Malliavin Calculus Approach to Statistical Inference for Lévy Driven SDE’s

By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX _t = a _θ (X _t )dt + dZ _t with a Lévy process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.     

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev–Schwartz distributions over the Wiener measure (Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to defining the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, but not with the classical Lévy Laplacian, under the embedding of the Sobolev space over the Wiener measure in the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected with the gauge fields.     

A Note on Malliavin Fractional Smoothness for Lévy Processes and Approximation

Assume a Lévy process (X _t ) _t?∈?[0,1] that is an L ₂-martingale and let Y be either its stochastic exponential or X itself. For certain integrands φ we investigate the behavior of $$ \bigg \|\int_{(0,1]} {{\varphi}}_t dX_t - \sum_{k=1}^N v_{k-1} (Y_{t_k}-Y_{t_{k-1}}) \bigg \|_{L_2}, $$ where v _k???1 is ${\mathcal{F}}_{t_{k-1}}$ -measurable, in dependence on the fractional smoothness in the Malliavin sense of $\int_{(0,1]} {{\varphi}}_t dX_t$ . A typical situation where these techniques apply occurs if the stochastic integral is obtained by the Galtchouk–Kunita–Watanabe decomposition of some f(X ₁). Moreover, using the example f(X ₁)?=?1_(K,?∞?)(X ₁) we show how fractional smoothness depends on the distribution of the Lévy process.     

Kato Classes for Lévy Processes

We prove that the definitions of the Kato class through the semigroup and through the resolvent of the Lévy process in \(\mathbb {R}^{d}\) coincide if and only if 0 is not regular for {0}. If 0 is regular for {0} then we describe both classes in detail. We also give an analytic reformulation of these results by means of the characteristic (Lévy-Khintchine) exponent of the process. The result applies to the time-dependent (non-autonomous) Kato class. As one of the consequences we obtain a simultaneous time-space smallness condition equivalent to the Kato class condition given by the semigroup.     

Perpetual Integrals for Lévy Processes

Given a Lévy process \(\xi \), we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral \(\int _0^\infty f(\xi _s)\hbox {d}s\), where \(f\) is a positive locally integrable function. If \(\mu =\mathbb {E}[\xi _1]\in (0,\infty )\) and \(\xi \) has local times we prove the 0–1 law

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned}$$

with the exact characterization

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned}$$

The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.

     

Precise Asymptotic Approximations for Kernels Corresponding to Lévy Processes

By using basic complex analysis techniques, we obtain precise asymptotic approximations for kernels corresponding to symmetric α-stable processes and their fractional derivatives. We use the deep connection between the decay of kernels and singularities of the Mellin transforms. The key point of the method is to transform the multi-dimensional integral to the contour integral representation. We then express the integrand as a combination of gamma functions so that we can easily find all poles of the integrand. We obtain various asymtotics of the kernels by using Cauchys Residue Theorem with shifting contour integration. As a byproduct, exact coefficients are also obtained. We apply this method to general Lévy processes whose characteristic functions are radial and satisfy some regularity and size conditions. Our approach is based on the Fourier analytic point of view.     

Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

Harnack Inequalities for some Lévy Processes

In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ ^d . We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process in ℝ ^d with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr ^{− d − α} , ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion with subordinator with Laplace exponent ϕ(λ) = λ ^α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2).     

A Decomposition Theorem for Lévy Processes on Local Fields

The study of Lévy processes on local fields has been initiated by Albeverio et al. (1985)–(1998) and Evans (1989)–(1998). In this paper, a decomposition theorem for Lévy processes on local fields is given in terms of a structure result for measures on local fields and a Lévy–Khinchine representation. It is shown that a measure on a local field can be decomposed into three parts: a spherically symmetric measure, a totally non-spherically symmetric measure and a singular measure. We show that if the Radon–Nikodym derivative of the absolutely continuous part of a Lévy measure on a local field is locally constant, the Lévy process is the sum of a spherically symmetric random walk, a finite or countable set of totally non, spherically symmetric Lévy processes with single balls as support of their Lévy measure, end a singular Lévy process. These processes are independent. Explicit formulae for the transition function are obtained.     

