Let $X$ be a Lévy process with absolutely continuous Lévy measure $ν$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_{t}\geq{}y)$ of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density $p_{t}$ of $X_{t}$, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order $n$ in $t$ are derived for the transition densities of the process. The conditions imposed on $p_{t}$ require that its derivatives remain uniformly bounded away from the origin, as $t\to{}0$; such conditions are shown to be satisfied for symmetric stable Lévy processes as well as for other related Lévy processes of relevance in mathematical finance. The expansions seem to correct asymptotics previously reported in the literature.
New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.
We derive a precise link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space. The results are applied to pathwise continuous Gaussian processes and a new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived. In the end an extension of this result to Gaussian stationary processes with convex covariance function is established.
New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of nonstationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.
In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described…
The paper characterizes uniform convergence rate for general classes of wavelet expansions of stationary Gaussian random processes. The convergence in probability is considered.
The article presents new results on convergence in $L_p([0,T])$ of wavelet expansions of $φ$-sub-Gaussian random processes. The convergence rate of the expansions is obtained. Specifications of the obtained results are discussed.
We introduce and study an alternative form of the chaotic expansion for counting processes using the Poisson imbedding representation; we name this alternative form \textit{pseudo-chaotic expansion}. As an application, we prove that the coefficients of this pseudo-chaotic expansion for any linear Hawkes process are obtained in closed form, whereas those of the usual chaotic expansion cannot be derived explicitly. Finally, we study further the structure of linear Hawkes processes by constructing an example of a process in a pseudo-chaotic form that satisfies the stochastic self-exciting intensi…
New algorithms for construction of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces are presented. These algorithms are based on a special technique of sequential phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces. Asymptotic expansions are given in two forms, without and with explicit bounds for remainders.
Asymptotic expansions with explicit upper bounds for remainders are given for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces. The corresponding algorithms are based on a special technique of sequential phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces.
In this paper we present the Edgeworth expansion for the Euler approximation scheme of a continuous diffusion process driven by a Brownian motion. Our methodology is based upon a recent work \cite{Yoshida2013}, which establishes Edgeworth expansions associated with asymptotic mixed normality using elements of Malliavin calculus. Potential applications of our theoretical results include higher order expansions for weak and strong approximation errors associated to the Euler scheme, and for studentized version of the error process.
In this paper we provide an expansion formula for Hawkes processes which involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the well-known integration by parts formula (or more precisely the Mecke formula) for Poisson functional. Our approach allows us to provide an expansion of the premium of a class of cyber insurance derivatives (such as reinsurance contracts including generalized Stop-Loss contracts) or risk management instruments (like Expected Shortfall) in terms of so-called shifted Hawkes processes. From the actuarial point of view, these proce…
In this paper, we study mixed power-exponential moment functionals of nonlinearly perturbed semi-Markov processes in discrete time. Conditions under which the moment functionals of interest can be expanded in asymptotic power series with respect to the perturbation parameter are given. We show how the coefficients in these expansions can be computed from explicit recursive formulas. In particular, the results of the present paper have applications for studies of quasi-stationary distributions.
New results on uniform convergence in probability for the most general classes of wavelet expansions of stationary Gaussian random processes are given.
In the present paper, a stochastic Taylor expansion of some functional applied to the solution process of an Itô or Stratonovich stochastic differential equation with a multi-dimensional driving Wiener process is given. Therefore, the multi-colored rooted tree analysis is applied in order to obtain a transparent representation of the expansion which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting. Further, some estimates for the mean--square and the mean truncation errors are given.
Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Final…
We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.
We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space. More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to $\mathbb{R}^d$, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.
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 second order 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.
Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is primarily useful for those processes for which we can identify the necessary components, i.e., a set of basis functions, and the distribution of an associated set of stochastic coefficients. Our ability to derive these components explicitly is limited to a handful processes. In this paper we derive all the necessary elements to implement the KLE for a square-integrable Levy process. We show that the eigenfunctions are sine functions, identica…
Cohen, Guyon, Perrin and Pontier have given assumptions under which the second-order quadratic variations of a Gaussian process converge almost surely to a deterministic limit. In this paper we present two new convergence results about these variations: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we apply these results to identify two-parameter fractional Brownian motion and anisotropic fractional Brownian motion.
For $γ>-{1/2}$, we provide the Karhunen-Loève expansion of the weighted mean-centered Wiener process, defined by \[W _γ(t)=\frac{1}{\sqrt{1+2γ}}\Big\{W\big(t^{1+2γ}\big)- \int_0^1W\big(u^{1+2γ}\big)du\Big\},\] for $t\in(0,1]$. We show that the orthogonal functions in these expansions have simple expressions in term of Bessel functions. Moreover, we obtain that the $L^2[0,1]$ norm of $W_γ$ is identical in distribution with the $L^2[0,1]$ norm of the weighted Brownian bridge $t^γB(t)$.
Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metr…
When the \textit{martingale representation property} holds, we call any local martingale which realizes the representation a \textit{representation process}. There are two properties of the \textit{representation process} which can greatly facilitate the computations under the \textit{martingale representation property}. Actually, on the one hand, the \textit{representation process} is not unique and there always exists a \textit{representation process} which is locally bounded and has pathwisely orthogonal components outside of a predictable thin set. On the other hand, the jump measure of a …
The solution of a (stochastic) differential equation (SDE) can be locally approximated by a stochastic expansion, a linear combination of iterated integrals. Quantities of interest, like moments, can then be approximated with the expansion. We present a formula for the case where the drivers of the equation are time and Wiener processes. We also present a Mathematica implementation of the result.
