Nacira Agram
We establish sufficient conditions for the existence and uniqueness of mean-field backward stochastic differential equations with time delayed generator in the sense that at t, the generator may depend on previous values up to a delay constant δ not on the hole past as in Delong and Imkeller [10], [13]. For sufficiently small delay constant δ and for any finite time horizon, we get a unique solution.
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We establish sufficient conditions for the existence and uniqueness of different types of delayed BSDEs in finite time horizon. We consider then infinite horizon, replacing the terminal value condition in the finite horizon case with a condition of strong decay at infinity.
Optimal Stopping of BSDEs with Constrained Jumps and Related Zero-Sum Games
本文提出非线性Snell包络刻画带约束跳跃BSDE的最优停止值，并证明其与零和随机微分博弈的值等价，通过惩罚技术与控制随机化建立存在性与正则性。
In this work, we study the problem of mean-variance hedging with a random horizon T ^ tau, where T is a deterministic constant and is a jump time of the underlying asset price process. We rst formulate this problem as a stochastic control problem and relate it to a system of BSDEs with jumps. We then provide a veri cation theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from ltration enlargement theory.
This work deals with backward stochastic differential equation (BSDE) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of filtrations. We show that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BDSEs. As applications, we study the…
We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is t…
In this paper, we consider a constrained stochastic linear-quadratic (LQ) optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the Itô-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations (BSDEs). The solvability of the BSDEs is obtained by solving a recursive sy…
We consider an optimal control problem for piecewise deterministic Markov processes (PDMPs) on a bounded state space. The control problem under study is very general: a pair of controls acts continuously on the deterministic flow and on the two transition measures (in the interior and from the boundary of the domain) describing the jump dynamics of the process. For this class of control problems, the value function can be characterized as the unique viscosity solution to the corresponding fully-nonlinear Hamilton-Jacobi-Bellman equation with a non-local type boundary condition. By means of the…
We investigate the optimal reinsurance problem when the loss process exhibits jump clustering features and the insurance company has restricted information about the loss process. We maximize expected exponential utility of terminal wealth and show that an optimal solution exists. By exploiting both the Kushner-Stratonovich and Zakai approaches, we provide the equation governing the dynamics of the (infinite-dimensional) filter and characterize the solution of the stochastic optimization problem in terms of a BSDE, for which we prove existence and uniqueness of solution. After discussing the o…
Portfolio Exponential Utility Maximization with Jump Signals
本文研究了在风险资产由布朗运动和独立同质泊松测度驱动的情况下，包含跳跃信号的投资组合效用最大化问题，通过建立跳跃BSDE并证明其解的存在性，验证了跳跃信号对效用提升的影响。
We study the discrete-time approximation for solutions of quadratic forward back- ward stochastic differential equations (FBSDEs) driven by a Brownian motion and a jump process which could be dependent. Assuming that the generator has a quadratic growth w.r.t. the variable z and the terminal condition is bounded, we prove the convergence of the scheme when the number of time steps n goes to infinity. Our approach is based on the companion paper [15] and allows to get a convergence rate similar to that of schemes of Brownian FBSDEs.
We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bou…
Comparison theorems for multi-dimensional BSDEs with jumps and applications to constrained stochastic linear-quadratic control
本文首次建立带跳多维倒向随机微分方程的比较定理，并用于构造带跳耦合二维随机Riccati方程的解，进而求解锥约束随机线性二次和带跳均值-方差投资组合问题，发现最优状态过程可能变号。
In this paper we look at ergodic BSDEs in the case where the forward dynamics are given by the solution to a non-autonomous (time-periodic coefficients) Ornstein-Uhlenbeck SDE with Lévy noise, taking values in a separable Hilbert space. We establish the existence of a unique bounded solution to an infinite horizon discounted BSDE. We then use the vanishing discount approach, together with coupling techniques, to obtain a Markovian solution to the EBSDE. We also prove uniqueness under certain growth conditions. Applications are then given, in particular to risk-averse ergodic optimal control an…
This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.
