On the construction and Malliavin differentiability of solutions of Lévy noise driven SDE's with singular coefficients

In this paper we introduce a new technique to construct unique strong solutions of SDE's with singular coefficients driven by certain Lévy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.     

Existence of densities of solutions of stochastic differential equations by Malliavin calculus

I considered if solutions of stochastic differential equations have their density or not when the coefficients are not Lipschitz continuous. However, when stochastic differential equations whose coefficients are not Lipschitz continuous, the solutions would not belong to Sobolev space in general. So, I prepared the class Vh which is larger than Sobolev space, and considered the relation between absolute continuity of random variables and the class Vh. The relation is associated to a theorem of N. Bouleau and F. Hirsch. Moreover, I got a sufficient condition for a solution of stochastic differential equation to belong to the class Vh, and showed that solutions of stochastic differential equations have their densities in a special case by using the class Vh.     

Malliavin calculus of Bismut type without probability

We translate in semigroup theory Bismut’s way of the Malliavin calculus. Dedicated to Professor Sinha     

Discretizing Malliavin calculus

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence $\left(X^n\right)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and sufficient conditions for strong $L^2$-convergence to a $\sigma \left(B\right)$-measurable random variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the continuous chaos coefficients.     

Asymptotic behavior for bi-fractional regression models via Malliavin calculus

Let B^H1,K1 and BH2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence Sn：=∑i=0^n-1K（n^αBi^H,K1）（Bi＋1^H2,K2-Bi^H2,K2）where K is a standard Gaussian kernel function and the bandwidth parameter α satisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion B^H1,K1. We also give the stable convergence of the sequence Sn by using the techniques of the Malliavin calculus.     

Malliavin Greeks without Malliavin calculus

We derive and analyze Monte Carlo estimators of price sensitivities (“Greeks”) for contingent claims priced in a diffusion model. There have traditionally been two categories of methods for estimating sensitivities: methods that differentiate paths and methods that differentiate densities. A more recent line of work derives estimators through Malliavin calculus. The purpose of this article is to investigate connections between Malliavin estimators and the more traditional and elementary pathwise method and likelihood ratio method. Malliavin estimators have been derived directly for diffusion processes, but implementation typically requires simulation of a discrete-time approximation. This raises the question of whether one should discretize first and then differentiate, or differentiate first and then discretize. We show that in several important cases the first route leads to the same estimators as are found through Malliavin calculus, but using only elementary techniques. Time-averaging of multiple estimators emerges as a key feature in achieving convergence to the continuous-time limit.     

On the Malliavin Calculus for SDE's on Hilbert Spaces

We consider a class of SDE's on Hilbert spaces and study the partial hypoellipticity of generators associated with these SDE's. We show that the Malliavin calculus with a new key lemma is efficient for the purpose. The partial Hörmander theorem is proved in this paper, and it is applied to the problem of propagation of absolute continuity of measures by stochastic flows given by those SDE's.     

Brownian and fractional Brownian stochastic currents via Malliavin calculus

By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in the Watanabe spaces.     

Canonical Lévy process and Malliavin calculus

A suitable canonical Lévy process is constructed in order to study a Malliavin calculus based on a chaotic representation property of Lévy processes proved by Itô using multiple two-parameter integrals. In this setup, the two-parameter derivative _Dt,x$D_{t,x}$ is studied, depending on whether x=0$x=0$ or x≠0$x\ne 0$; in the first case, we prove a chain rule; in the second case, a formula by trajectories.     

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.     

Discrete Malliavin calculus and computations of greeks in the binomial tree

This paper proposes new methods for computation of greeks using the binomial tree and the discrete Malliavin calculus. In the last decade, the Malliavin calculus has come to be considered as one of the main tools in financial mathematics. It is particularly important in the computation of greeks using Monte Carlo simulations. In previous studies, greeks were usually represented by expectation formulas that are derived from the Malliavin calculus and these expectations are computed using Monte Carlo simulations. On the other hand, the binomial tree approach can also be used to compute these expectations. In this article, we employ the discrete Malliavin calculus to obtain expectation formulas for greeks by the binomial tree method. All the results are obtained in an elementary manner.     

Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

We prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H>0.5$H>0.5$. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy condition.     

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 $L^2$-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.     

On generalized Malliavin calculus

The Malliavin derivative, the divergence operator (Skorokhod integral), and the Ornstein-Uhlenbeck operator are extended from the traditional Gaussian setting to nonlinear generalized functionals of white noise. These extensions are related to the new developments in the theory of stochastic PDEs, in particular elliptic PDEs driven by spatial white noise and quantized nonlinear equations.     

Malliavin calculus for infinite-dimensional systems with additive noise

We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the dynamics, we develop in this setting a partial counterpart of Hörmander's classical theory of Hypoelliptic operators. We study the distributions of finite-dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the Lebesgue measure. We also apply our results to concrete SPDEs such as a Stochastic Reaction Diffusion Equation and the Stochastic 2D Navier-Stokes System.     

Normal convergence using Malliavin calculus with applications and examples

We prove the chain rule in the more general framework of the Wiener–Poisson space, allowing us to obtain the so-called Nourdin–Peccati bound. From this bound, we obtain a second-order Poincaré-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener–Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein–Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many “small” jumps (particularly fractional Lévy processes), and the product of two Ornstein–Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.     

Wiener integrals, Malliavin calculus and covariance measure structure

We introduce the notion of covariance measure structure for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratic variation processes with stationary increments and the bifractional Brownian motion.     

A numerically efficient closed-form representation of mean-variance hedging for exponential additive processes based on Malliavin calculus

ABSTRACT

We focus on mean-variance hedging problem for models whose asset price follows an exponential additive process. Some representations of mean-variance hedging strategies for jump-type models have already been suggested, but none is suited to develop numerical methods of the values of strategies for any given time up to the maturity. In this paper, we aim to derive a new explicit closed-form representation, which enables us to develop an efficient numerical method using the fast Fourier transforms. Note that our representation is described in terms of Malliavin derivatives. In addition, we illustrate numerical results for exponential Lévy models.