THEORETICAL STUDY OF STOCHASTIC DIFFERENTIAL EQUATIONS AND PROBABILITY DENSITY FUNCTION FOR LINEAR AND NONLINEAR GEOMETRIC BROWNIAN MOTION
KAJIAN TEORITIK PERSAMAAN DIFERENSIAL STOKASTIK DAN PROBABILITY DENSITY FUNCTION BAGI GERAK BROWNIAN GEOMETRIK LINEAR DAN NONLINEAR
Abstract
A theoretical study has been conducted on stochastic differential equations for both linear and nonlinear Geometric Brownian Motion, focusing on deriving the Probability Density Function (PDF) and its characteristics. The research is purely theoretical. The study involved formulating stochastic differential equations for both linear and nonlinear Geometric Brownian Motion. The solution for the probability density function of Geometric Brownian Motion was obtained using the Fokker-Planck equation, which corresponds to its stochastic differential equation. Linear Geometric Brownian Motion with a constant drift term does not yield a probability density function that is valid for all x and t. In contrast, nonlinear Geometric Brownian Motion can have a probability density function valid for all x and ttt under certain conditions.
References
[1] Horsthemke W and Lefever R 1984 Noise-Induced Transitions ~Springer-Verlag Berlin
[2] Black F and Scholes M1973 The Pricing of Option and Corporate Liabilities The Journal of Political Economy 3 81 637-654.
[3] Dragulescu AA and Yakovenko VM 2002 Probability Distribution of Returns in The Heston Model with Stochastic Volatility Quantitative Finance 6 2 443-453
[4] Fuchs A, Herbert C, Rolland J, Wächte M, Bouchet F and Peinke J 2022 Instantons and the Pathto Intermittency in Turbulent Flows , Physical Review Letters 129, 034502
[5] Kuo HH 2006, Introduction to Stochastic Integration,Springer, USA.
[6] Oksendal B 2003 Stochastic Differential Equations Spinger-Verlag Heidelberg New York USA
[7] Giordano S, Cleri F, Blossey R 2023 Infinite ergodicity in generalized geometric Brownian motions with nonlinear drift Physical Review E 107 044111,
