A novel nonlinear stochastic differential equation model with quadratic and logarithmic drift

Abstract

This study introduces a new class of nonlinear stochastic differential equations (SDEs) with a logarithmic drift term, given by dX_t = (αX_t + βX_t^2 + γ log(X_t)) dt + σ dW_t . Assuming a lognormal distribution, we derive expressions for the mean and variance and also determine the stationary distribution. Furthermore, the proposed model is compared with existing SDE models using economic data, including exchange rates, inflation rates, and interest rates. Model performance is evaluated using the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The results show that the proposed SDE outperforms the competing models.

J. Bangladesh Acad. Sci. 50(1); 15-22: March 2026