Convergence Of The Grünwald-Letnikov Scheme For A Class Of Black-scholes Time-fractional Diffusion

During the last 20 years many numerical methods have been developed using discrete approximations to fractional differential equations, which make them can be applied to many different topics, e.g., in physics, chemistry, financial and so on. In some of these methods the essential role is played by the Gr(?)nwald-Letnikov approximation of fractional derivatives.The Black-Scholes equation is well-known, which plays the key role in the financial mathematics. In this thesis, instead, we use the Caputo time-fractional derivative in the Black-Scholes equation and then prove convergence of the Griinwald-Letnikov scheme.In this thesis, we prove convergence of the following initial & boundary problem under the the Griinwald-Letnikov scheme:where the operator D_t~βdenotes the Caputo time-fractional derivative, and the coefficients r,σsatisfy 2r-σ~2≥0 andδ(x) generalized function.We will use bivariate generating functions and Fourier-Laplace transform to prove convergence of (1) under the the Griinwald-Letnikov scheme...