On the Numerical Solution of Some Non-Linear Stochastic Differential Equations Using the Semi-Discrete Method
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Nikolaos Halidias
Abstract
We are interested in the numerical solution of stochastic differential equations with non-negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super-linear stochastic differential equations. It is well known that the usual Euler scheme diverges on super-linear problems and the tamed Euler method does not preserve positivity. In that direction, we use the semi-discrete method that the first author has proposed in two previous papers. We propose a new numerical scheme for a class of stochastic differential equations which are super-linear with non-negative solution. The Heston 3/2-model appearing in financial mathematics belongs to this class of stochastic differential equations. For this model we prove, through numerical experiments, the “optimal” order of strong convergence at least 1/2 of the semi-discrete method.
The authors thank an anonymous referee and Prof. Dr. Peter E. Kloeden for their helpful comments.
© 2016 by De Gruyter
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Articles in the same Issue
- Frontmatter
- Dual-Dual Formulation for a Contact Problem with Friction
- A Fitted Finite-Volume Method Combined with the Lagrangian Derivative for the Weather Option Pricing Model
- Comparison of the Analytical Approximation Formula and Newton's Method for Solving a Class of Nonlinear Black–Scholes Parabolic Equations
- Mollification in Strongly Lipschitz Domains with Application to Continuous and Discrete De Rham Complexes
- Numerical Methods for Genetic Regulatory Network Identification Based on a Variational Approach
- On the Numerical Solution of Some Non-Linear Stochastic Differential Equations Using the Semi-Discrete Method
- Numerical Computation of the Inverse Born Approximation for the Nonlinear Schrödinger Equation in Two Dimensions
- Efficient Computation of Highly Oscillatory Integrals by Using QTT Tensor Approximation
- A Splitting Scheme to Solve an Equation for Fractional Powers of Elliptic Operators
- Predictor-Corrector Balance Method for the Worst-Case 1D Option Pricing
