Malliavin Calculus Research Articles

A note on Malliavin smoothness on the Lévy space

We consider Malliavin calculus based on the Itô chaos decomposition of square integrable random variables on the Lévy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and fractional differentiability can be determined by weighted Lebesgue spaces. The measurability condition is satisfied for all random variables if the underlying Lévy process is a compound Poisson process on a finite time interval.

Computation of Greeks Using Binomial Tree

This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that Greeks for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithm of Greeks for American options using the binomial tree is also given in this article. There are three advantageous points to use binomial tree approach for the computation of Greeks. First, mathematics is much simpler than using the continuous time Malliavin calculus approach. Second, we can construct a simple algorithm to obtain the Greeks for American options. Third, this algorithm is very efficient because one can compute the price and Greeks (delta, gamma, vega, and rho) at once. In spite of its importance, only a few previous studies on the computation of Greeks for American options exist, because performing sensitivity analysis for the optimal stopping problem is difficult. We believe that our method will become one of the popular ways to compute Greeks for options.

Computation of conditional expectation based on the multidimensional J-process using Malliavin calculus related to pricing American options

Computation of conditional expectation based on the multidimensional J-process using Malliavin calculus related to pricing American options

The Implied Volatility of Forward-Start Options: ATM Short-Time Level, Skew and Curvature

Using Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths.

On optimal mean-field control problem of mean-field forward-backward stochastic system with jumps under partial information

This paper considers the problem of partially observed optimal control for forward-backward stochastic systems driven by Brownian motions and an independent Poisson random measure with a feature that the cost functional is of mean-field type. When the coefficients of the system and the objective performance functionals are allowed to be random, possibly non-Markovian, Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed. The authors also investigate the mean-field type optimal control problem for the system driven by mean-field type forward-backward stochastic differential equations (FBSDEs in short) with jumps, where the coefficients contain not only the state process but also its expectation under partially observed information. The maximum principle is established using convex variational technique. An example is given to illustrate the obtained results.

Intrinsic expansions for averaged diffusion processes

We show that the convergence rate of asymptotic expansions for solutions of SDEs is higher in the case of degenerate diffusion compared to the elliptic case, i.e. it is higher when the Brownian motion directly acts only along some directions. In the scalar case, this phenomenon was already observed in Gobet and Miri 2014 using Malliavin calculus techniques. Here, we provide a general and detailed analysis by employing the recent study of intrinsic functional spaces related to hypoelliptic Kolmogorov operators in Pagliarani et al. 2016. Applications to finance are discussed, in the study of path-dependent derivatives (e.g. Asian options) and in models incorporating dependence on past information.

Edgeworth expansion for functionals of continuous diffusion processes

This paper presents new results on the Edgeworth expansion for high frequency functionals of continuous diffusion processes. We derive asymptotic expansions for weighted functionals of the Brownian motion and apply them to provide the Edgeworth expansion for power variation of diffusion processes. Our methodology relies on martingale embedding, Malliavin calculus and stable central limit theorems for semimartingales. Finally, we demonstrate the density expansion for Studentized statistics of power variations.

Computation of the Delta of European Options Under Stochastic Volatility Models

We employ Malliavin calculus techniques to compute the Delta of European type options in the presence of stochastic volatility. We obtain a general formula for the Malliavin weight and apply the derived formula to the well known models of Stein-Stein and Heston in order to show the numerical accuracy and efficiency of our approach.

Functional Cramér–Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes

We investigate the problems of drift estimation for a shifted Brownian motion and intensity estimation for a Cox process on a finite interval [ 0 , T ] , when the risk is given by the energy functional associated to some fractional Sobolev space H 0 1 ⊂ W α , 2 ⊂ L 2 . In both situations, Cramér–Rao lower bounds are obtained, entailing in particular that no unbiased estimators (not necessarily adapted) with finite risk in H 0 1 exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).

