Mean-Variance Hedging Under Multiple Defaults Risk

We solve a mean-variance hedging problem in an incomplete market where multiple defaults can occur. For this purpose, we use a default-density modeling approach. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of the default times is modelled using a conditional density hypothesis. We prove the quadratic form of each value process between consecutive default times and recursively solve systems of coupled quadratic backward stochastic differential equations (BSDEs). We demonstrate the existence of these solutions using BSDE techniques. Then, using a verification theorem, we prove that the solutions of each subcontrol problem are related to the solution of our global mean-variance hedging problem. As a byproduct, we obtain an explicit formula for the optimal trading strategy. Finally, we illustrate our results for certain specific cases and for a multiple defaults case in particular.     

Multi-player stopping games with redistribution of payoffs and BSDEs with oblique reflection

We examine the connections between a novel class of multi-person stopping games with redistribution of payoffs and multi-dimensional reflected BSDEs in discrete- and continuous-time frameworks. Our goal is to provide an essential extension of classic results for two-player stopping games (Dynkin games) to the multi-player framework. We show the link between certain multi-period m$m$-player stopping games and a new kind of m$m$-dimensional reflected BSDEs. The existence and uniqueness of a solution to continuous-time reflected BSDEs are established. Continuous-time redistribution games are constructed with the help of reflected BSDEs and a characterization of the value of such stopping games is provided.     

由Lévy过程驱动的BSDE定义的一般非线性期望

本文研究了平凡可积随机变量的一类非线性期望f-期望.利用陈增敬推广g-期望的方法,扩张了f-期望的定义空间.     

由Levy过程驱动的倒向随机微分方程在随机单调条件下解的存在唯一性

本文讨论了如下的由Levy过程驱动的倒向随机微分方程适应解的存在唯一性■其中W_s是一Wiener过程,H_s为由Levy过程构成Teugels鞅.我们通过构造函数逼近序列的方法证明了,在漂移系数f关于Y满足随机单调,f关于Z和U满足随机Lipschitz条件下,方程存在唯一适应解.     

Linear-quadratic optimal control under non-Markovian switching

We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which turned out to be a backward stochastic differential equation driven by the Brownian motion and by the random measure associated with the marked point process.     

倒向随机微分方程的Malliavin微分和共单调定理

本文利用Malliavin微分的理论研究了倒向随机微分方程的解$(y,z)$, 首先利用$y$的Malliavin微分得到了一种比较$z$的方法, 然后利用该方法得到了含有随机生成元的倒向随机微分方程的共单调定理.     

Discrete-Time BSDEs with Random Terminal Horizon

This article studies the theory of discrete-time backward stochastic differential equations (also called BSDEs) with a random terminal time, which is not a stopping time. We follow Cohen and Elliott [2 Cohen , S.N. , and Elliott , R.J. 2010 . A general theory of finite state backward stochastic difference equations . Stochastic Processes and Their Applications 120 ( 4 ): 442 – 466 .[Crossref], [Web of Science ®] , [Google Scholar]] and consider a reference filtration generated by a general discrete-time finite-state process. The martingale representation theorem for essentially bounded martingales under progressively enlarged filtration is established. Then we prove the existence and uniqueness theorem of BSDEs under enlarged filtration using some weak assumptions of the driver. We also present conditions for a comparison theorem. Applications to nonlinear expectations and optimal design of dynamic default risk are explored.     

Modeling host-parasitoid interactions with correlated events

In this paper, we present a host-parasitoid model with correlated events. We apply a block-structured state-dependent (BSDE) approach that provides a methodological tool to model state-dependent Markovian transitions operating in the presence of phases. A particularly appealing feature of the resulting BSDE host-parasitoid model is that it allows us to deal with non-exponential distributional assumptions on a host birth, a parasitoid death, and parasitism, but keeping the dimensionality of the underlying block-structured Markov chain tractable. Numerical examples are presented to illustrate the effects of the correlation structure on the expected extinction times and the extinction probabilities.     

Importance Sampling for Backward SDEs

In this article, we explain how the importance sampling technique can be generalized from simulating expectations to computing the initial value of backward stochastic differential equations (SDEs) with Lipschitz continuous driver. By means of a measure transformation we introduce a variance reduced version of the forward approximation scheme by Bender and Denk [4 Bender , C. , and Denk , R. 2007 . A forward scheme for backward SDEs . Stochastic Processes and their Applications 117 ( 12 ): 1793 – 1812 . [Google Scholar]] for simulating backward SDEs. A fully implementable algorithm using the least-squares Monte Carlo approach is developed and its convergence is proved. The success of the generalized importance sampling is illustrated by numerical examples in the context of Asian option pricing under different interest rates for borrowing and lending.