Wiener Functionals Research Articles

New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.

SUPPORT THEOREM FOR PINNED DIFFUSION PROCESSES

AbstractIn this paper, we prove a support theorem of Stroock–Varadhan type for pinned diffusion processes. To this end, we use two powerful results from stochastic analysis. One is quasi-sure analysis for Brownian rough path. The other is Aida–Kusuoka–Stroock’s positivity theorem for the densities of weighted laws of non-degenerate Wiener functionals.

High order asymptotic expansion for Wiener functionals

Combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for general Wiener functionals. Our method gives an expansion of the characteristic functional and of the local density of a Wiener functional up to an arbitrary order. The asymptotic expansion is distributional. Except for the non-degeneracy of the limit covariance matrix, we do not assume any condition of non-degeneracy of the Malliavin covariance like a non-degeneracy condition for temporally local characteristic functions so far assumed in the theory for mixing processes, that corresponds to the Cramér condition in the classical setting. Moreover, our method does not require the Markovian property used in the mixing approach. An application to the stochastic wave equation with space–time white noise is discusses.

Asymptotic expansion and estimates of Wiener functionals

Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and other random symbols. To specify these random symbols, it is necessary to classify the level of the effect of each term appearing in the stochastic expansion of the variable in question. To solve this problem, we consider a class L of certain sequences (In)n∈N of Wiener functionals and give a systematic way of estimation of the order of (In)n∈N. Based on this method, we introduce a notion of exponent of the sequence (In)n∈N, and investigate the stability and contraction effect of the operators Dun and D on L, where un is the integrand of a Skorohod integral. After constructed these machineries, we derive asymptotic expansion of the variation having anticipative weights. An application to robust volatility estimation is mentioned.

The Boué–Dupuis formula and the exponential hypercontractivity in the Gaussian space

This paper concerns a variational representation formula for Wiener functionals. Let $B=\{ B_{t}\} _{t\ge 0}$ be a standard $d$-dimensional Brownian motion. Bou\'e and Dupuis (1998) showed that, for any bounded measurable functional $F(B)$ of $B$ up to time $1$, the expectation $\mathbb{E}\!\left[ e^{F(B)}\right] $ admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also $F(B)$ to be a functional of $B$ over the whole time interval, we prove that the Bou\'e--Dupuis formula holds true provided that both $e^{F(B)}$ and $F(B)$ are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein--Uhlenbeck semigroup in $\mathbb{R}^{d}$, and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the $d$-dimensional Gaussian space.

Higher-Order Error Estimates of the Discrete-Time Clark–Ocone Formula

In this article, we investigate the convergence rate of the discrete-time Clark–Ocone formula provided by Akahori–Amaba–Okuma (J Theor Probab 30: 932–960, 2017). In that paper, they mainly focus on the \(L_{2}\)-convergence rate of the first-order error estimate related to the tracking error of the delta hedge in mathematical finance. Here, as two extensions, we estimate “the higher order error” for Wiener functionals with an integrability index 2 and “an arbitrary differentiability index.”

Applications of Lévy Differential Operators in the Theory of Gauge Fields

This paper is a survey of results on the relationship between gauge fields and infinitedimensional equations for parallel transport that contain the Levy Laplacian or the divergence associated with this Laplacian. Also we analyze the deterministic case where parallel transports are operator-valued functionals on the space of curves and the case of the Malliavin calculus where (stochastic) parallel transports are operator-valued Wiener functionals.

Structure–Property Correlation for Calculating the Critical Pressures of Liquid–Vapor Phase Transitions from the Topological Characteristics of Alkene Molecules

A multidimensional QSPR model is proposed for calculating the critical pressure of normal and substituted alkenes from combinations of the topological descriptors of molecular graphs, i.e., the Wiener and Randic indices and function of the eigenvalues of the topological matrix, and indices that consider the differences between cis- and trans-isomers. It is shown that the QSPR model adequately describes the critical pressure of olefins for the liquid–vapor phase transition. It is concluded that the obtained results can be recommended for determining the critical properties of known and novel alkenes. They can be used when making scientific and engineering calculations in petroleum chemistry and the technology of supercritical fluids.

Life Prediction and Long-Term Durability of Coated Steel Bars in Magnesium Oxychloride Concrete

The solution to the problem of corrosion of the steel bars in magnesium oxychloride cement concrete (MOCC) is to apply a coating that protects the steel. To investigate the coating's long- term protective effect on the steel, a solution immersion accelerated corrosion test was carried out on coated reinforced MOCC for up to 2,160 days. Electrochemical tests were conducted every 90 days to measure the polarization curve and AC impedance during the steel bars' corrosion process. Then, the corrosion current density that characterizes the corrosion of a coated steel bar was used as a degradation index, and the time to the coated steel bar's corrosion was predicted based on the Wiener function. It was concluded that after 2,160 days of solution immersion, the coated steel bars' corrosion current density was 0.063 µA·cm−2, indicating that the coating protected the steel bars in MOCC from corrosion well. Thdroughout the immersion process, the specimen's corrosion current density showed a steady and increasing trend, and its polarization impedance showed a volatile growth trend. The degradation of the coated steel bars also obeyed the Wiener degradation process, and the bars reached a severely rusted state at 21,800d with a corrosion time of R0.6(t) is 10,650d.

