Taylor Based Jump Diffusion Model of Fractional Brownian Motion of Stock Price

Abstract

Assuming that the stock price follows the stochastic differential equation driven by fractional Brownian motion and jump process, under the condition that the interest rate and volatility are constant. It has nothing to do with the relative importance of the information causing the jump. Generally speaking, the relative jump height of stock price depends on the relative importance of the important information. Fractional Brownian motion not only has various properties of fractional Brownian motion, but also has nonstationary increment. On the basis of fractional Brownian motion model, the fractional Brownian motion model with jump, namely fractional jump-diffusion process, is investigated. It is proposed to use Taylor expansion to deal with the control items in the model. For large-scale numerical simulation, it can reduce the amount of calculation required by the algorithm and save computer memory. The method of guessing value function and Taylor series expansion is introduced to solve the value function. The dialectical relationship between mathematics and finance is shown from one side: on the one hand, mathematics is a powerful tool for financial research; on the other hand, financial practice has promoted the development of mathematical theory itself.

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