[Paper Review] Functional Itō calculus in Hilbert spaces and application to path-dependent Kolmogorov equations

Mauro Rosestolato|arXiv (Cornell University)|Jun 20, 2016

Stochastic processes and financial applications12 references3 citations

TL;DR

This paper extends functional Itō calculus to Hilbert space-valued diffusions, establishing a path-dependent Itō formula and applying it to solve path-dependent Kolmogorov equations. It derives a Clark-Ocone type representation and proves regularity for solutions to SDEs with path-dependent drift and constant diffusion, validating the framework on a class of delay SDEs with Radon measure kernels.

ABSTRACT

Recently, functional Itō calculus has been introduced and developed in finite dimension for functionals of continuous semimartingales. With different techniques, we develop a functional Itō calculus for functionals of Hilbert spacevalued diffusions. In this context, we first prove a path-dependent Itō's formula, then we show applications to classical solutions of path-dependent Kolmogorov equations in Hilbert spaces and derive a Clark-Ocone type formula. Finally, we explicitly verify that all the theory developed can be applied to a class of diffusions driven by SDEs with a path-dependent drift (suitably regular) and constant diffusion coefficient.

Motivation & Objective

• To develop functional Itō calculus in infinite-dimensional Hilbert spaces for non-anticipative functionals of continuous semimartingales.
• To establish a path-dependent Itō formula for functionals of Hilbert space-valued diffusions driven by stochastic differential equations.
• To show that the solution to a path-dependent Kolmogorov equation with terminal condition corresponds to the conditional expectation of a functional of the diffusion process.
• To derive a Clark-Ocone type representation formula for the solution of the path-dependent Kolmogorov equation.
• To verify the regularity of the solution map to the SDE with path-dependent drift and constant diffusion coefficient in the Hilbert space of paths.

Proposed method

• Formalizes a functional Itō calculus for non-anticipative functionals of Hilbert space-valued diffusions using time and space derivatives in the pathwise sense.
• Derives a path-dependent Itō formula for functionals $ u(s, X^{t,oldsymbol{x}}_s) $, where $ X^{t,oldsymbol{x}} $ is a diffusion with path-dependent drift and constant diffusion coefficient.
• Applies the Itō formula to show that the function $ \varphi(t, \mathbf{x}) = \mathbb{E}[f(X^{t,\mathbf{x}})] $ solves a path-dependent backward Kolmogorov equation with terminal condition $ f $.
• Establishes a Clark-Ocone type formula by identifying the stochastic integrand in the martingale representation via the vertical derivative of the solution $ \varphi $.
• Analyzes the Fréchet differentiability of the solution map $ (t, \mathbf{x}) \mapsto X^{t,\mathbf{x}} $ in the space $ \mathbb{B}^1(H) $, under regularity assumptions on the drift and diffusion coefficients.
• Uses composition rules and boundedness of differentials to verify that the solution map $ \varphi $ belongs to the class $ \mathcal{G}^2(\mathbb{B}^1_\infty(H), \mathbb{R}) $ with uniformly bounded first and second-order derivatives.

Experimental results

Research questions

• RQ1 Can functional Itō calculus be extended from finite-dimensional to infinite-dimensional Hilbert spaces for path-dependent functionals of diffusions?
• RQ2 Does a path-dependent Itō formula hold for functionals of Hilbert space-valued diffusions with path-dependent drift and constant diffusion coefficient?
• RQ3 Can the solution to a path-dependent Kolmogorov equation be represented as the conditional expectation of a functional of the diffusion process?
• RQ4 Is a Clark-Ocone type representation formula derivable for the solution of the path-dependent Kolmogorov equation in this infinite-dimensional setting?
• RQ5 What regularity properties does the solution map $ (t, \mathbf{x}) \mapsto X^{t,\mathbf{x}} $ possess in the Hilbert space of paths, particularly when the drift involves a convolution with a Radon measure?

Key findings

• A path-dependent Itō formula is established for non-anticipative functionals of Hilbert space-valued diffusions driven by SDEs with path-dependent drift and constant diffusion coefficient.
• The solution $ \varphi(t, \mathbf{x}) = \mathbb{E}[f(X^{t,\mathbf{x}})] $ to the path-dependent Kolmogorov equation is shown to be in $ \mathcal{G}^2(\mathbb{B}^1_\infty(H), \mathbb{R}) $ with uniformly bounded first and second-order Fréchet derivatives.
• A Clark-Ocone type formula is derived, where the stochastic integrand in the martingale representation is identified with the vertical derivative of $ \varphi $.
• The solution map $ (t, \mathbf{x}) \mapsto X^{t,\mathbf{x}} $ is proven to be Fréchet differentiable in $ \mathbb{B}^1(H) $, with differentials bounded uniformly in $ \omega, t, \mathbf{x} $, under the assumption of a drift with convolution structure involving a Radon measure.
• The theory is explicitly verified for SDEs with path-dependent drift of the form $ b(t, \mathbf{x}) = \int \tilde{g}(t-s, \mathbf{x}(t-s)) \mu(ds) $, where $ \tilde{g} $ is $ C^2_b $ with uniformly continuous derivatives.
• The differentiability of the composition $ \mathbf{x} \mapsto f(X^{t,\mathbf{x}}) $ is established in $ \mathbb{B}^1(H) $, with bounded Fréchet derivatives derived via chain rule and uniform boundedness of the differentials of $ X^{t,\mathbf{x}} $.