Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps
2014
To realize the applications of stochastic differential equations with jumps, much attention has recently been paid to the construction of efficient numerical solutions of the equations. Considering the fact that the use of the explicit methods often results in instability and inaccurate approximations in solving stochastic differential equations, we propose two implicit methods, the θ-Taylor method and the balanced θ-Taylor method, for numerically solving the stochastic differential equation with jumps and prove that the numerical solutions are convergent with strong order 1.0. For a linear scalar test equation, the mean-square stability regions of the methods are derived. Finally, numerical examples are given to evaluate the performance of the methods.
Keywords:
- Mathematical analysis
- Stochastic partial differential equation
- Exponential integrator
- Mathematical optimization
- Backward differentiation formula
- Explicit and implicit methods
- Mathematics
- Numerical methods for ordinary differential equations
- Numerical stability
- Runge–Kutta method
- Numerical partial differential equations
- Stochastic differential equation
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