Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Relative entropy rate based multiple hidden Markov Model Approximation

This paper proposes a novel relative entropy rate (RER) based approach for multiple HMM (MHMM) approximation of a class of discrete-time uncertain processes. Under different uncertainty assumptions, the model design problem is posed either as a min-max optimisation problem or stochastic minimisation problem on the RER between joint laws describing the state and output processes (rather than the more usual RER between output processes). A suitable filter is proposed for which performance results are established which bound conditional mean estimation performance and show that estimation performance improves as the RER is reduced. These filter consistency and convergence bounds are the first results characterising multiple HMM approximation performance and suggest that joint RER concepts provide a useful model selection criteria. The proposed model design process and MHMM filter are demonstrated on an important image processing dim-target detection problem.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.