Stochastic Calculus Of Variations Research Articles

A Stochastic Fractional Calculus with Applications to Variational Principles

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

Stochastic and variational approach to finite difference approximation of Hamilton-Jacobi equations

The author presented a stochastic and variational approach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic setting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. (2015)]. In the current paper, we extend these results to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equations, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks; a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition; the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and variational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs (1969)].

The Malliavin calculus and hypoelliptic differential operators

This article is intended as an introduction to Malliavin's stochastic calculus of variations and his probabilistic approach to hypoellipticity. Topics covered include an elementary derivation of the basic integration by parts formulae, a proof of the probabilistic version of Hörmander's theorem as envisioned by Malliavin and completed by Kusuoka and Stroock, and an extension of Hörmander's theorem valid for operators with degeneracy of exponential type due to the author and S. Mohammed.

Stein’s method for invariant measures of diffusions via Malliavin calculus

Given a random variable F regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and any probability measure with a density function which is continuous, bounded, strictly positive on an interval in the real line and admits finite variance. The bounds are given in terms of the Malliavin derivative of F. Our approach is based on the theory of Itô diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.

Stochastic quantization of finite dimensional systems with electromagnetic interactions

We study the stochastic quantization of finite dimensional systems via path-wise calculus of variations with the mean discretized classical action in the general case of electromagnetic interactions. We show that there exists a unique choice of the mean discretized action corresponding to the minimal classical magnetic coupling and derive the general equations of motion by means of a path-wise stochastic calculus of variations. In the case of purely scalar interactions, the total mean energy of the system (which gives the usual quantum mechanical expectation of the Hamiltonian in the canonical limit) works as a Lyapunov functional and the system relaxes on the canonical solutions, represented by Nelson's diffusions, which act as an attracting set. We show that, in presence of a minimal magnetic coupling, the mean energy is no longer a Lyapunov functional. We construct for a simple example, a new Lyapunov functional, and we show that the system can reach the dynamical equilibrium also by absorbing energy from the external magnetic field.

Blind Deconvolution of the Aortic Pressure Waveform Using the Malliavin Calculus

Multichannel Blind Deconvolution (MBD) is a powerful tool particularly for the identification and estimation of dynamical systems in which a sensor, for measuring the input, is difficult to place. This paper presents an MBD method, based on the Malliavin calculus MC (stochastic calculus of variations). The arterial network is modeled as a Finite Impulse Response (FIR) filter with unknown coefficients. The source signal central arterial pressure CAP is also unknown. Assuming that many coefficients of the FIR filter are time-varying, we have been able to get accurate estimation results for the source signal, even though the filter order is unknown. The time-varying filter coefficients have been estimated through the proposed Malliavin calculus-based method. We have been able to deconvolve the measurements and obtain both the source signal and the arterial path or filter. The presented examples prove the superiority of the proposed method, as compared to conventional methods.

A Stochastic Poisson Structure

We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra.

Renormalized stochastic calculus of variations for a renormalized infinite-dimensional Brownian motion

We build a stochastic calculus of variations for some renormalized infinite-dimensional Brownian motions.

Estimating Multidimensional Density Functions Using the Malliavin–Thalmaier Formula

The Malliavin–Thalmaier formula was introduced in [P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance, Springer-Verlag, Berlin, 2006] as an alternative expression for the density of a multivariate smooth random variable in Wiener space. In comparison with classical integration by parts formulae, this alternative formulation requires the application of the integration by parts formula only once to obtain an expression that can be simulated. Therefore, this expression is free from the curse of dimensionality. Unfortunately, when this formula is applied directly in computer simulation, it exhibits unstable behavior. We propose an approximation to the Malliavin–Thalmaier formula in the spirit of the theory of kernel density estimation to solve this problem. In the first part of this paper, we obtain a central limit theorem for the estimation error. And in the latter part, we apply the Malliavin–Thalmaier formula for the calculation of Greeks in finance.

