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

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985823%3A_____%2F22%3A00557426" target="_blank" >RIV/67985823:_____/22:00557426 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1016/j.cam.2021.113951" target="_blank" >https://doi.org/10.1016/j.cam.2021.113951</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.cam.2021.113951" target="_blank" >10.1016/j.cam.2021.113951</a>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10103 - Statistics and probability

Project

<a href="/en/project/GF20-21030L" target="_blank" >GF20-21030L: Stochastic models and methods for the study of olfaction</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2022

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Journal of Computational and Applied Mathematics

ISSN

0377-0427

e-ISSN

1879-1778

Volume of the periodical

406

Issue of the periodical within the volume

May 1

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

29

Pages from-to

113951

UT code for WoS article

000789740200019

EID of the result in the Scopus database

2-s2.0-85121879507