Analytical solutions for stochastic differential equations via Martingale processes

Abstract

In this paper, we propose some analytical solutions of stochastic differential equations related to Martingale processes. In the first resolution, the answers of some stochastic differential equations are connected to other stochastic equations just with diffusion part (or drift free). The second suitable method is to convert stochastic differential equations into ordinary ones that it is tried to omit diffusion part of stochastic equation by applying Martingale processes. Finally, solution focuses on change of variable method that can be utilized about stochastic differential equations which are as function of Martingale processes like Wiener process, exponential Martingale process and differentiable processes.

A computational method for solving stochastic Itô–Volterra integral equation with multi-stochastic terms

“…where and are stochastic MHFs coefficient vectors and and j , j = 1, 2, … , n are stochastic MHFs coefficient matrices. Substituting (9)- (12) in relation (8), we obtain Using the 6-th property in relation (13), we get Utilizing operational matrices defined in relations (5) and (6) in (14), we have…”

“…Nowadays, modelling different problems in different issues of science leads to stochastic equations [1]. These equations arise in many fields of science such as mathematics and statistics [2][3][4][5][6][7], finance [8][9][10], physics [11][12][13], mechanics [14,15], biology [16][17][18], and medicine [19,20]. Whereas most of them do not have an exact solution, the role of numerical methods and finding a reliable and accurate numerical approximation have become highlighted [21].…”

The Impact of Fourier Series Expansion on the Analysis of Asset Value Function and its Return Rates for Capital Markets

“…In trying to study stochastic model [8] considered the stability analysis of stochastic model for stock market prices and did analysis of the unstable nature of stock market forces applying a new differential equation model that can impact the expected returns of investors in stock exchange market with a stochastic volatility in the equation. While [9] suggested in their study, some analytical solutions of stochastic differential equations with respect to Martingale processes and discovered that the solutions of some SDEs are related to other stochastic equations with diffusion part. The second technique is to change SDE to ODE that are tried to omit diffusion part of stochastic equation by using Martingale processes.…”

A Two Model Approach of Assessing Asset Value Functions for Capital Investments

“…In trying to study stochastic model Davis, et al [5] considered "the stability analysis of stochastic model for stock market prices and did analysis of the unstable nature of stock market forces applying a new differential equation model that can impact the expected returns of investors in stock exchange market with a stochastic volatility in the equation". While Farnoosh, et al [6] suggested in their study, "some analytical solutions of stochastic differential equations with respect to Martingale processes and discovered that the solutions of some SDEs are related to other stochastic equations with diffusion part. The second technique is to change SDE to ODE that are tried to omit diffusion part of stochastic equation by using Martingale processes".…”