倒向随机微分方程及其在证券投资组合中的应用

在理论与实践中,对投资组合的研究已取得了辉煌的成果。本文通过运用倒向随机微分方程,研究当投资者以将来某一时刻获取一定数额的适应性收益为投资目标时,如何确定当前证券投资组合中各证券的投资比例。文中运用倒向随机微分方程给出证券组合的模型,并给出了相应的解的形式。

确定收益模式下证券投资组合研究

在理论与实践中,证券投资组合问题的研究已取得了辉煌的成果。本文通过运用非线性倒向随机微分方程,研究当投资者以将来某一时刻获取一定数额确定性收益为投资目标时,如何确定当前证券投资组合中各证券的投资比例。文中运用非线性倒向随机微分方程给出证券组合的模型,并给出了相应的解的形式。

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.

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.

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.

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.

BACKWARD LINEAR-QUADRATIC STOCHASTIC OPTIMAL CONTROL AND NONZERO-SUM DIFFERENTIAL GAME PROBLEM WITH RANDOM JUMPS

This paper studies the existence and uniqueness of solutions of fully coupled forward-backward stochastic differential equations with Brownian motion and random jumps.The result is applied to solve a linear-quadratic optimal control and a nonzero-sum differential game of backward stochastic differential equations.The optimal control and Nash equilibrium point are explicitly derived. Also the solvability of a kind Riccati equations is discussed.All these results develop those of Lim, Zhou(2001) and Yu,Ji(2008).

Forward-backward stochastic differential equation with subdifferential operator and associated variational inequality

We study the existence and uniqueness of the solution to a forward-backward stochastic differential equation with subdifferential operator in the backward equation. This kind of equations includes, as a particular case, multi-dimensional forward-backward stochastic differential equation where the backward equation is reflected on the boundary of a closed convex(time-independent) domain. Moreover, we give a probabilistic interpretation for the viscosity solution of a kind of quasilinear variational inequalities.