[Paper Review] Filtering equations for partially observable diffusion processes with Lipschitz continuous coefficients

Н. В. Крылов|ArXiv.org|Aug 13, 2009

Advanced Mathematical Modeling in Engineering14 references8 citations

TL;DR

This paper establishes $L_p$-smoothness of filtering densities for partially observable diffusion processes with Lipschitz continuous coefficients by reformulating the filtering equation in divergence form. Under nondegeneracy and Lipschitz assumptions, it proves the filtering density belongs to $H^1_p$, achieving almost Lipschitz continuity in space and $1/2$-Hölder continuity in time, even under minimal regularity conditions on coefficients.

ABSTRACT

We present several results on smoothness in $L_{p}$ sense of filtering densities under the Lipschitz continuity assumption on the coefficients of a partially observable diffusion processes. We obtain them by rewriting in divergence form filtering equation which are usually considered in terms of formally adjoint to operators in nondivergence form.

Motivation & Objective

• To establish $L_p$-smoothness of filtering densities for partially observable diffusion processes under minimal regularity assumptions on coefficients.
• To remove the need for smoothness assumptions on the diffusion coefficient $\theta\theta^*$ by using analytic methods instead of filtering theory.
• To prove that the filtering density $\pi_t$ is almost Lipschitz continuous in $x$ and $1/2$-Hölder continuous in $t$ under Lipschitz continuity of coefficients.
• To show that the solution to the filtering equation is stable under data perturbations and independent of the choice of $p \geq 2$.
• To establish embedding results that yield pathwise Hölder regularity of the filtering density in time and space.

Proposed method

• Rewriting the filtering equation in divergence form to leverage analytic tools from stochastic PDEs.
• Using a priori estimates in $\mathbb{H}^1_p$-spaces for SPDEs with coefficients satisfying Lipschitz and nondegeneracy conditions.
• Applying maximal $L_p$-estimates and embedding theorems to derive Hölder continuity of the solution in time and space.
• Establishing continuous dependence of solutions on data via convergence in $L_p$-norms and predictable measures.
• Proving uniqueness of solutions in $\mathcal{H}^1_p$-spaces across different $p$ values using interpolation and embedding techniques.
• Using the maximum principle to ensure nonnegativity of the filtering density under nonnegative initial and forcing data.

Experimental results

Research questions

• RQ1 Under what minimal regularity assumptions on the coefficients of a partially observable diffusion process can the filtering density be shown to be smooth in $L_p$?
• RQ2 Can the $L_p$-regularity of the filtering density be established without relying on filtering theory, using only SPDE techniques?
• RQ3 What is the optimal regularity of the filtering density in terms of Hölder continuity in time and space under Lipschitz continuous coefficients?
• RQ4 How does the solution to the filtering equation behave under perturbations of the coefficients and initial data?
• RQ5 Is the solution to the filtering SPDE unique across different $L_p$-spaces, and does it preserve positivity?

Key findings

• The filtering density $\pi_t$ belongs to $H^1_p$ for any $p \geq 2$ under Lipschitz continuous coefficients and nondegeneracy.
• If the initial data is sufficiently regular, $\pi_t(x)$ is almost Lipschitz continuous in $x$ and $1/2$-Hölder continuous in $t$.
• The solution to the filtering equation is stable under data perturbations: $\|u^n - u\|_{\mathbb{H}^1_p(\tau \wedge T)} \to 0$ as $n \to \infty$.
• The solution is unique in $\mathcal{H}^1_p(\tau)$ across different $p \geq 2$, with the solution lying in $\mathcal{H}^1_{p_1}(\tau) \cap \mathcal{H}^1_{p_2}(\tau)$.
• Embedding theorems yield pathwise Hölder regularity: $E[\|u\|_{C^{\alpha/2 - 1/p}([0,\tau], H^{1-\beta}_p)}]^p \leq NT^{(\beta - \alpha)/p} a^{\beta - 1} I(a)$ for $2/p < \alpha < \beta \leq 1$.
• When $p(1 - \beta) > d$, the solution satisfies $E[\sup_x |u(\cdot,x)|_{C^{\alpha/2 - 1/p}}]^p \leq NT^{(\beta - \alpha)/p} a^{\beta - 1} I(a)$, implying spatial Hölder continuity.