论文标题
线性和非线性局部局部差异方程是由金融产生的
Linear and Nonlinear Partial Integro-Differential Equations arising from Finance
论文作者
论文摘要
这篇评论论文的目的是介绍我们对现代金融数学产生的非线性和非本地数学模型的最新结果。它基于我们的四篇论文，由J. Cruz，M。Grossinho，D。Sevcovic和C. Udeani共同撰写，以及J. Cruz的Phd论文的一部分。我们研究了由期权定价和投资组合选择问题引起的线性和非线性部分局部差异方程式（PIDE），并研究了与选项定价理论和黑色 - choles模型之间的系统关系。首先，我们放宽了液体和完整的市场假设，并将研究市场不流动性的模型扩展到了基础资产价格遵循lévy随机工艺随着跳跃而进行的情况。然后，我们在适当的假设下建立了对期权定价的相应PIDE。使用抽象半线性抛物线方程的理论在贝塞尔电位空间的规模中提出了非局部线性和非线性pide的定性特性。在多维空间中，还建立了对无穷大和原点满足适当生长条件的所谓的可允许的Lévy措施的PIDE解决方案的存在和独特性。
The purpose of this review paper is to present our recent results on nonlinear and nonlocal mathematical models arising from modern financial mathematics. It is based on our four papers written jointly by J. Cruz, M. Grossinho, D. Sevcovic, and C. Udeani, as well as parts of PhD thesis by J. Cruz. We investigated linear and nonlinear partial integro-differential equations (PIDEs) arising from option pricing and portfolio selection problems and studied the systematic relationships between the PIDEs with option pricing theory and Black--Scholes models. First, we relax the liquid and complete market assumptions and extend the models that study the market's illiquidity to the case where the underlying asset price follows a Lévy stochastic process with jumps. Then, we establish the corresponding PIDE for option pricing under suitable assumptions. The qualitative properties of solutions to nonlocal linear and nonlinear PIDE are presented using the theory of abstract semilinear parabolic equation in the scale of Bessel potential spaces. The existence and uniqueness of solutions to the PIDE for a general class of the so-called admissible Lévy measures satisfying suitable growth conditions at infinity and origin are also established in the multidimensional space.
