An extended existence result for quadratic BSDEs with jumps with application to the utility maximization problem

BSDEs and log-utility maximization for Lévy processes

In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for Lévy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary Lévy process.

Quadratic Exponential Semimartingales and Application to BSDEs with jumps

In this paper, we study a class of Quadratic Backward Stochastic Differential Equations (QBSDE in short) with jumps and unbounded terminal condition. We extend the class of quadratic semimartingales introduced by Barrieu and El Karoui (2013) in the jump diffusion model. The properties of these class of semimartingales lead us to prove existence result for the solution of a quadratic BSDEs.

Robust optimal problem for dynamic risk measures governed by BSDEs with jumps and delayed generator

The aim of this paper is to study an optimal stopping problem for dynamic risk measures induced by backward stochastic differential equations with jumps and delayed generator. Firstly, we connect the value function of this problem to reflected BSDEs with jump and delayed generator. Furthermore, after establishing existence and uniqueness result for this reflected BSDE, we use its to address through a mixed/optimal stopping game problem for the previous dynamic risk measure in ambiguity case.

L^p -solution for BSDEs with jumps in the case p \textless{} 2. Corrections to the paper "BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration"

In [8] we established existence and uniqueness of solutions of backward stochastic differential equations in L^p under a monotonicity condition on the generator and in a general filtration. There was a mistake in the case 1 \textless{} p \textless{} 2. Here we give a corrected proof. Moreover the quasi-left continuity condition on the filtration is removed.

Simulation of BSDEs with jumps by Wiener Chaos Expansion

We present an algorithm to solve BSDEs with jumps based on Wiener Chaos Expansion and Picard's iterations. This paper extends the results given in Briand-Labart (2014) to the case of BSDEs with jumps. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. Concerning the error, we derive explicit bounds with respect to the number of chaos, the discretization time step and the number of Monte Carlo simulations. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.

Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping proble…

Finite and infinite time horizon for BSDE with Poisson jumps

This paper is devoted to solving a real valued backward stochastic differential equation with jumps where the time horizon may be finite or infinite. Under linear growth generator, we prove existence of a minimal solution. Using a comparison theorem we show existence and uniqueness of solution to such equations when the generator is uniformly continuous and satisfies a weakly monotonic condition.

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 ana…

Quadratic BSDEs driven by a continuous martingale and application to utility maximization problem

In this paper, we study a class of quadratic Backward Stochastic Differential Equations (BSDEs) which arises naturally when studying the problem of utility maximization with portfolio constraints. We first establish existence and uniqueness results for such BSDEs and then, we give an application to the utility maximization problem. Three cases of utility functions will be discussed: the exponential, power and logarithmic ones.

Convergence of BSΔEs driven by random walks to BSDEs: the case of (in)finite activity jumps with general driver

In this paper we present a weak approximation scheme for BSDEs driven by a Wiener process and an (in)finite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BSΔEs) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BSΔEs. By way of illustration we analyse explicitly a scheme with discrete step-si…

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 is 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 affect both drift …

Existence, Uniqueness and Comparison Results for BSDEs with Lévy Jumps in an Extended Monotonic Generator Setting

We show existence of a unique solution and a comparison theorem for a one-dimensional backward stochastic differential equation with jumps that emerge from a Lévy process. The considered generators obey a time-dependent extended monotonicity condition in the y-variable and have linear time-dependent growth. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008) and, in the $L^2$-case with linear growth, those of Kruse and Popier (2016). Moreover, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we conside…

On the Solution of Locally Lipschitz BSDE Associated to Jump Markov Process

In this study, we consider a class of backward SDE driven by jump Markov process. An existence and uniqueness result to this kind of equations is obtained in a locally Lipschitz case. We essentially approximate the initial problem by constructing a convenient sequence of globally Lipschitz BSDEs having the existence and the uniqueness propriety. Then, we show, by passing to the limits, the existence, and uniqueness of a solution to the initial problem. After that, a stability theorem is also proved in the local Lipschitz setting. Applying the aforementioned result, we give an application to Eu…

Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles

We study a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DBBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodski's condition. Our main contribution consists in the construction of an implementable numerical sheme, based on two random binomial trees and the penalization method, which is shown to converge to the so…

具有跳跃和时间延迟生成函数的BSDEs的Feynman-Kac公式及与路径依赖非线性柯尔莫戈罗夫方程的相关性

Feynman-Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations

本文提出了一种将具有跳跃和时间延迟生成函数的BSDEs解与路径依赖非线性柯尔莫戈罗夫方程解关联的非线性Feynman-Kac公式，并应用于研究大型投资者问题的扩展。

Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison

It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the…

Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps

We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation in [12] by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity con…

$\boldsymbol{\mathbb{L}^p(p\ge2)}$-solutions of generalized BSDEs with jumps and monotone generator in a general filtration

In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of $\mathbb{L}^p(p\ge2)$-solutions in the case of a fixed terminal time under suitable $p$-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension $1$.

Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles

We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in Dumitrescu-Labart (2014), we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this paper, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps $n$ (contrary to the scheme proposed in Dumit…

BSDEs in Utility Maximization with BMO Market Price of Risk

This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum of distinct square-integrable solutions. This feature occurs since, contrary to the classical Ito representation theorem, a representation of random variables in terms of stochastic exponentials is not unique. We study in detail when the BSDE has a bounded solution and derive a n…

The Stability of the Constrained Utility Maximization Problem - A BSDE Approach

This article studies the sensitivity of the power utility maximization problem with respect to the investor's relative risk aversion, the statistical probability measure, the investment constraints and the market price of risk. We extend previous descriptions of the dual domain then exploit the link between the constrained utility maximization problem and continuous semimartingale quadratic BSDEs to reduce questions on sensitivity to results on stability for such equations. This then allows us to prove appropriate convergence of the primal and dual optimizers in the semimartingale topology.

Limit behaviour of BSDE with jumps and with singular terminal condition

We study the behaviour at the terminal time T of the minimal solution of a backward stochastic differential equation when the terminal data can take the value +$\infty$ with positive probability. In a previous paper, we have proved existence of this minimal solution (in a weak sense) in a quite general setting. But two questions arise in this context and were still open: is the solution c{à}d{ì}{à}g on [0,T] ? In other words does the solution have a left limit at time T ? The second question is: is this limit equal to the terminal condition? In this paper, under additional conditions on the ge…

A New Result for Second Order BSDEs with Quadratic Growth and its Applications

In this paper, we study a class of second order backward stochastic differential equations (2BSDEs) with quadratic growth in coefficients. We first establish solvability for such 2BSDEs and then give their applications to robust utility maximization problems.

Progressive enlargement of filtrations and Backward SDEs with jumps

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…

Generalized Dynkin Games and Doubly Reflected BSDEs with Jumps

We introduce a generalized Dynkin game problem with non linear conditional expectation ${\cal E}$ induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let $ξ, ζ$ be two RCLL adapted processes with $ξ\leq ζ$. The criterium is given by \begin{equation*} {\cal J}_{τ, σ}= {\cal E}_{0, τ\wedge σ} \left(ξ_τ\textbf{1}_{\{ τ\leq σ\}}+ζ_σ\textbf{1}_{\{σ<τ\}}\right) \end{equation*} where $τ$ and $ σ$ are stopping times valued in $[0,T]$. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. $\inf_σ\sup_τ {\cal J}_{τ, σ} = \sup_τ \inf_σ {\ca…

Non-Markovian optimal stopping problems and constrained BSDEs with jump

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…

Robustness of quadratic hedging strategies in finance via backward stochastic differential equations with jumps

We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate- approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poi…

随机Lipschitz系数下一般滤子中带跳反射BSDE的$\mathbb{L}^p$-解

$\mathbb{L}^p$-solutions for Reflected BSDEs with jumps in a general filtration under stochastic Lipschitz coefficients

本文在一般滤子下，对同时含布朗运动和独立泊松随机测度的反射BSDE，在随机Lipschitz系数和$\mathbb{L}^p$可积条件下，建立了$p\in(1,2)$时$\mathbb{L}^p$-解的存在唯一性。

$\mathbb{L}^p$-解 $(1<p<2)$：具有一般跳跃和随机单调生成元的反射倒向随机微分方程

$\mathbb{L}^p$-solutions $(1 <p< 2)$ for reflected BSDEs with general jumps and stochastic monotone generators

本文研究具有一般跳跃和随机单调生成元的反射倒向随机微分方程，通过惩罚方法证明了$\mathbb{L}^p$（$1<p<2$）解的存在唯一性。

Second-order BSDEs with jumps: Formulation and uniqueness

In this paper, we define a notion of second-order backward stochastic differential equations with jumps (2BSDEJs for short), which generalizes the continuous case considered by Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190]. However, on the contrary to their formulation, where they can define pathwise the density of quadratic variation of the canonical process, in our setting, the compensator of the jump measure associated to the jumps of the canonical process, which is the counterpart of the density in the continuous case, depends on the underlying probability measu…

Backward SDEs with constrained jumps and quasi-variational inequalities

We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stoch…

A BSDE arising in an exponential utility maximization problem in a pure jump market model

We consider the problem of utility maximization with exponential preferences in a market where the traded stock/risky asset price is modelled as a Lévy-driven pure jump process (i.e. the driving Lévy process has no Brownian component). In this setting, we study the terminal utility optimization problem in the presence of a European contingent claim. We consider in detail the BSDE (backward stochastic differential equation) characterising the value function when using an exponential utility function. First we analyse the well-definedness of the generator. This leads to some conditions on the ma…

