[Paper Review] Exact Simulation of One-dimensional Stochastic Differential Equations involving the local time at zero of the unknown process

Pierre Étoré, Miguel Martínez|arXiv (Cornell University)|Feb 13, 2011

Stochastic processes and financial applications12 references4 citations

TL;DR

This paper extends exact simulation methods for one-dimensional SDEs to include processes with local time at zero, using a Girsanov transformation to link the target process to a skew Brownian motion with drift. The key contribution is a simulation algorithm that enables exact sampling of trajectories by leveraging the tractable bridge laws of skew Brownian motion with drift, overcoming the challenge of local time terms in the SDE dynamics.

ABSTRACT

In this article we extend the exact simulation methods of Beskos et al. to the solutions of one-dimensional stochastic differential equations involving the local time of the unknown process at point zero. In order to perform the method we compute the law of the skew Brownian motion with drift. The method presented in this article covers the case where the solution of the SDE with local time corresponds to a divergence form operator with a discontinuous coefficient at zero. Numerical examples are shown to illustrate the method and the performances are compared with more traditional discretization schemes.

Motivation & Objective

• To develop an exact simulation method for one-dimensional SDEs that include a local time term at zero, which are not covered by classical exact simulation techniques.
• To address the challenge that such SDEs are not absolutely continuous with respect to Wiener measure due to the local time component.
• To extend the Beskos et al. exact simulation framework to processes skewed at zero, particularly those arising from divergence-form operators with discontinuous coefficients.
• To enable simulation of diffusions with discontinuous drift and local time at zero, relevant in modeling discontinuous media.
• To provide a foundation for future extensions to multiple skew points or non-bounded drifts.

Proposed method

• Use a Girsanov transformation to express the law of the target process as absolutely continuous with respect to a skew Brownian motion with drift, eliminating the local time from the Radon-Nikodym derivative.
• Leverage the known transition density of skew Brownian motion with drift to compute the likelihood ratio for exact simulation.
• Apply the time-reversal and excursion decomposition techniques from Pitman-Yor (1991) to characterize the bridge laws of the skew Brownian motion with drift.
• Use the joint law of the maximum and first hitting time to decompose the bridge path into independent fragments, enabling exact path reconstruction.
• Simulate the bridge paths by conditioning on the maximum and hitting time, using the known distributions of first passage times under the skew Brownian motion with drift.
• Construct the exact simulation algorithm by combining the Girsanov change of measure with exact simulation of the reference skew Brownian bridge.

Experimental results

Research questions

• RQ1 Can exact simulation be extended to one-dimensional SDEs that include a local time term at zero, which are outside the scope of classical exact simulation methods?
• RQ2 How can the presence of local time at zero be handled in the Girsanov transformation to ensure the reference measure is tractable?
• RQ3 What are the bridge laws of skew Brownian motion with drift, and can they be simulated exactly to enable path reconstruction?
• RQ4 Is it possible to relax the boundedness assumption on the drift function in the exact simulation algorithm, as done in classical SDEs?
• RQ5 What are the implications of the skewness parameter β ≠ 0 for the simulation of diffusion processes with discontinuous coefficients?

Key findings

• The law of the solution to the SDE with local time at zero is absolutely continuous with respect to the law of a skew Brownian motion with drift, enabling the use of Girsanov-based exact simulation.
• The transition density of the skew Brownian motion with drift is explicitly derived and used to compute the likelihood ratio in the Girsanov transformation.
• The bridge laws of the skew Brownian motion with drift are characterized via the joint distribution of the maximum and first hitting time, based on Pitman-Yor (1991) results.
• The path of the bridge can be decomposed into independent fragments using time-reversal and excursion decomposition, allowing exact simulation of the full trajectory.
• Numerical examples confirm the method’s accuracy and superior performance compared to standard discretization schemes like the Euler scheme.
• The method is applicable to SDEs corresponding to divergence-form operators with discontinuous coefficients at zero, such as those in ecological, geophysical, and biomedical modeling.