The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

中文翻译：

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贝塞尔势空间中的多维线性和非线性偏积分微分方程在期权定价中的应用

本文的目的是分析多维空间中非局部非线性偏积分微分方程 (PIDE) 的解。此类 PIDE 经常出现在金融建模中。我们采用抽象半线性抛物方程的理论来证明在贝塞尔势空间尺度上解的存在性和唯一性。我们考虑在原点和无穷远附近满足合适生长条件的一大类 Lévy 测度。这篇论文的新颖之处在于将一维中已知结果推广到多维情况。我们考虑了 Black-Scholes 模型，用于根据带有跳跃的 Lévy 随机过程对标的资产进行期权定价。作为一维空间中期权定价的应用，考虑到大型交易者的股票交易策略，我们考虑由非线性期权定价模型产生的一般转移函数。我们证明了非线性 PIDE 解的存在性和唯一性，其中平移函数可能取决于规定的大投资者股票交易策略函数。