Existence and Uniqueness of the Nonlinear BSDEs with a Small Parameter under Locally Lipschitz Condition

In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coefficient f is locally Lipschitz in y and z,the coefficient g 1 is locally Lipschitz in y,and the coefficient g 2 is uniformly Lipschitz in y and z.Let L N be the locally Lipschitz constant of the coefficients on the ball B（0,N） of R d × R d×r.We prove the existence and uniqueness of the solution when L N ～ √ log N and the parameter ε is small.

Penalization schemes for BSDEs and reflected BSDEs with generalized driver

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations(BSDE)and pre-default reflected backward stochastic differential equations(RBSDE).The goal of this work is twofold.First,we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver A.Second,we study penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a constrained optimal stopping problem or a constrained Dynkin game in which the set of minimizer's admissible exercise times is constrained to the right support of the measure generated by A.

BSDES IN GAMES,COUPLED WITH THE VALUE FUNCTIONS.ASSOCIATED NONLOCAL BELLMAN-ISAACS EQUATIONS

We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.

L^(p)Solution of Reflected BSDEs with One Continuous Barrier and Quasi-linear Growth Generators

This paper is devoted to solving a reflected backward stochastic differential equation(BSDE in short)with one continuous barrier and a quasi-linear growth generator g,which has a linear growth in(y,z),except the upper direction in case of y<0,and is more general than the usual linear growth generator.By showing the convergence of a penalization scheme we prove existence and comparison theorem of the minimal L^(p)(p>1)solutions for the reflected BSDEs.We also prove that the minimal Lpsolution can be approximated by a sequence of Lpsolutions of certain reflected BSDEs with Lipschitz generators.

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z.The existence and uniqueness results are given to these BSDEs.As an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.

Capital allocation for cash-subadditive risk measures:From BSDEs toBSVIEs

In the context of risk measures,the capital allocation problem is widely studied in the literature where different approaches have been developed,also in connection with cooperative game theory and systemic risk.Although static capital allocation rules have been extensively studied in the recent years,only few works deal with dynamic capital allocations and its relation with BSDEs.Moreover,all those works only examine the case of an underneath risk measure satisfying cash-additivity and,moreover,a large part of them focuses on the specific case of the gradient allocation where Gateaux differentiability is assumed.The main goal of this paper is,instead,to study general dynamic capital allocations associated to cash-subadditive risk measures,generalizing the approaches already existing in the literature and motivated by the presence of(ambiguity on)interest rates.Starting from an axiomatic approach,we then focus on the case where the underlying risk measures are induced by BSDEs whose drivers depend also on the yvariable.In this setting,we surprisingly find that the corresponding capital allocation rules solve special kinds of Backward Stochastic Volterra Integral Equations(BSVIEs).

L^(p)-solutions of Multi-dimensional Oblique Reflected BSDEs and Optimal Switching Problem on Finite or Infinite Time Horizon

In this paper,we study mulit-dimensional oblique reflected backward stochastic differential equations(RBSDEs)in a more general framework over finite or infinite time horizon,corresponding to the pricing problem for a type of real option.We prove that the equation can be solved uniquely in L^(p)(1<p≤2)-space,when the generators are uniformly continuous but each component taking values independently.Furthermore,if the generator of this equation fulfills the infinite time version of Lipschitzian continuity,we can also conclude that the solution to the oblique RBSDE exists and is unique,despite the fact that the values of some generator components may affect one another.

Mean-field BSDEs with jumps and dual representation for global risk measures

We study mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher-order interactions.We interpret the BSDE solution as a dynamic risk measure for a representative bank whose risk attitude is influenced by the system.This influence can come in a wide class of choices,including the average system state or average intensity of system interactions.Using Fenchel−Legendre transforms,our main result is a dual representation for the expectation of the risk measure in the convex case.In particular,we exhibit its dependence on the mean-field operator.

Moment Inequality and Hlder Inequality for BSDEs

Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.