Estimates of the Difference Between Two Probability Densities of Wiener Functionals and Its Application

This study investigates precise estimates of the difference between two probability densities of Wiener functionals in the space of continuously differentiable functions and the Holder continuous functions. As an application, the convergence rate of the density of the solution to non-Markovian stochastic differential equations is derived utilizing these precise estimates.

Pricing Path-Dependent Payoff Using Quadrature on Wiener Space

Solution to stochastic differential equation that defines the dynamics of asset prices is interpreted as a function of underlying Brownian motion when permissible, and the expectation values of various functionals of the solution – therefore functionals of Brownian motion (Wiener functionals) – are estimated from numerical approximations to high-dimensional Gaussian integrals using quadrature. The simple method is exemplified for a few standard asset price dynamics used in practice for structured and contingent claim pricing, such as local volatility, Heston stochastic volatility, SABR, and LIBOR market model (LMM/BGM). The quadrature in these concrete implementations is performed in m-dimensional space where m is between 10 and 30 depending on the specific model or a payoff structure. The methodology appears to be a broadly viable alternative to Monte Carlo for short- to medium-dated financial products, with potential uses outside those examined in this paper.

Generalized diffusion of quantum Brownian motion.

This article discusses the numerical result predicted by the quantum Langevin equation of the generalized diffusion function of a Brownian particle immersed in an Ohmic quantum bath of harmonic oscillators. The time dependence of the standard deviation of the reduced Wigner function of the system, obtained by integrating the whole function in the momentum space, is determined as well. The complexity of the equations leads to resort to a much simpler calculation based in the position correlation function. They are done for the three possible regimes of the system, namely, periodic, aperiodic, and overdamped. It is found in the periodic case that the generalized diffusion is a discontinuous function exhibiting negative values during short time periods of time. This counterintuitive result, found theoretically in other systems and waiting for its experimental confirmation, can be perfectly explained in the framework of the quantum Langevin equation. Its oscillatory behavior is primordially due to the response to the external field while its quantum origin contributes only in its magnitude. The results are compared to those of the continuum limit which exhibits similar behavior.

On the Smallness of a Paley–Wiener Function Near Its Integer Zeros

We prove that a function from the Paley–Wiener class PWπ vanishing on some subset in ℤ cannot be concentrated near this set.

Malliavin calculus for non-colliding particle systems

In this paper, we use Malliavin calculus to show the existence and continuity of density functions of d-dimensional non-colliding particle systems such as hyperbolic particle systems and Dyson Brownian motion with smooth drift. For this purpose, we apply results proved by Florit and Nualart (1995) and Naganuma (2013) on locally non-degenerate Wiener functionals.

Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data

This paper establishes an upper bound for the Kolmogorov distance between the maximum of a high-dimensional vector of smooth Wiener functionals and the maximum of a Gaussian random vector. As a special case, we show that the maximum of multiple Wiener–Itô integrals with common orders is well approximated by its Gaussian analog in terms of the Kolmogorov distance if their covariance matrices are close to each other and the maximum of the fourth cumulants of the multiple Wiener–Itô integrals is close to zero. This may be viewed as a new kind of fourth moment phenomenon, which has attracted considerable attention in the recent studies of probability. This type of Gaussian approximation result has many potential applications to statistics. To illustrate this point, we present two statistical applications in high-frequency financial econometrics: One is the hypothesis testing problem for the absence of lead-lag effects and the other is the construction of uniform confidence bands for spot volatility.

Fine properties of fractional Brownian motions on Wiener space

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter H under the (p,r)-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener functionals via the integral representation discovered by Decreusefond and Üstünel, and show non-differentiability, modulus of continuity, law of the iterated logarithm (LIL) and self-avoiding properties of fractional Brownian motion sample paths using Malliavin calculus as well as the tools developed in the previous work by Fukushima, Takeda and etc. for Brownian motion case.

Applying Multiple Stochastic iterated integral to moment inequalities for Wiener Functional

Applying Multiple Stochastic iterated integral to moment inequalities for Wiener Functional

A weak version of path-dependent functional Itô calculus

We introduce a variational theory for processes adapted to the multidimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the “sensitivities” of processes, namely derivatives of martingale components and a weak notion of infinitesimal generator, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t. Brownian motion driving noise.

Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates

This article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity λ>0. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of O(λ−1∕4) for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.

Weak differentiability of Wiener functionals and occupation times

In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Leão, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t. Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite p-variation (p≥2) regularity is also discussed. Under stronger regularity conditions in the sense of finite (p,q)-variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established.