The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed technique are shown for several classes of stochastic processes. In particular, the main theorem is adjusted to the fractional Brownian motion case. New results on the rate of convergence of the wavelet expansions in the space $C([0,T])$ are also presented.
For an expansionary process, the size of the expansion space will increase. If the expansionary process is time-dependent, time (t) will increase as a function of the increase in the size of the expansion space. A statistical information entropy methodology was used to investigate the properties of time-dependent expansionary processes both in theory and through examples. The primary objective of this paper was to investigate whether there is a universal measure of time (T) and how it relates to process related time (t), that is specific to any given time-dependent expansionary process. It was…
Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Lévy processes are studied. The Lévy measure is assumed to have completely monotone density function, and a scaling-type condition inf z Psi"(z) / Psi'(z) > 0 is imposed on the Lévy-Khintchine exponent Psi. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin.
We give sufficient conditions for a discrete set of points in any dimensional real hyperbolic space to have positive anchored expansion. The first condition is a bounded mean density property, ensuring not too many points can accumulate in large regions. The second is a bounded mean vacancy condition, effectively ensuring there is not too much space left vacant by the points over large regions. These properties give as an easy corollary that stationary Poisson--Delaunay graphs have positive anchored expansion, as well as Delaunay graphs built from stationary determinantal point processes. We i…
We provide a detailed derivation of the Karhunen-Loève expansion of a stochastic process. We also discuss briefly Gaussian processes, and provide a simple numerical study for the purpose of illustration.
In this paper, we study quasi-stationary distributions of nonlinearly perturbed semi-Markov processes in discrete time. This type of distributions is of interest for the analysis of stochastic systems which have finite lifetimes, but are expected to persist for a long time. We obtain asymptotic power series expansions for quasi-stationary distributions and it is shown how the coefficients in these expansions can be computed from a recursive algorithm. As an illustration of this algorithm, we present a numerical example for a discrete time Markov chain.
When expanding a filtration with a stochastic process it is easily possible for semimartingale no longer to remain semimartingales in the enlarged filtration. Y. Kchia and P. Protter indicated a way to avoid this pitfall in 2015, but they were unable to give the semimartingale decomposition in the enlarged filtration except for special cases. We provide a way to compute such a decomposition, and moreover we provide a sufficient condition for Itô processes to remain Itô processes in the enlarged filtration. This has significance in applications to Mathematical Finance.
We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fern{á}ndez and Procacci (2007), the proof is based on the Kirkwood-Salsburg integral equations and is close in spirit to the approach by Bissacot, Fern{á}ndez and Procacci (2010). In addition, we provide formul…
We lead the algorithm of expansion of sojourn probability of many-dimensional diffusion processes in small domain. The principal member of this expansion defines normalizing coefficient for special limit theorems.
We prove that certain asymptotic moments exist for some random distance expanding dynamical systems and Markov chains in random dynamical environment, and compute them in terms of the derivatives at the $0$ of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments $\gam_k=\lim_{n\to\infty}n^{-[\frac k2]}\bbE(\sum_{i=1}^n X_i)^k$ of sums of independent and identically distributed random variables satisfy. We will also obtain certain (Edgeworth) asymptotic expansions related to the central limit theorem for such processes. Our proofs…
We show that the rate of convergence of asymptotic expansions for solutions of SDEs is generally higher in the case of degenerate (or partial) diffusion compared to the elliptic case, i.e. it is higher when the Brownian motion directly acts only on some components of the diffusion. In the scalar case, this phenomenon was already observed in (Gobet and Miri 2014) using Malliavin calculus techniques. In this paper, 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). Relev…
Many important index transforms can be constructed via the spectral theory of Sturm-Liouville differential operators. Using the spectral expansion method, we investigate the general connection between the index transforms and the associated parabolic partial differential equations. We show that the notion of Yor integral, recently introduced by the second author, can be extended to the class of Sturm-Liouville integral transforms. We furthermore show that, by means of the Feynman-Kac theorem, index transforms can be used for studying Markovian diffusion processes. This gives rise to new applic…
We obtain a first order extension of the large deviation estimates in the Gärtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special family of functions mentioned above is based on heat kernel expansions. Some of the ideas employed in the paper come from the theory of affine stochastic processes. For instance, we provide an explicit expansion with respect to the homogenization parameter of the rescaled cumulant g…
We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci, and Riga (2013), we derive a family of asymptotic expansions for the transition density of the underlying as well as for European-style option prices and defaultable bond prices. For the density expansion, we also provide error bounds for the truncated asy…
Motivated from option and derivative pricing, this note develops Edgeworth expansions both in the Kolmogorov and Wasserstein metric for many different types of discrete time volatility models and their possible transformations. This includes, among others, Hölder-type functions of (augmented) Garch processes of any order, iterated random functions or Volterra-processes.
We consider a Poisson process $η$ on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of $η$. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (st…
Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=\int_Ω \mathbb{P}^x (X_t\inΩ^c) \, \mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $α$-stable Lévy processes and more general Lévy processes, under mild assumptions on the characteristic exponent.
We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectr…
By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we ill…
This article introduces adaptive Fourier decomposition (AFD) type methods, emphasizing on those that can be applied to stochastic processes and random fields, mainly including stochastic adaptive Fourier decomposition and stochastic pre-orthogonal adaptive Fourier decomposition. We establish their algorithms based on the covariant function and prove that they enjoy the same convergence rate as the Karhunen-Loève (KL) decomposition. The AFD type methods are compared with the KL decomposition. In contrast with the latter, the AFD type methods do not need to compute eigenvalues and eigenfunctions…