Mc Kean–Vlasov BSDEs with Locally Monotone Coefficient

We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally monotone in y,locally Lipschitz with respect to z and law's variable,and the monotonicity and Lipschitz constants κ_(n),L_(n) are such that L_(n)^(2)+κ_(n)^(+)=O(log(N)),then the MVBSDE has a unique stable solution.

A Note on Jensen’s Inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.

Gradient Convergence of Deep Learning-Based Numerical Methods for BSDEs

The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations(SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method,quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the O(K^(-1/4)) rate of gradient convergence with K being the total number of iterative steps.

Quadratic mean-field reflected BSDEs

In this paper,we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown z.Using a linearization technique and the BMO martingale theory,we first apply a fixed-point argument to establish the uniqueness and existence result for the case with bounded terminal condition and obstacle.Then,with the help of theθ-method,we develop a successive approximation procedure to remove the boundedness condition on the terminal condition and obstacle when the generator is concave(or convex)with respect to the second unknown.

Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection

We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations(BSDEs)with bounded terminal data.By virtue of bounded mean oscillation martingale and change of measure techniques,we obtain stability estimates for the variation of the solutions with different underlying forward processes.In addition,we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.

Explicit solutions for a class of nonlinear BSDEs and their nodal sets

In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explicit solutions to an interesting class of nonlinear BSDEs,including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.Moreover,we show its applications in PDEs and contingent pricing in an incomplete market.

不确定环境下再装股票期权的稳健定价模型

研究具有Knight不确定性的金融市场,假定标的资产(股票)价格过程服从几何布朗运动,建立了再装股票期权在一个概率测度集合上的最大、最小定价模型。并借助于倒向随机微分方程(BSDE)以及偏微分方程(PDE)的重要理论完成了对模型的转化。最后利用随机过程的有关知识求出了该模型的显示表达式,并通过具体的数值分析揭示了Knight不确定性对再装股票期权定价的重要影响。

Knight不确定环境下欧式股票期权的最小定价模型

研究具有Knight不确定性的金融市场,假定标的资产(股票)价格过程服从几何布朗运动,建立了欧式期权在一个概率测度集合上的最小定价模型,并借助于倒向随机微分方程(BSDE)的重要理论以及鞅方法求出了该模型的显示表达式;通过研究一个避险参数揭示了Knight不确定性对欧式期权定价的影响。

Optimal consumption-leisure, portfolio and retirement selection based on α-maxmin expected CES utility with ambiguity

This article studies optimal consumption-leisure, portfolio and retirement selection of an infinitely lived investor whose preference is formulated by a-maxmin expected CES utility which is to differentiate ambiguity and ambiguity attitude. Adopting the recursive multiple- priors utility and the technique of backward stochastic differential equations （BSDEs）, we transform the （^-maxmin expected CES utility into a classical expected CES utility under a new probability measure related to the degree of an investor＇s uncertainty. Our model investi- gates the optimal consumption-leisure-work selection, the optimal portfolio selection, and the optimal stopping problem. In this model, the investor is able to adjust her supply of labor flex- ibly above a certain minimum work-hour along with a retirement option. The problem can be analytically solved by using a variational inequality. And the optimal retirement time is given as the first time when her wealth exceeds a certain critical level. The optimal consumption-leisure and portfolio strategies before and after retirement are provided in closed forms. Finally, the distinctions of optimal consumption-leisure, portfolio and critical wealth level under ambiguity from those with no vagueness are discussed.

无穷水平上的倒向随机微分方程和g-期望

证明了无穷水平上BSDEs的系数唯一性定理,并利用此定理将平方可积随机变量的g-期望扩张到可积随机变量的g-期望。

条件g-期望与相关风险测度

利用倒向随机微分方程(BSDE)理论中的条件g-期望来定义风险测度及动态风险测度,证明了它们都满足相关风险测度及动态相关风险测度的公理化定义,并且给出了所定义的相关风险测度及动态相关风险测度的表示定理.最后,给出了相关风险测度及动态相关风险测度的一些重要性质.