On the Euler–Maruyama Scheme for Degenerate Stochastic Differential Equations with Non-sticky Condition

Dai Taguchi; Akihiro Tanaka

doi:10.1007/978-3-030-28535-7_9

On the Euler–Maruyama Scheme for Degenerate Stochastic Differential Equations with Non-sticky Condition

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

The aim of this paper is to study weak and strong convergence of the Euler–Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation dX_t = σ(X_t)dW_t with non-sticky condition. For proving this, we first prove that the Euler–Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation dX_t = |X_t|^αdW_t, α ∈ (0, 1∕2) with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.

Original language	English
Title of host publication	Lecture Notes in Mathematics
Publisher	Springer
Pages	165-185
Number of pages	21
DOIs	https://doi.org/10.1007/978-3-030-28535-7_9
Publication status	Published - 2019

Publication series

Name	Lecture Notes in Mathematics
Volume	2252
ISSN (Print)	0075-8434
ISSN (Electronic)	1617-9692

Keywords

CEV models
Euler–Maruyama scheme
Hölder continuous diffusion coefficient
Mathematical finance
Non-sticky condition
Stochastic differential equations

ASJC Scopus subject areas

Algebra and Number Theory

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10.1007/978-3-030-28535-7_9

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title = "On the Euler–Maruyama Scheme for Degenerate Stochastic Differential Equations with Non-sticky Condition",

abstract = "The aim of this paper is to study weak and strong convergence of the Euler–Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation dXt = σ(Xt)dWt with non-sticky condition. For proving this, we first prove that the Euler–Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation dXt = |Xt|αdWt, α ∈ (0, 1∕2) with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.",

keywords = "CEV models, Euler–Maruyama scheme, H{\"o}lder continuous diffusion coefficient, Mathematical finance, Non-sticky condition, Stochastic differential equations",

author = "Dai Taguchi and Akihiro Tanaka",

note = "Funding Information: Acknowledgements The authors would like to thank Professor Masatoshi Fukushima for his valuable comments. The authors would also like to thank an anonymous referee for his/her careful readings and advices. The first author was supported by JSPS KAKENHI Grant Number 17H06833. The second author was supported by Sumitomo Mitsui Banking Corporation.",

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N1 - Funding Information: Acknowledgements The authors would like to thank Professor Masatoshi Fukushima for his valuable comments. The authors would also like to thank an anonymous referee for his/her careful readings and advices. The first author was supported by JSPS KAKENHI Grant Number 17H06833. The second author was supported by Sumitomo Mitsui Banking Corporation.

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AB - The aim of this paper is to study weak and strong convergence of the Euler–Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation dXt = σ(Xt)dWt with non-sticky condition. For proving this, we first prove that the Euler–Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation dXt = |Xt|αdWt, α ∈ (0, 1∕2) with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.

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