Jayaraman Amitesh S, Campolo Domenico, Chirikjian Gregory S
Department of Mechanical Engineering, National University of Singapore, Singapore 117575, Singapore.
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore.
Entropy (Basel). 2020 Apr 17;22(4):455. doi: 10.3390/e22040455.
The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker-Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.
来自数理金融的布莱克 - 斯科尔斯偏微分方程（PDE）已得到广泛分析，众所周知，通过变量的对数变换，该方程可简化为欧几里得空间上的热方程。然而，本文提出了一种替代解释，即将该偏微分方程重新构建为在李群上演变。此方程可转化为一个扩散过程，并使用先前在李群上求解福克 - 普朗克方程的背景下开发的均值和协方差传播技术来求解。具有耦合资产动态的布莱克 - 斯科尔斯理论的扩展产生了一个仿射群上的扩散方程，而仿射群不是单模群。在本文中，我们表明，对于中心平凡的群的情况，本文构建的赋予半直积群运算的李群的余切丛总是单模的，并且将偏微分方程视为单模余切丛群上的扩散过程允许直接应用先前开发的均值和协方差传播技术，从而提供了一种求解偏微分方程的替代方法。最终，本文在数理金融中的偏微分方程背景下给出的这些结果可应用于各种不同领域中出现的偏微分方程，并为新的求解方法提供思路。