On Anticipative Girsanov Transformations for Lévy Processes

The smooth approach to Malliavin calculus for Lévy processes in (Osswald in J. Theor. Probab., 2008) is used to study time-anticipative Girsanov transformations for a large class of Lévy processes by means of the substitution rule in finite-dimensional analysis. Dedicated to Wolfram Pohlers on the occasion of his 65th birthday.     

A Distributional Equality for Suprema of Spectrally Positive Lévy Processes

Let \(Y\) be a spectrally positive Lévy process with \({\mathbb {E}}Y_1\!<\!0\) and \(C\) an independent subordinator with finite expectation, and let \(X\!=\!Y\!+\!C\). A curious distributional equality proved in Huzak et al. (Ann Appl Probab 14:1278–1397, 2004) states that if \({\mathbb {E}}X_1<0\), then \(\sup _{0\le t <\infty }Y_t\) and the supremum of \(X\) just before the first time its new supremum is reached by a jump of \(C\) have the same distribution. In this paper, we give an alternative proof of an extension of this result and offer an explanation why it is true.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

Harnack’s Inequality for Stable Lévy Processes

We prove Harnacks inequality for harmonic functions of a symmetric stable Lévy process on R^d without the assumption that the density function of its Lévy measure is locally bounded from below. Mathematics Subject Classifications (2000) Primary 60J45, 31C05; Secondary 60G51.Research partially supported by KBN (2P03A 041 22) and RTN (HPRN-CT-2001-00273-HARP).     

Ratio Limit Theorems for Random Walks and Lévy Processes

In this paper, we study quasi-symmetric random walks and Lévy processes, a property first introduced by C.J. Stone, discuss the -invariant Radon measures for random walks and Lévy processes, and formulate some nice ratio limit theorems which are closely related to -invariant Radon measures. Mathematics Subject Classifications (2000) 60G51, 60G50.Research supported in part by NSFC 10271109.     

On Subordinated Multivariate Gaussian Lévy Processes

We discuss criteria for the selfdecomposability of multivariate Lévy processes. We consider in detail Thorin subordinated multivariate Gaussian Lévy processes. Partially on the basis of the author’s recent results (MII preprint No. 2004-33, 2004), in this paper, we consider the properties of the Pólya subordinated multivariate Gaussian Lévy processes. We define, as a special class, the multivariate generalized z-processes. The one-dimensional case was investigated in (Grigelionis, B.: Liet. Mat. Rink. 41(3), 303–309, 2001).     

Occupation Times of Refracted Lévy Processes

A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation $$\begin{aligned} {\mathrm{d}}U_t=-\delta \mathbf 1 _{\{U_t>b\}}{\mathrm{d}}t +{\mathrm{d}}X_t,\quad t\ge 0 \end{aligned}$$ where \(X=(X_t, t\ge 0)\) is a Lévy process with law \(\mathbb{P }\) and \(b,\delta \in \mathbb{R }\) such that the resulting process \(U\) may visit the half line \((b,\infty )\) with positive probability. In this paper, we consider the case that \(X\) is spectrally negative and establish a number of identities for the following functionals $$\begin{aligned} \int \limits _0^\infty \mathbf 1 _{\{U_t where \(\kappa ^+_c=\inf \{t\ge 0: U_t> c\}\) and \(\kappa ^-_a=\inf \{t\ge 0: U_t< a\}\) for \(a . Our identities extend recent results of Landriault et al. (Stoch Process Appl 121:2629–2641, 2011) and bear relevance to Parisian-type financial instruments and insurance scenarios.     

Occupation Times of General Lévy Processes

For an arbitrary Lévy process X which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of X and its occupation times. Our formulas are compact, and more importantly, the forms of the formulas clearly demonstrate the essential quantities for the calculation of occupation times of X. It is believed that our results are important not only for the study of stochastic processes, but also for financial applications.     

The Coding Complexity of Lévy Processes

We introduce a new coding scheme for general real-valued Lévy processes and control its performance with respect to L _p [0,1]-norm distortion under different complexity constraints. We also establish lower bounds that prove the optimality of our coding scheme in many cases.