Image Solution of Stochastic Differential Equation of Diffusion Type Driven by Brownian Motion

Abstract

Consider the stochastic differential equation of diffusion type driven by Brownian motion dX(t, ω) = μX(t, ω)dt + σX(t, ω)dB (t, ω) where B(t, ω) = limn→∞ B n (t, ω) is a Brownian motion, n is a positive integer, t is time variable, ω is state variable, μ and σ are constants. The solution X(t, ω) is represented by images. Solution contains a term of Brownian motion. Therefore, the image of a solution needs the image of Brownian motion. We have obtained the images of Brownian motion and solution X(t, ω) for different combinations of parameters (μ, σ, n and p. Note that p controls the degree of randomness in Brownian motion. Degree of randomness in Brownian motion is maximum for p = 0.5). The key observations from image analysis are 1. less randomness is visualized for p values away from 0.5, 2. colors in images for n = 10, 000 is more than the color in images for n = 10, 000, 00, and 3. randomness in solution depends on μ and σ also. More randomness is visualized as μ − 1 2 σ 2 is away from 0. The observations are consistent with mathematical analysis of the solution X(t, ω).

Time and Error Analysis of Neural Network Model of Rectangular Vibrating Membrane

“…The solution of vibrating membrane are solved by Seperation of variables, Stochastic calculus [5,25], Finite difference, Finite element and Finite volume method [7,14,16,32].…”