The mixture of Wiener and a Poisson processes are the primary tools used in creating jump-diffusion process which is very popular in mathematical modeling. In financial mathematics, they are used to describe the change of stock rates and bonanzas, and they are often used in mathematical biology modeling and population dynamics. In this thesis, we extended the Lie point symmetry theory of deterministic differential equations to the class of jump-diffusion stochastic differential equations, i.e., a stochastic process driven by both Wiener and Poisson processes. The Poisson process generates the jumps whereas the Brownian motion path is continuous. The determining equations for a stochastic differential equation with finite jump are successfully derived in an Itô calculus context and are found to be deterministic, even though they represent a stochastic process. This work leads to an understanding of the random time change formulae for Poisson driven process in the context of Lie point symmetries without having to consult much of the intense Itô calculus theory needed to formally derive it. We apply the invariance methodology of Lie point transformation together with the more generalized Itô formulae, without enforcing any conditions to the moments of the stochastic processes to derive the determining equations and apply it to few models. In the first part of the thesis, point symmetry of Poisson-driven stochastic differential equations is discussed, by considering the infinitesimals of not only spatial and temporal variables but also infinitesimals of the Poisson process variable. This was later extended, in the second part, to define the symmetry of jumpdiffusion stochastic differential equations (i.e., stochastic differential equations driven by both Wiener and Poisson processes).
Reference:
Nass, A. 2017. Point symmetry methods for Itô Stochastic Differential Equations (SDE) with a finite jump process. University of Cape Town.
Nass, A. M. (2017). Point symmetry methods for Itô Stochastic Differential Equations (SDE) with a finite jump process. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/25387
Nass, Aminu Ma'aruf. "Point symmetry methods for Itô Stochastic Differential Equations (SDE) with a finite jump process." Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2017. http://hdl.handle.net/11427/25387
Nass AM. Point symmetry methods for Itô Stochastic Differential Equations (SDE) with a finite jump process. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2017 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/25387