Malliavin Calculus Research Articles

Integration theory for infinite dimensional volatility modulated Volterra processes

We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an It\^{o} formula. Moreover, we derive an alternative definition using a random-field approach and relate both concepts. We present examples related to fundamental solutions to partial differential equations.

Corrigendum and addendum to "Chaos expansion methods in Malliavin calculus: A survey of recent results"

Corrigendum and addendum to "Chaos expansion methods in Malliavin calculus: A survey of recent results"

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin’s calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.

Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives

In this article we consider a stochastic optimal control problem where the dynamics of the state process, X(t), is a controlled stochastic differential equation with jumps, delay and noisy memory. The term noisy memory is, to the best of our knowledge, new. By this we mean that the dynamics of X(t) depend on ∫t−δtX(s)dB(s) (where B(t) is a Brownian motion). Hence, the dependence is noisy because of the Brownian motion, and it involves memory due to the influence from the previous values of the state process.We derive necessary and sufficient maximum principles for this stochastic control problem in two different ways, resulting in two sets of maximum principles. The first set of maximum principles is derived using Malliavin calculus techniques, while the second set comes from reduction to a discrete delay optimal control problem, and application of previously known results by Øksendal, Sulem and Zhang. The maximum principles also apply to the case where the controller has only partial information, in the sense that the admissible controls are adapted to a sub-σ-algebra of the natural filtration.

Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise

We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and also provide the full weak convergence rate for non-self-adjoint linear operator with additive noise. Key part of the proof does not rely on Malliavin calculus. For non-self-adjoint operators, we analyse the optimal strong error for spatially semi-discrete approximations for both multiplicative and additive noise with truncated and non-truncated noise. Depending on the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.

Anticipating random periodic solutions—I. SDEs with multiplicative linear noise

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as solutions of coupled forward–backward infinite horizon stochastic integral equations (IHSIEs), using the “substitution theorem” of stochastic differential equations with anticipating initial conditions. In general, random periodic solutions and the solutions of IHSIEs, are anticipating. For the linear noise case, with the help of the exponential dichotomy given in the multiplicative ergodic theorem, we can identify them as the solutions of infinite horizon random integral equations (IHSIEs). We then solve a localised forward–backward IHRIE in C(R,Lloc2(Ω)) using an argument of truncations, the Malliavin calculus, the relative compactness of Wiener–Sobolev spaces in C([0,T],L2(Ω)) and Schauder's fixed point theorem. We finally measurably glue the local solutions together to obtain a global solution in C(R,L2(Ω)). Thus we obtain the existence of a random periodic solution and a periodic measure.

Berry–Esseen Type bound of a sequence [formula omitted] and its application

Using the recent results obtained by combining Malliavin calculus and Stein’s method, we obtain the Berry–Esseen type bound of a sequence of the random variables of the form {XNYN,N∈N}, where XN and YN are square integrable random variables such that their Malliavin derivatives also are square integrable. The aim of this paper is to develop the new techniques, allowing us to obtain sharp Berry–Esseen bound. As an application, we will discuss the rate of convergence of the distribution of the maximum likelihood estimator of a parameter appearing in a stochastic partial differential equation.

Computation of Greeks in LIBOR models driven by time–inhomogeneous Lévy processes

ABSTRACTThe aim of this article is to compute Greeks, i.e. price sensitivities in the framework of the Lévy LIBOR model. Two approaches are discussed. The first approach is based on the integration-by-parts formula, which lies at the core of the application of the Malliavin calculus to finance. The second approach consists of using Fourier-based methods for pricing derivatives. We illustrate the result by applying the formula to a caplet price where the jump part of the driving process of the underlying model is given by a time–inhomogeneous Gamma process and alternatively by a Variance Gamma process.

Analysis of the density of the solution to a semilinear SPDE with fractional noise

In this paper, we consider a general class of semilinear stochastic partial differential equations driven by a two-parameter fractional-coloured noise with Hurst index bigger than one half. Using the techniques of Malliavin calculus, we analyze the properties of the density of the solution. In particular, we establish lower and upper Gaussian-type bounds for the probability density of the mild solution at any fixed point and we prove the smoothness of this density.

