Malliavin Calculus Research Articles

Asymptotic analysis of higher-order scattering transform of Gaussian processes

We analyze the scattering transform with the quadratic nonlinearity (STQN) of Gaussian processes without depth limitation. STQN is a nonlinear transform that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the deep convolutional neural network. We prove that with a proper normalization, the output of STQN converges to a chi-square process with one degree of freedom in the finite dimensional distribution sense, and we provide a total variation distance control of this convergence at each time that converges to zero at an exponential rate. To show these, we derive a recursive formula to represent the intricate nonlinearity of STQN by a linear combination of Wiener chaos, and then apply the Malliavin calculus and Stein’s method to achieve the goal.

Weak convergence of delay SDEs with applications to Carathéodory approximation

<p style='text-indent:20px;'>In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well.</p>

Solving a class of high-order fractional stochastic heat equations with fractional noise

<abstract><p>This paper is concerned with a class of high-order fractional stochastic partial differential equations driven by fractional noise. We firstly prove the existence and uniqueness of the mild solution and then study the Hölder continuity of the solution with respect to space and time variables. In addition, we also prove the existence and Gaussian-type estimates for the density of the solution by using the techniques of Malliavin calculus.</p></abstract>

Using moment approximations to study the density of jump driven SDEs

In order to study the regularity of the density of a solution of a infinite activity jump driven stochastic differential equation we consider the following two-step approximation method. First, we use the solution of the moment problem in order to approximate the small jumps by another whose Lévy measure has finite support. In a second step we replace the approximation of the first two moments by a small noise Brownian motion based on the Assmussen-Rosiński approach. This approximation needs to satisfy certain properties in order to apply the “balance” method which allows the study of densities for the solution process based on Malliavin Calculus for the Brownian motion. Our results apply to situations where the Lévy measure is absolutely continuous with respect to the Lebesgue measure or purely atomic measures or combinations of them.

LAMN property for multivariate inhomogeneous diffusions with discrete observations

We consider a class of multidimensional inhomogeneous diffusions whose drift coefficient depends on a multidimensional unknown parameter. Under some appropriate assumptions, we prove the local asymptotic mixed normality property for the drift parameter from high-frequency observations when the length of the observation window tends to infinity. To obtain the result, we use the Malliavin calculus techniques and the change of measures. Our approach is applicable for both ergodic and non-ergodic diffusions.

Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift

In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.

Bounds for the expected supremum of some non-stationary Gaussian processes

The aim of this article is to obtain closed-form lower and upper bounds for the expectation of the supremum of some non-stationary Gaussian processes. This supremum is an important quantity in applications such as finance and queueing systems. The covariance function structure, decomposition of the processes and probability inequalities are applied for some examples of non-stationary Gaussian processes, including sub-fractional, bifractional and multifractional Brownian motions. Malliavin calculus operators are also applied in some cases to find bounds for the density and expectation of the supremum of these processes.

Tail distribution estimates of the mixed-fractional CEV model

The aim of this paper is to study the tail distribution of the CEV model driven by Brownian motion and fractional Brownian motion. Based on the techniques of Malliavin calculus and a result established recently in [<a href="#1">1</a>], we obtain an explicit estimate for tail distributions.

Influence of risk tolerance on long-term investments: a Malliavin calculus approach

This study investigates the influence of risk tolerance on the expected utility in the long run. We estimate the extent to which the expected utility of optimal portfolios is affected by small changes in the risk tolerance. For this purpose, we adopt the Malliavin calculus method and the Hansen–Scheinkman decomposition, through which the expected utility is expressed in terms of the eigenvalues and eigenfunctions of an operator. We conclude that the influence of risk aversion on the expected utility is determined by these eigenvalues and eigenfunctions in the long run.

Central limit theorems for spatial averages of the stochastic heat equation via Malliavin–Stein’s method

Suppose that \(\{u(t, x)\}_{t >0, x \in {\mathbb {R}}^d}\) is the solution to a d-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang’s condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form \(N^{-d} \int _{[0,N]^d} g(u(t,x))\, \mathrm {d}x\), as \(N\rightarrow \infty \), where g is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein’s method for normal approximations. Our results include a central limit theorem for the Hopf–Cole solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.

Density estimates for jump diffusion processes

We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solution in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution and techniques of the Malliavin calculus in order to bound the density by the tails of the solution. We also extend the lower bounds to the multidimensional case.

