Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit positivity preserving numerical method for linear stochastic volatility models driven by $α$-stable process

Xiaotong Li, Wei Liu, Xuerong Mao, Hongjiong Tian, Yue Wu

TL;DR

This work analyzes a linear stochastic volatility model driven by an $\\alpha$-stable process, addressing positivity and numerical approximation challenges. The authors prove that the SDE $dx(t)=(\\mu-\\lambda x)\\,dt+\\kappa x\\,dL^{\\alpha}(t)$ has a unique positive solution under suitable conditions and introduce a positivity-preserving Euler–Maruyama scheme. They establish a strong convergence rate of $1/\\alpha$ in $L^q$, $q\\in[1,\\alpha)$, for the discrete approximation and provide rigorous moment bounds and positivity results. Numerical simulations corroborate the theoretical rate and demonstrate robustness across parameter choices, highlighting the method’s practical potential for modeling volatility with heavy-tailed jumps.

Abstract

In this paper, we introduce a linear stochastic volatility model driven by $α$-stable processes, which admits a unique positive solution. To preserve positivity, we modify the classical forward Euler-Maruyama scheme and analyze its numerical properties. The scheme achieves a strong convergence order of $1/α$. Numerical simulations are presented at the end to verify theoretical results.

Explicit positivity preserving numerical method for linear stochastic volatility models driven by $α$-stable process

TL;DR

This work analyzes a linear stochastic volatility model driven by an -stable process, addressing positivity and numerical approximation challenges. The authors prove that the SDE has a unique positive solution under suitable conditions and introduce a positivity-preserving Euler–Maruyama scheme. They establish a strong convergence rate of in , , for the discrete approximation and provide rigorous moment bounds and positivity results. Numerical simulations corroborate the theoretical rate and demonstrate robustness across parameter choices, highlighting the method’s practical potential for modeling volatility with heavy-tailed jumps.

Abstract

In this paper, we introduce a linear stochastic volatility model driven by -stable processes, which admits a unique positive solution. To preserve positivity, we modify the classical forward Euler-Maruyama scheme and analyze its numerical properties. The scheme achieves a strong convergence order of . Numerical simulations are presented at the end to verify theoretical results.

Paper Structure

This paper contains 6 sections, 10 theorems, 92 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Properties of the underlying solution
  4. Positivity preserving EM method
  5. Numerical simulations
  6. Conclusion and future work

Key Result

Lemma 2.3

For a function $V(x)\in C^2(\mathbb{R})$, the Itô formula D.Applebaum2009OksendalSulem2000 of $V(x)$ associated with SDE mainSDE is defined by where $f(x)=\mu-\lambda x$ and $g(x)=\kappa x$.

Figures (4)

  • Figure 1: Three paths generated by the positivity preserving EM numerical method
  • Figure 2: Errors versus stepsize $\Delta$ on log-log scale with $\mu=1.5$, $\lambda=2$, $\kappa=0.5$
  • Figure 3: Errors versus stepsize $\Delta$ on log-log scale with $\mu=2$, $\lambda=3$, $\kappa=0.5$
  • Figure 4: Errors versus stepsize $\Delta$ on log-log scale with $\mu=2$, $\lambda=3$, $\kappa=0.2$

Theorems & Definitions (19)

  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • ...and 9 more