We consider a classical finite horizon optimal control problem for continuous-time pure jump Markov processes described by means of a rate transition measure depending on a control parameter and controlled by a feedback law. For this class of problems the value function can often be described as the unique solution to the corresponding Hamilton-Jacobi-Bellman equation. We prove a probabilistic representation for the value function, known as nonlinear Feynman-Kac formula. It relates the value function with a backward stochastic differential equation (BSDE) driven by a random measure and with a …
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…
Singular limit of BSDEs and optimal control of two scale systems with jumps in infinite dimensional spaces
本文研究无限维双尺度随机系统在连续与泊松跳噪声共同驱动下的最优控制问题，利用BSDE理论证明快变量速度发散时值函数收敛到约化正倒向系统，并建立其与约化最优控制问题的联系。
Optimal Stopping of BSDEs with Constrained Jumps and Related Double Obstacle PDEs
本文研究带约束跳的BSDE最优停止问题与双障碍PDE的关系，通过Feynman-Kac表示和压缩论证证明非局部情形下粘性解的存在唯一性，并建立新型非线性Snell包络。
Convergence of a Deep BSDE solver with jumps
该研究针对带跳解耦FBSDE及相关PIDE的深度学习数值逼近误差，拓展已有经典工作，给出有限和无限活动情形下的先验与后验误差估计。
Aggregation of value processes for semi-martingale BSDEs with jumps
本文为带跳半鞅BSDE值过程构造可测聚合器，适用于非支配或不确定控制问题，并建立时间一致非线性条件期望系统、路径正则化及动态规划原理，统一处理扩散、纯跳和离散时间过程。
This paper is concerned with a linear quadratic (LQ, for short) optimal control problem for mean-field backward stochastic differential equations (MF-BSDE, for short) driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control is established. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a …
The classical maximum principle for optimal stochastic control states that if a control $\hat{u}$ is optimal, then the corresponding Hamiltonian has a maximum at $u=\hat{u}$. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first order der…
Two person non-zero-sum linear-quadratic differential game with Markovian jumps in infinite horizon
本文研究带马尔可夫跳的无限时域两人非零和线性二次微分博弈，通过耦合代数Riccati方程与倒向随机微分方程刻画最优控制并给出闭环Nash均衡。
Multi-Asset Utility Maximization with Jump Signals
本文研究带跳跃信号的多资产投资组合效用最大化问题，通过半鞅表示与带跳跃BSDE刻画最优策略，证明解的存在唯一性并处理异质信号情形，给出高斯信号下的数值示例。
A deep solver for BSDEs with jumps
该工作将深度求解器扩展到带跳的FBSDE，用神经网络参数化控制过程，处理有限与无限跳活动，并成功应用于低维和高维期权定价及对手方信用风险。
In this paper, we study one-dimensional backward stochastic differential equation with jump under logarithmic growth assumption in the z-variable (|z|\sqrt{|\ln|z|}|) and an L^p terminal value (for a suitable p>2). We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the stochastic control problem.
In this article, we establish a propagation of chaos result for weakly interacting nonlinear Snell envelopes which converge to a class of mean-field reflected backward stochastic differential equations (BSDEs) with jumps and right-continuous and left-limited obstacle, where the mean-field interaction in terms of the distribution of the $Y$-component of the solution enters both the driver and the lower obstacle. Under mild Lipschitz and integrability conditions on the coefficients, we prove existence and uniqueness of the solution to both the mean-field reflected BSDEs with jumps and the corres…
A class of $L_1$-to-$L_1$ and $L_\infty$-to-$L_\infty$ interval observers for (delayed) Markov jump linear systems
本文基于正马尔可夫跳跃线性系统稳定性与性能分析结果，设计了具有和无延迟的马尔可夫跳跃线性系统区间观测器，所有条件均以线性规划形式高效求解。
This paper investigates state feedback control laws for Markov jump linear systems with state and mode-observation delays. An assumption in this study is that the delay of mode observation obeys an exponential distribution. Also, we raise an unknown time-varying state delay applied in the composition of the state feedback controller. A method of remodeling the closed-loop system as a standard Markov jump linear system with state delay is shown. Furthermore, on the basis of this remodeling, several Linear Matrix Inequalities (LMI) for designing feedback gains for stabilization and mixed $H_2/H_…
Optimal reinsurance in a dynamic contagion model: comparing self-exciting and externally-exciting risks
本文在具有跳跃聚类的风险模型中研究最优再保险，结合Hawkes与Cox过程，通过HJB和BSDE方法求解指数效用最大化，并比较自激与外激风险的影响。
Constrained Zero-Sum Stochastic Linear-Quadratic Differential Game for Jump-Diffusion Systems with Random Coefficients
本文研究了具有随机系数和 regime switching 的跳扩散系统中带锥约束的零和线性二次微分博弈，通过FBSDEs建立了开环解的存在性并推导了闭式解表达式。
This paper deals with the finite horizon optimal control problem for discrete-time Markov jump linear system with input delay. The correlation among the jumping parameters and the input delay are considered simultaneously, which forms the basic difficulty of the design. one of the key techniques is to solve a delayed forward and backward jumping parameter difference equation which is obtained by an improved maximum principle, and the other is the introduction of a "d-step backward formula". Based on the proposed techniques, a necessary and sufficient condition for the existence of the optimal …
Recursive Optimal Stopping with Poisson Stopping Constraints
针对带泊松停时约束的递归最优停时问题，提出基于Jacod-Pham的分解方法拆分问题，结合带跳BSDE比较定理表示值函数，并应用于非线性市场美式期权定价。
In this paper, we prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semi-explicitly) constructed for an incomplete financial market with external risk factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) processes. Analytical and numerical examples are both presented to illustrate the effectiveness of our optimal strategy. Our study establishes the connection between our financial system and existing general semimartingale based discussions by justifying required conditions. More precisely, there are three steps involved. First, we firmly…
A randomisation method for mean-field control problems with common noise
本文提出一种控制随机化方法，将带公共噪声的平均场控制问题转化为控制泊松点过程强度的问题，并通过BSDE和随机化动态规划原理刻画值函数。
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