金融数学中的非线性抛物型方程
Nonlinear Parabolic Equations arising in Mathematical Finance
摘要：
本文主要研究金融数学中抛物型完全非线性偏微分方程的定性和数值分析。主要目的是回顾经典的Black-Scholes定价理论的各种非线性扩展，以及随机动态投资组合优化模型导致的Hamilton-Jacobi-Bellman（HJB）方程。经过适当的变换，这两个问题都可以用非线性抛物方程的解来表示。定性分析将集中在有关问题的存在性和唯一性的解决方案。数值部分讨论了求解完全非线性抛物型方程的稳定的有限体积格式和有限差分格式。
英文摘要：
This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.
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