Simulation of multi-asset option Greeks under a special Lévy model by Malliavin calculus

Simulation of multi-asset option Greeks under a special Lévy model by Malliavin calculus

A weak approximation with asymptotic expansion and multidimensional Malliavin weights

This paper develops a new efficient scheme for approximations of expectations of the solutions to stochastic differential equations (SDEs). In particular, we present a method for connecting approximate operators based on an asymptotic expansion with multidimensional Malliavin weights to compute a target expectation value precisely. The mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin calculus. Moreover, numerical experiments for option pricing under local and stochastic volatility models confirm the effectiveness of our scheme. Especially, our weak approximation substantially improves the accuracy at deep Out-of-The-Moneys (OTMs).

Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

In this paper we study short time asymptotics of a density function of the solution of a stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H$ (1/2 < $H$ < 1) when the coefficient vector fields satisfy an ellipticity condition at the starting point. We prove both on-diagonal and off-diagonal asymptotics under mild additional assumptions. Our main tool is Malliavin calculus, in particular, Watanabe's theory of generalized Wiener functionals.

Solving a stochastic heat equation driven by a bi-fractional noise

In this paper, we consider a stochastic heat equation with multiplicative bi-fractional Brownian sheet. Using the technique of Feynman-Kac formula and Malliavin calculus, we give an explicit formula of the weak solution and study the regularity.

Central limit theorem for the solution to the heat equation with moving time

We consider the solution to the stochastic heat equation driven by the time-space white noise and study the asymptotic behavior of its spatial quadratic variations with “moving time”, meaning that the time variable is not fixed and its values are allowed to be very big or very small. We investigate the limit distribution of these variations via Malliavin calculus.

Existence and regularity of the density for solutions of stochastic differential equations with boundary noise

We study the existence and regularity of densities for the solution of a nonlinear heat diffusion with stochastic perturbation of Brownian and fractional Brownian motion type: we use the Malliavin calculus in order to prove that, if the nonlinear term is suitably regular, then the law of the solution has a smooth density with respect to the Lebesgue measure.

Intermittency for the wave and heat equations with fractional noise in time

In this article, we consider the stochastic wave and heat equations driven by a Gaussian noise which is spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index $H>1/2$. The solutions of these equations are interpreted in the Skorohod sense. Using Malliavin calculus techniques, we obtain an upper bound for the moments of order $p\geq2$ of the solution. In the case of the wave equation, we derive a Feynman-Kac-type formula for the second moment of the solution, based on the points of a planar Poisson process. This is an extension of the formula given by Dalang, Mueller and Tribe [Trans. Amer. Math. Soc. 360 (2008) 4681-4703], in the case $H=1/2$, and allows us to obtain a lower bound for the second moment of the solution. These results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time.

Malliavin calculus for Markov chains using perturbations of time

In this article, we develop a Malliavin calculus associated to a time-continuous Markov chain with finite state space. We apply it to get a criterion of density for solutions of stochastic differential equation involving the Markov chain and also to compute greeks.

Berry–Esseen bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion

Abstract Let B be a bifractional Brownian motion with parameters H ∈ (0,1) and K ∈ (0,1]. For any n ≥ 1, set Z n = ∑ i = 0 n - 1 [ n 2 H K ( B ( i + 1 ) / n - B i / n ) 2 - 𝔼 ( ( B i + 1 - B i ) 2 ) ] ${Z_n = \sum _{i=0 }^{n-1} [n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\mathbb {E} ((B_{i+1}-B_{i})^2 ) ]}$ . We use Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati [11] to derive, in the case when 0 &lt; H K ≤ 3/4, Berry–Esseen-type bounds for the Kolmogorov distance between the law of the correct renormalization Vn of Zn and the standard normal law. Finally, we study almost sure central limit theorems for the sequence Vn .

Integration by parts formula and applications for SPDEs with jumps

By using the Malliavin calculus and finite jump approximations, the Driver-type integration by parts formula is established for the semigroup associated to stochastic (partial) differential equations with noises containing a subordinate Brownian motion. As applications, the shift Harnack inequality and heat kernel estimates are derived. The main results are illustrated by SDEs driven by -stable like processes.

Joint distributions for stochastic functional differential equations

Consider stochastic functional differential equations, whose coefficients depend on past histories. The solution determines a non-Markov process. In the present paper, we shall obtain the existence of smooth densities for joint distributions of solutions, under the uniformly elliptic condition on the diffusion coefficients, via the Malliavin calculus. As an application, we shall study the computations of the Greeks on options associated with the asset price dynamics models with delayed effects.