What is it about?

In this paper, we propose an efficient general simulation method for diffusions which are solutions to stochastic differential equations with discontinuous coefficients and local time terms. The proposed method is based on sampling from the corresponding continuous-time Markov chain (CTMC) approximation. In contrast to existing time discretization schemes, the Markov chain approximation method corresponds to a spatial discretization scheme, and is demonstrated to be particularly suited for simulating diffusion processes with discontinuities in their state space. We establish the theoretical convergence order and also demonstrate the accuracy and robustness of the method in numerical examples by comparing it to the known benchmarks in terms of root mean squared error (RMSE), run time and the parameter sensitivity.

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Why is it important?

The main contributions of this paper are as follows: (1) A simulation framework for SDEs with local time term and discontinuity occurring at fixed known points is developed, which will cover many practical applications. (2) The convergence rate is faster than the traditional Euler time-stepping scheme, which is of first order. The (weak) convergence order of our state-discretization scheme, (or CTMC approximation scheme) is O(n^{-2}) with n being the number of finite states of the approximating CTMC. (3) Our scheme allows one to obtain the distribution function in closed-form once the state space is discretized, which is more flexible for further computations. (4) We also simulate the integral functional of the original process for fixed terminal value conditional on its initial value, based on the approximating distribution function with closed-form expression. This outperforms the alternative time-discretization technique, whose accuracy cannot be easily improved since the time integral part of the algorithm itself also need to be discretized.

Perspectives

Writing this article was a great pleasure as it has co-authors with whom I have had long standing collaborations. In this paper, we propose a spatial-discretization method to simulate stochastic differential equations with discontinuous coefficients and local times. The spatial discretization algorithms based on the CTMC approximation are provided and the distribution functions are given in closed-form expressions. We also illustrate two potential applications: ranked diffusion and bang-bang control problem. I hope this article makes people pay more attention on discontinuous coefficients diffusions and their potential values.

Kailin Ding

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This page is a summary of: A General Framework to Simulate Diffusions with Discontinuous Coefficients and Local Times, ACM Transactions on Modeling and Computer Simulation, August 2022, ACM (Association for Computing Machinery), DOI: 10.1145/3559541.
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