MALLIAVIN CALCULUS FOR THE ESTIMATION OF TIME-VARYING REGRESSION MODELS USED IN FINANCIAL APPLICATIONS

The paper introduces a new method for the estimation of time-varying regression coefficients employed in financial modeling. We use Malliavin calculus (stochastic calculus of variations) to estimate the time-varying regression coefficients that appear in linear regression models, and the generalized Clark–Ocone formula to derive a closed-form solution for the estimates of the time-varying coefficients. While this approach can be applied to any signal model, we present its application to signals modeled as a Brownian motion and an Ornstein–Uhlenbeck process. Simulation results prove the superiority of the proposed method, as compared to conventional methods.

Book Review: Stochastic calculus of variations in mathematical finance

Book Review: Stochastic calculus of variations in mathematical finance

Bismut Type Formulae for Diffusion Semigroups on Riemannian Manifolds

Applying the stochastic calculus of variations on frame bundles along tangent processes, we derive Bismut type formulae for the derivatives of diffusion semigroups on Riemannian manifold in both variables. We also obtain the Bismut formulae expressed in terms of the Ricci and torsion tensors for the connection with torsion.

Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold

We introduce Sobolev spaces and capacities on the path space P m 0 (M) over a compact Riemannian manifold M. We prove the smoothness of the Ito map and the stochastic anti-development map in the sense of stochastic calculus of variation. We establish a Sobolev norm comparison theorem and a capacity comparison theorem between the Wiener space and the path space P m 0 (M). Moreover, we prove the tightness of (r, p)-capacities on P m 0 (M), , which generalises a result due to Airault-Malliavin and Sugita on the Wiener space. Finally, we extend our results to the fractional Holder continuous path space .

Stochastic calculus of variations and Harnack inequality on Riemannian path spaces

We describe the tangent space of Riemannian path space as a space of tangent processes localized on Brownian sheets; the bundle of adapted frames above a Riemannian path space and its structural equation are given. The stochastic calculus of variations allows us to derive Harnack–Bismut inequality for the Norris semigroup. To cite this article: A.-B. Cruzeiro, P. Malliavin, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 817–820.

Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach

We consider a two-dimensional Kac equation without cutoff,which we relate to a stochastic differential equation.We prove the existence of a solution for this SDE, and we use the Malliavin calculus (or stochastic calculus of variations) to prove that the law of this solution admits a smooth density with respect to the Lebesgue measure on $\mathbf{R}^2$.This density satisfies the Kac equation.

Anticipating Integral Equations

In this paper we deduce some estimates of the Lp(Ω)-norm of the Skorohod and the forward integrals. These estimates allow us to study the existence of a unique solution to anticipating Volterra equations of the Skorohod and forward type. The coefficients Fi(t,s,x),t≥s, are \(F_t \)-measurable and satisfy some differentiability conditions (in the sense of the stochastic calculus of variations).

Existence and Regularity of a Solution of a Kac Equation Without Cutoff Using the Stochastic Calculus of Variations

We consider a generalized Kac equation without cutoff, with which we associate a non-standard nonlinear stochastic differential equation. We adapt the techniques in Bichteler and Jacod [2] to prove that the law of a solution of the stochastic differential equation has a density, which is a solution of the Kac equation. The initial law is very general: we only assume it has second order moments and is not the Dirac mass at 0. We thus generalize the analytical results of existence of a solution of this equation. If we furthermore assume existence of all moments for the initial law, we obtain as a corollary using the proof in Desvillettes [6] that the density is smooth. We prove a slightly better regularity result under more stringent assumptions using the stochastic calculus of variations, adapting the methods in [1].

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Ito formula, the Ito–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.

Malliavin calculus with time dependent coefficients applied to a class of stochastic differential equations

In this paper, we consider stochastic differential equations with time dependent coefficients driven by an infinite dimensional Brownian motion. Using the stochastic calculus of variations (Malliavin calculus), we prove, that under a local Hörmander condition, the law of the solution possesses a smooth density with respect to Lebesgue measure.

Estimation of Densities and Applications

In this paper we show some estimates for the density of a random variable on the Wiener space that satisfies a nondegeneracy condition using the stochastic calculus of variations. The case of a diffusion process is considered, and an application to the solution of a stochastic partial differential equation is discussed.