Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Revisiting Stochastic Realization Theory using Functional Itô Calculus

Tanya Veeravalli, Maxim Raginsky

TL;DR

The paper addresses the problem of constructing finite-dimensional state-space realizations for outputs of causal stochastic systems driven by continuous semimartingales, by leveraging Dupire's functional Itô calculus to define Dupire-differentiability. It extends Hijab’s Brownian-driven approach and integrates Chen–Fliess-type formal expansions with Lie and Hankel ranks to characterize when finite-dimensional realizations exist, providing a blueprint for bilinear and nonlinear realizations. The work formalizes a stochastic realization theory for Dupire-smooth processes, connects to deterministic nonlinear realization via Lie derivatives, and presents canonical examples including memoryless systems, linear filters, nonlinear state-space models, and nonlinear filtering. It also discusses the relationship to Wiener/Volterra representations and outlines crucial future work on convergence and extension to broader càdlàg inputs. Overall, the framework offers a principled route to finite-dimensional, realizable models for diffusion-driven outputs with potential impact on nonlinear filtering and stochastic control.

Abstract

This paper considers the problem of constructing finite-dimensional state space realizations for stochastic processes that can be represented as the outputs of a certain type of a causal system driven by a continuous semimartingale input process. The main assumption is that the output process is infinitely differentiable, where the notion of differentiability comes from the functional Itô calculus introduced by Dupire as a causal (nonanticipative) counterpart to Malliavin's stochastic calculus of variations. The proposed approach builds on the ideas of Hijab, who had considered the case of processes driven by a Brownian motion, and makes contact with the realization theory of deterministic systems based on formal power series and Chen-Fliess functional expansions.

Revisiting Stochastic Realization Theory using Functional Itô Calculus

TL;DR

The paper addresses the problem of constructing finite-dimensional state-space realizations for outputs of causal stochastic systems driven by continuous semimartingales, by leveraging Dupire's functional Itô calculus to define Dupire-differentiability. It extends Hijab’s Brownian-driven approach and integrates Chen–Fliess-type formal expansions with Lie and Hankel ranks to characterize when finite-dimensional realizations exist, providing a blueprint for bilinear and nonlinear realizations. The work formalizes a stochastic realization theory for Dupire-smooth processes, connects to deterministic nonlinear realization via Lie derivatives, and presents canonical examples including memoryless systems, linear filters, nonlinear state-space models, and nonlinear filtering. It also discusses the relationship to Wiener/Volterra representations and outlines crucial future work on convergence and extension to broader càdlàg inputs. Overall, the framework offers a principled route to finite-dimensional, realizable models for diffusion-driven outputs with potential impact on nonlinear filtering and stochastic control.

Abstract

This paper considers the problem of constructing finite-dimensional state space realizations for stochastic processes that can be represented as the outputs of a certain type of a causal system driven by a continuous semimartingale input process. The main assumption is that the output process is infinitely differentiable, where the notion of differentiability comes from the functional Itô calculus introduced by Dupire as a causal (nonanticipative) counterpart to Malliavin's stochastic calculus of variations. The proposed approach builds on the ideas of Hijab, who had considered the case of processes driven by a Brownian motion, and makes contact with the realization theory of deterministic systems based on formal power series and Chen-Fliess functional expansions.
Paper Structure (9 sections, 2 theorems, 63 equations)

This paper contains 9 sections, 2 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Causal stochastic systems
  3. Functional Itô calculus
  4. Dupire-differentiable causal stochastic systems
  5. Examples
  6. Hijab's formulation as a special case
  7. Realization theory for Dupire-smooth processes
  8. Relation to Wiener and Volterra series
  9. Conclusion

Key Result

Theorem 1

Dupire2009FunctionalICCont_2010 Let $F$ be continuous causal system $F$ with continuous first- and second-order derivatives $\partial_0 F, \dots, \partial_m F$ and $\partial_i \partial_j F$, $i,j = 1,\dots,m$. Let $W$ be a continuous semimartingale satisfying the conditions eq:W_qvar_1 and eq:W_qvar

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 3
  • Example 1
  • ...and 7 more