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

S P Tampubolon
D S Palupi

PROSIDING SEMINAR NASIONAL FISIKA (E-JOURNAL)
Vol. 13 (2025): PROSIDING SEMINAR NASIONAL FISIKA (E-JOURNAL) SNF2024

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.