Sensitivity of option prices via fuzzy Malliavin calculus

In this paper, we define the Malliavin derivative and the Skorohod integral for fuzzy stochastic processes for the sensitivity analysis of the option prices. The price dynamics could be modelled by equations involving uncertainties that lead to modelling with fuzzy processes in equations. We introduce a fuzzy stochastic differential equation for financial pricing models under an asset price that follows a fuzzy stochastic process. If there is no explicit solution to the equations, the sensitivity analysis, and the estimates are computationally expensive, and the result will be inaccurate due to the estimates and the derivative of the payoff function. The Malliavin calculus and the integration by parts formula in fuzzy space can be exploited to bypass the derivative of the payoff function.

Strong solutions of forward–backward stochastic differential equations with measurable coefficients

This paper investigates solvability of fully coupled systems of forward–backward stochastic differential equations (FBSDEs) with irregular coefficients. In particular, we assume that the coefficients of the FBSDEs are merely measurable and bounded in the forward process. We crucially use compactness results from the theory of Malliavin calculus to construct strong solutions. Despite the irregularity of the coefficients, the solutions turn out to be differentiable, at least in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.

Regularity for distribution-dependent SDEs driven by jump processes

In this paper, the following [Formula: see text]-dimensional distribution-dependent stochastic differential equation driven by a pure jump process is studied: [Formula: see text] where [Formula: see text] denotes the distribution of [Formula: see text]. The differentiability of the map [Formula: see text] is investigated in the sense of [Formula: see text]. By the Malliavin calculus for jump processes, the following Bismut type derivative formula is established, [Formula: see text] where [Formula: see text] is a test function and [Formula: see text] is a random variable depending on the initial value [Formula: see text]. Sharp gradient estimates in short time are also obtained.

A Clark-Ocone Type Formula via Itô Calculus and its Application to Finance

An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Ito's formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Levy processes.

Option pricing under the fractional stochastic volatility model

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

Approximation of fractional local times: Zero energy and derivatives

We consider empirical processes associated with high-frequency observations of a fractional Brownian motion (fBm) X with Hurst parameter H∈(0,1), and derive conditions under which these processes verify a (possibly uniform) law of large numbers, as well as a second order (possibly uniform) limit theorem. We devote specific emphasis to the “zero energy” case, corresponding to a kernel whose integral on the real line equals zero. Our asymptotic results are associated with explicit rates of convergence, and are expressed either in terms of the local time of X or of its derivatives: in particular, the full force of our finding applies to the “rough range” 0<H<1/3, on which the previous literature has been mostly silent. The use of the derivatives of local times for studying the fluctuations of high-frequency observations of a fBm is new, and is the main technological breakthrough of the present paper. Our results are based on the use of Malliavin calculus and Fourier analysis, and extend and complete several findings in the literature, for example, by Jeganathan (Ann. Probab. 32 (2004) 1771–1795; (2006); (2008)) and Podolskij and Rosenbaum (J. Financ. Econom. 16 (2018) 588–598).

A weak approximation method for irregular functionals of hypoelliptic diffusions

The paper introduces a new weak approximation scheme for Kolmogorov type hypoelliptic diffusions based on an operator splitting method and a Malliavin calculus approach. The approximation works even if the test function is not smooth, in other words, irregular functionals of hypoelliptic diffusions appearing in various fields are effectively approximated. A simple numerical algorithm is provided, which is easily implemented by a simulation method with minimum cost on random number generation. The effectiveness of the scheme is confirmed through numerical examples for irregular functionals of hypoelliptic diffusions.

On the sensitivity analysis of energy quanto options

In recent years there has been an advent of quanto options in energy markets. The structure of the payoff is rather a different type from other markets since it is written as a product of an underlying energy index and a measure of temperature. In the Heath-Jarrow-Morton (HJM) framework, by adopting the futures energy dynamics and model with stochastic volatility, we use the Malliavin calculus to derive the energy delta, temperature delta and cross-gamma formulae. The results reveal that these quantities are expressed in terms of expectations of the payoff and a random variable only depending on the underlying dynamics. This work can be viewed as a generalization of the work done, for example, by Benth et al. (2015).

Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u ( t , x ) ∈ D ∞ u(t,x)\in \mathbb {D}^\infty for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.