deep learning and forward backward stochastic differential equations Master’s thesis in engineering mathematics and computational science ... 6.1 Upper: The BSDE control and the analytic control applied to two samplepathsofanOrnstein-Uhlenbeckprocess. 14 Some new BSDE results for an infinite-horizon stochastic control problem Ying Hu1 and Martin Schweizer2,3 1 Universit´e Rennes 1, IRMAR, Campus de Beaulieu, F – 35042 Rennes Cedex, France, [email protected] 2 ETH Zu¨rich, Departement Mathematik, CH – 8092 Zu¨rich, Switzerland, and Swiss Finance Institute, Walchestrasse 9, CH – 8006 Zu¨rich, Switzerland, A BSDE Approach to Stochastic Differential Games Involving Impulse Controls and HJBI Equation. [` 14]. Introduction. wealth. This problem is solved in a closed form by the stochastic linear-quadratic (LQ) theory developed recently. Elena Bandini, Fulvia Confortola, and Andrea Cosso Full-text: Open access. The martingale term in the original BSDE is regarded as the control, and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the BSDE. Key words: Stochastic control, Switching problems, Re ected BSDE. So the stochastic control systems with delay are more complex. BSDE Approach Multi-Dim Reflective BSDE Based on two preprints: Martingale Interpretation to a Non-Zero-Sum Stochastic Differential Game of Controls and Stoppings I. Karatzas, Q. Li, 2009 A BSDE Approach to Non-Zero-Sum Stochastic Differential Games … ... Eq. Singular stochastic control goes back to [2, 3] and has subsequently been studied by e.g. Through an impulse control approximation scheme we construct a solution to the control … ... •This is the Pontryagin maximum principle for stochastic control. 105-153 (2012) No Access BSDE approach to non-zero-sum stochastic differential games of control … In the second part of this talk, we propose the deep BSDE method, in a similar vein, to solve general high-dimensional parabolic PDEs. … BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. I. Formulation of Stochastic Control Model dynamics: s t+1 = s t+ b t(s t,a A saddle point for the Dynkin game is given by the pair of first action times of an optimal control. In the context of BSDE for control, Y tdenotes theoptimal value and Z tdenotes theoptimal control(up to a constant scaling). Let (β,γ) be a bounded (R,Rn)-valued pre-dictable process, φ ∈ H2 [4, 16, 17, 19, 18, 20, 9, 1, 23, 11, 21, 15]. We prove that the value, i.e. Our results extend earlier work by Skiadas (2003) and are based on a difierent approach. From that one can derive properties of the solution to the control problem. Firstly I will show that a certain stochastic control problem is connecte to a LBSDE. It leads to a simple and natural approach for the existence and uniqueness of an optimal strategy of the game problem without Markov assumptions. (1) is called the anticipated BSDE. The upper and the lower value functions are proved to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. Øksendal and Sulem (2000) discussed a certain class of stochastic control systems with delay in the state variable, and they gave the sufficient conditions for the stochastic maximum principle. We prove that a sequence of solutions to BSDEs driven by birth/death processes converges to a BSDE driven by a one dimen-sional Brownian motion in tractable spaces to investigate the scaling limits of solutions to stochastic control problems. This paper studies a class of non-Markovian singular stochastic control problems, for which we provide a novel probabilistic representation. STOCHASTIC OPTIMAL CONTROL • The state of the system is represented by a controlled stochastic process. BSDE Approach to Non-Zero-Sum Stochastic Differential Games of Control and Stopping ∗ Ioannis Karatzas † INTECH Investment Management One Palmer Square, Suite 441 Princeton, NJ 08542 [email protected] Qinghua Li Statistics Department, Columbia University 1255 Amsterdam Avenue, 1009 SSW New York, NY 10027 [email protected] June 7, 2011 This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. Nonlinear backward stochastic daerential equations (BSDE's in short) have been independently introduced by Pardoux and Peng [18] and DdEe and Epstein [7]. In G. di Nunno and B. Øksendal, editors, Advanced Mathematical Methods for Finance , pages 367–395.

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