Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion

Irene Tubikanec,Massimiliano Tamborrino,Petr Lansky,Evelyn Buckwar

doi:10.1016/j.cam.2021.113951

Abstract

We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller’s boundary classification. We compare the frequently used Euler–Maruyama and Milstein methods, two Lie–Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong–Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler–Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler–Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie–Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features.

Highlights

The inhomogeneous geometric Brownian motion (IGBM), described by the Itô stochastic differential equation (SDE) (1 )dY (t) = − τ Y (t) + μ dt + σ Y (t)dW (t), t ≥ 0, Y (0) = Y0, is frequently applied in mathematical and computational finance, neuroscience and other fields
It is often used to describe price fluctuations in finance [1,2] or changes in the neuronal membrane voltage in neuroscience [3]. This process is known as geometric Brownian motion (GBM) with affine drift [4], geometric Ornstein–Uhlenbeck (OU) process [5] or mean reverting GBM [6] in real option theory, as Brennan–Schwarz model [7,8] in the interest rate literature, as GARCH model [9,10] in stochastic volatility and energy markets, as Lognormal diffusion with exogenous factors [11] in growth analysis and forecasting or as reciprocal gamma diffusion in [12]
Among the class of strong approximations, we focus on numerical methods based on the splitting and ordinary differential equation (ODE) approaches, and on their comparison with the commonly used Euler–Maruyama and Milstein schemes

Summary

Introduction

It is often used to describe price fluctuations in finance [1,2] or changes in the neuronal membrane voltage in neuroscience [3] This process is known as geometric Brownian motion (GBM) with affine drift [4], geometric Ornstein–Uhlenbeck (OU) process [5] or mean reverting GBM [6] in real option theory, as Brennan–Schwarz model [7,8] in the interest rate literature, as GARCH model [9,10] in stochastic volatility and energy markets, as Lognormal diffusion with exogenous factors [11] in growth analysis and forecasting or as reciprocal gamma diffusion in [12]. We need to rely on numerical methods that accurately reproduce the features of the process, making its analysis and investigation via simulations possible and reliable

Objectives

Methods

Conclusion