Momentum-Space Asymptotic Expansions for Stochastic Filtering

Summary

This paper describes an asymptotic expansion method for stochastic filtering, aimed at approximating a conditional distribution in nonlinear settings. It transforms the problem into momentum space using a Fourier transform and approximates nonlinear terms with polynomial functions. The resulting formulation is a closed recursive system of ordinary differential equations, which can be solved in sequence to calculate higher-order approximations.

For numerical implementation, the method advances through short sub-periods, updating initial conditions at each step. The document reports that this substepping improves performance in cases where the approximation otherwise fails badly. It proposes applications to financial models with unobserved parameters and nonlinear measure-valued processes, but provides no specific datasets, benchmarks, or trading results in the excerpt.

Key ideas

• A Fourier transform moves the filtering problem into momentum space.
• Polynomial approximations to nonlinear terms yield recursive ordinary differential equations.
• Sequential equation solving supports higher-order asymptotic calculations.
• Substepping with updated initial conditions can improve numerical performance where an approximation fails.
• The excerpt suggests financial filtering applications but gives no benchmark details.

Tags

# Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering


# Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering









This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distribution. Thanks to the simplicity of the ODE system, higher order calculation can be performed easily. Furthermore, solving ODEs sequentially with small sub-periods with updated initial conditions makes it possible to implement a substepping method for asymptotic expansion in a numerically efficient way. This is found to improve the performance significantly where otherwise the approximation fails badly. The method is expected to provide a useful tool for more realistic financial modeling with unobserved parameters, and also for problems involving nonlinear measure-valued processes.

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