带跳平均场反射倒向随机微分方程的混沌传播

Propagation of chaos for mean-field reflected BSDEs with jumps

本文研究标记点过程与 Poisson 过程驱动的平均场反射倒向随机微分方程，证明粒子系统的存在唯一性、收敛性及收敛速率。

BSDEs with default jump

We study the properties of nonlinear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale measure associated with a default jump with intensity process $(λ_t)$. We give a priori estimates for these equations and prove comparison and strict comparison theorems. These results are generalized to drivers involving a singular process. The special case of a $λ$-linear driver is studied, leading to a representation of the solution of the associated BSDE in terms of a conditional expectation and an adjoint exponential semi-martingale. We then apply these resu…

Optimal reinsurance via BSDEs in a partially observable model with jump clusters

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…

Credit risk premia and quadratic BSDEs with a single jump

This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equations (BSDE). The class of BSDEs needed for that representation allows for quadratic growth generators and jumps at random times. Since the existence and uniqueness theory for this class of BSDEs has not yet been developed to the required generality, the first part of the paper is de…

A decomposition approach for the discrete-time approximation of BSDEs with a jump II: the quadratic case

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.

On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

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…

带跳和随机单调生成元的BSDE的$\mathbb{L}^p$ $(p>1)$-解

$\mathbb{L}^p$ $(p>1)$-solutions for BSDEs with jumps and stochastic monotone generator

研究带跳和多维不连续倒向随机微分方程，在生成元满足随机单调性和随机Lipschitz条件下，建立了$p\geq2$和$p\in(1,2)$时$\mathbb{L}^p$解的存在唯一性。

带跳多维倒向随机微分方程的比较定理及其在约束随机线性二次控制中的应用

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。

Ergodic BSDEs with jumps and time dependence

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…

Quadratic BSDEs with jumps: related non-linear expectations

In this article, we follow the study of quadratic backward SDEs with jumps,that is to say for which the generator has quadratic growth in the variables (z; u), started in our accompanying paper [15]. Relying on the existence and uniqueness result of [15], we define the corresponding g-expectations and study some of their properties. We obtain in particular a non-linear Doob-Meyer decomposition for g-submartingales and a downcrossing inequality which implies their regularity in time. As a consequence of these results, we also obtain a converse comparison theorem for our class of BSDEs. Finally,…

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 on a general state space K. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K, that generalize the Kolmogorov equation of 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…

Quadratic BSDEs with jumps: a fixed-point approach

In this article, we prove the existence of bounded solutions of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables (z,u). From a technical point of view, we use a direct fixed point approach as in Tevzadze [38], which allows us to obtain existence and uniqueness of a solution when the terminal condition is small enough. Then, thanks to a well-chosen splitting, we recover an existence result for general bounded solution. Under additional assumptions, we can obtain stability results and a comparison theorem, which as usual implies uni…

Second Order BSDEs with Jumps: Existence and probabilistic representation for fully-nonlinear PIDEs

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [15]. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations, which is the point of the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs.

Existence and uniqueness results for BSDEs with jumps: the whole nine yards

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.

二次均值场BSDEs与指数效用最大化

Quadratic Mean-Field BSDEs and Exponential Utility Maximization

本文研究了具有二次增长生成元的均值场BSDEs，提出了一种结合马尔可夫算子与BMO估计的新方法，解决了指数效用最大化问题，并扩展了终端条件的适用范围。

Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes

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 …

Reflected Backward SDEs with General Jumps

In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. The reflecting process is right continuous with left limits (rcll for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the result in the general framework we use a fixed point argument in an appropriate s…

Optimal investment under multiple defaults risk: A BSDE-decomposition approach

We study an optimal investment problem under contagion risk in a financial model subject to multiple jumps and defaults. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of default times is modeled by a conditional density hypothesis. In this Ito-jump process model, we give a decomposition of the corresponding stochastic control problem into stochastic control problems in the default-free filtration, which are determined in a backward induction. The dynamic programming method leads to a backward recursive system …

无限维空间中带跳双尺度系统的倒向随机微分方程奇异极限与最优控制

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。

Utility Maximization in a jump market model

In this paper, we consider the classical problem of utility maximization in a financial market allowing jumps. Assuming that the constraint set is a compact set, rather than a convex one, we use a dynamic method from which we derive a specific BSDE. We then aim at showing existence and uniqueness results for the introduced BSDE. This allows us to give an explicit expression of the value function and characterize optimal strategies for our problem.

带约束跳的倒向随机微分方程的最优停止问题及相关双障碍偏微分方程

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。

BSDE Representation and Randomized Dynamic Programming Principle for Stochastic Control Problems of Infinite-Dimensional Jump-Diffusions

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. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (kn…