非利普希茨条件下由G-布朗运动驱动的倒向随机微分方程的比较定理

研究了由G布朗运动驱动的倒向随机微分方程Y t=ξ+∫T tf(s,Y s,Z s)d s+∫T tg(s,Y s,Z s)d〈B〉s-∫T tZ s d B s-(K T-K t),0≤t≤T的解的比较定理。其中f(s,y,z),g(s,y,z)关于变量y单调且线性增长,关于变量z利普希茨连续。

市场经济体制下金融机制及其数学建模机理的可拓性分析

本文在简要阐述市场经济体制下金融机制的含义、特征、条件的基础上，重点对其典型的数学建模机理进行可拓性分析。

A General Converse Comparison Theorem for Backward Stochastic Differential Equation with Non-lipschitz Coefficient

In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establish a general converse comparison theorem for backward stochastic differential equation with non-Lipschitz coefficient.

The Domination for Evaluation of the Contingent Claims by Nonlinear Generator

In this paper,through applying the result of backward stochastic differential equations,it investigates a domination for pricing of the contingent claims by the use of nonlinear infinitesimal generator of process X. This domination provides a guide for valuing the price of the position on the financial market.

Research on problem about technology insurance pricing based on backward stochastic differential equation theory

The development of Backward Stochastic Differential Equation Theory is just a thing happened in the past years. Although its development and application is far behind Forward Stochastic Differential Equation, its wide application prospect on financial mathematics gets more and more attention. The meaning of Backward Stochastic Differential Equation is that change a already-known final （usually uncertain） goal into a present certain answer to make a present resolution. But Insurance Pricing happens to know the final result, it＇ s certain that the result is uncertain, that is to say, to get out of danger or not. And then make present insurance price according to the future uncertain result. The Insurance Pricing just follows the meaning of Backward Stochastic Differential Equation. Insurance Pricing itself is also a research field sprang up in past scores of years, because insurance pricing is the indisputable core of insurance work, and gets quite a few researchers＇ attention. This article adopts backward stochastic differential equation theory and do research on problem about technology insurance pricing.

Existence of Solutions for Forward-Backward Stochastic Differential Equations with Jumps and Non-Lipschitzian Coefficients

This paper studies for ward-back ward differential equations with Poisson jumps and with stopping time as termination. Under some weak monotonicity conditions and for non-Lipschitzian coefficients, the existence and uniqueness of solutions are proved via a purely probabilistic approach, while a priori estimate is given. Here, we allow the forward equation to be degenerate.

BSDE,path-dependent PDE and nonlinear Feynman-Kac formula

We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.

Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations （FBDSDEs）. It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistie interpretation for the solutions to a class of quasilinear stochastic partial differential equations （SPDEs） combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backvzard variable.

Necessary and sufficient condition for the comparison theorem of multidimensional anticipated backward stochastic differential equations

Anticipated backward stochastic differential equations, studied the first time in 2007, are equations of the following type:{-dY t = f(t1, Y t1 , Z t1 , Y t+δ(t) , Z t+ζ(t) )dt Z t dB t1 , t ∈ [0, T ], Y t = ξ t1 , t ∈ [T, T + K], Z t = η t1 , t ∈ [T, T + K].In this paper, we give a necessary and sufficient condition under which the comparison theorem holds for multidimensional anticipated backward stochastic differential equations with generators independent of the anticipated term of Z.

Second-order schemes for solving decoupled forward backward stochastic differential equations

In this paper,by using trapezoidal rule and the integration-by-parts formula of Malliavin calculus,we propose three new numerical schemes for solving decoupled forward-backward stochastic differential equations.We theoretically prove that the schemes have second-order convergence rate.To demonstrate the effectiveness and the second-order convergence rate,numerical tests are given.

STOCHASTIC DIFFERENTIAL EQUATIONS AND STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM WITH LEVY PROCESSES

In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.

BSDEs with Jumps and Path-Dependent Parabolic Integro-differential Equations

This paper deals with backward stochastic differential equations with jumps,whose data(the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.

Successful couplings for a class of stochastic differential equations driven by Lvy processes

By constructing proper coupling operators for the integro-differential type Markov generator, we establish the existence of a successful coupling for a class of stochastic differential equations driven by Levy processes. Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups, and it is sharp for Ornstein-Uhlenbeck processes driven by s-stable Levy processes.

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differentiM equation （BSDE） driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, where Teugel＇s martingales are a family of pairwise strongly orthonormal martingales associated with L6vy processes （see e.g., Nualart and Schoutens＇ paper in 2000）. We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria （or backward linear-quadratic problem, or BLQ problem for short） is discussed and characterized by a stochastic Hamilton system.

Backward Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process （BDSDEP） with non-Lipschitz coefficients on random time interval are studied. The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations （SPDIEs） is treated with BDSDEP. Under non-Lipschitz conditions, the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique. Then, the continuous depen- dence for solutions to BDSDEP is derived. Finally, the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.

A Type of General Forward-Backward Stochastic Differential Equations and Applications

The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with Ito's stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations.The existence and uniqueness results of the general FBSDEs are obtained.In the framework of the general FBSDEs in this paper,the explicit form of the optimal control for linear-quadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.

Large deviation principle for diffusion processes under a sublinear expectation

We represent the exponential moment of the Brownian functionals under a nonlinear expectation according to the solution to a backward stochastic differential equation.As an application,we establish a large deviation principle of the Freidlin and Wentzell type under the corresponding nonlinear probability for diffusion processes with a small diffusion coefficient.

Indifference pricing and hedging in a multiple-priors model with trading constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein(2002). The price is determined by two optimal stochastic control problems(mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations.By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates.The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （e.g., the multiple Malliavin integration-by-parts formula）, and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.

FOUR STEP SCHEME FOR GENERAL MARKOVIAN FORWARD-BACKWARD SDES

This paper studies a class of forward-backward stochastic differential equations (FBSDE)in a general Markovian framework.The forward SDE represents a large class of strong Markov semimartingales,and the backward generator requires only mild regularity assumptions.The authors showthat the Four Step Scheme introduced by Ma,et al.(1994) is still effective in this case.Namely,the authors show that the adapted solution of the FBSDE exists and is unique over any prescribedtime duration;and the backward components can be determined explicitly by the forward componentvia the classical solution to a system of parabolic integro-partial differential equations.An importantconsequence the authors would like to draw from this fact is that,contrary to the general belief,in aMarkovian set-up the martingale representation theorem is no longer the reason for the well-posednessof the FBSDE,but rather a consequence of the existence of the solution of the decoupling integralpartialdifferential equation.Finally,the authors briefly discuss the possibility of reducing the regularityrequirements of the coefficients by using a scheme proposed by F.Delarue (2002) to the current case.