Parameter estimation for a partially observed Ornstein–Uhlenbeck process with long-memory noise

We consider the parameter estimation problem for the Ornstein–Uhlenbeck process X driven by a fractional Ornstein–Uhlenbeck process V, i.e. the pair of processes defined by the non-Markovian continuous-time long-memory dynamics , with , where and are unknown parameters, and is a fractional Brownian motion of Hurst index . We study the strong consistency as well as the asymptotic normality of the joint least squares estimator of the pair , based either on continuous or discrete observations of as the horizon T increases to +. Both cases qualify formally as partial-observation questions since V is unobserved. In the latter case, several discretization options are considered. Our proofs of asymptotic normality based on discrete data, rely on increasingly strict restrictions on the sampling frequency as one reduces the extent of sources of observation. The strategy for proving the asymptotic properties is to study the case of continuous-time observations using the Malliavin calculus, and then to exploit the fact that each discrete-data estimator can be considered as a perturbation of the continuous one in a mathematically precise way, despite the fact that the implementation of the discrete-time estimators is distant from the continuous estimator. In this sense, we contend that the continuous-time estimator cannot be implemented in practice in any naïve way, and serves only as a mathematical tool in the study of the discrete-time estimators’ asymptotics.

On the non-commutative fractional Wishart process

We investigate the process of eigenvalues of a fractional Wishart process defined by N=B⁎B, where B is the matrix fractional Brownian motion recently studied in [18]. Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter H∈(1/2,1), we derive a stochastic differential equation in the Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally, a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the non-commutative fractional Wishart process, which constitutes the family of fractional dilations of the free Poisson distribution.

On stochastic conservation laws and Malliavin calculus

For stochastic conservation laws driven by a semilinear noise term, we propose a generalization of the Kružkov entropy condition by allowing the Kružkov constants to be Malliavin differentiable random variables. Existence and uniqueness results are provided. Our approach sheds some new light on the stochastic entropy conditions put forth by Feng and Nualart [17] and Bauzet, Vallet, and Wittbold [2], and in our view simplifies some of the proofs.

Malliavin calculus for regularity structures: The case of gPAM

Malliavin calculus is implemented in the context of Hairer (2014) [16]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust, and purely deterministic, translation operator, in L2-directions, between “models”. In the concrete context of the generalized parabolic Anderson model in 2D – one of the singular SPDEs discussed in the afore-mentioned article – we establish existence of a density at positive times.

The transport equation and zero quadratic variation processes

We analyze the transport equation driven by a zero quadratic variation process. Using the stochastic calculus via regularization and the Malliavin calculus techniques, we prove the existence, uniqueness and absolute continuity of the law of the solution. As an example, we discuss the case when the noise is a Hermite process.

An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach

This paper proposes a new analytical approximation scheme for the representation of the forward–backward stochastic differential equations (FBSDEs) of Ma and Zhang (Ann Appl Probab, 2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment is taken into account.

Tail probability estimates for additive functionals

In this paper, based on techniques of Malliavin calculus, we obtain an explicit bound for tail probabilities of a general class of additive functionals. Applications to fractional Brownian motion and Cox–Ingersoll–Ross process are given to illustrate the theory.

A general expression of the price of an American options using the Malliavin derivative

The aim of this paper is to elaborate a general expression of the conditional expectation related to pricing problem of the American options using the stochastic integration by parts ( Malliavin derivative). This work is a generalization of the Jerbi and Kharrat’s work (2013). Basing on the density function of the asset price, Jerbi and Kharrat used the Malliavin calculus to give the expression of the conditional expectation related to pricing American option problem, but in this work we use the stochastic integration by parts in order to resolve the previous problem.

An integral inequality for the invariant measure of a stochastic reaction–diffusion equation

We consider a reaction–diffusion equation perturbed by noise (not necessarily white). We prove an integral inequality for the invariant measure $${\nu}$$ of a stochastic reaction–diffusion equation. Then, we discuss some consequences as an integration by parts formula which extends to $${\nu}$$ a basic identity of the Malliavin Calculus. Finally, we prove the existence of a surface measure for a ball and a half-space of H.

Optimal insider control and semimartingale decompositions under enlargement of filtration

ABSTRACTWe combine stochastic control methods, white noise analysis, and Hida–Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations. Some of the expressions are more explicit than previously known. The results are illustrated by examples.

Stochastic differential games with inside information

We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida–Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.