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Numerical method for solving stochastic differential equations with Poissonian white shot noise
Changho Kim1, Eok Kyun Lee, Peter Hänggi
1Department of Chemistry and School of Molecular Science (BK21), Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea.
Summary
We developed a numerical method for stochastic differential equations with Poissonian noise. This accurate and efficient algorithm works well even when noise approaches Gaussian characteristics.
Area of Science:
- Computational Physics
- Stochastic Processes
- Numerical Analysis
Background:
- Stochastic differential equations (SDEs) are crucial for modeling systems with inherent randomness.
- Solving SDEs driven by non-Gaussian noise, such as Poissonian white shot noise, presents significant numerical challenges.
- Existing methods often struggle with the unique characteristics of shot noise, limiting their applicability.
Purpose of the Study:
- To introduce a novel numerical integration scheme specifically designed for SDEs driven by Poissonian white shot noise.
- To demonstrate the scheme's convergence to the well-known Euler formula in the limit of Gaussian white noise.
- To validate the accuracy and efficiency of the proposed method through a relevant physical system.
Main Methods:
- Development of a discrete-time numerical integration formula based on an integral equation equivalent to the SDE.
- Approximation of the continuous-time SDE using a fixed time step.
- Analysis of the dynamics of an overdamped particle in a periodic potential subjected to Poissonian white shot noise.
Main Results:
- The proposed numerical integration scheme accurately solves SDEs driven by Poissonian white shot noise.
- The scheme demonstrates convergence towards the Euler formula as Poissonian noise approaches Gaussian white noise.
- The algorithm's accuracy shows weak dependence on the specific parameters of the Poissonian noise.
Conclusions:
- The developed numerical scheme offers a robust and efficient tool for simulating SDEs with Poissonian white shot noise.
- The method's stability and accuracy across varying noise parameters make it broadly applicable in computational physics and related fields.
- This work provides a valuable advancement in the numerical treatment of non-Gaussian stochastic processes.
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