Let Z be an Rd-valued Levy process with strong finite p-variation for some p<2. We prove that the ''decompensated'' process Y obtained from Z by annihilating its generalized drift has a small deviations property in p-variation. This property means that the null function belongs to the support of the law of Y with respect to the p-variation distance. Thanks to the continuity results of T. J. Lyons/D. R. E. Williams, this allows us to prove a support theorem with respect to the p-Skorohod distance for canonical SDE driven by Z without any assumption on Z, improving the results of H. Kunita. We also give a criterion ensuring the small deviation property for Z itself, noticing that the characterization under the uniform distance, which we had obtained in a previous paper, no more holds under the p-variation distance.
For a general Multidimensional Lévy process (satisfying some moment conditions), we introduce the Multidimensional power jump processes and the related Multidimensional Teugels martingales. Furthermore, we orthogonalize the Multidimensional Teugels martingales by applying Gram-Schmidt process. We give a chaotic representation for every square integral random variable in terms of these orthogonalized Multidimensional Teugels martingales. The predictable representation with respect to the same set of Multidimensional orthogonalized martingales of square integrable random variables and of square …
In this paper we show the existence and form uniqueness of a solution for multidimensional backward stochastic differential equations driven by a multidimensional Lévy process with moments of all orders. The results are important from a pure mathematical point of view as well as in the world of finance: an application to Clark-Ocone and Feynman-Kac formulas for multidimensional Lévy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equations provide an analogue of the famous Black-Scholes partial differential equation and thus can be used f…
We consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follows a multidimensional exponential Levy model. We carefully examine the relation between the option prices, related partial integro-differential variational inequalities and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for broad class of payoff functions.
In this short article we show how the techniques presented in arXiv:1207.4469 can be extended to a variety of non continuous and multivariate processes. As examples, we prove uniqueness of the location of the maximum for spectrally positive Lévy processes, Ornstein-Uhlenbeck process, fractional Brownian Motion and the Brownian sheet among others gaussian processes.
Using a very simple argument based on the indepenence of increments and the fact that in a finite dimensional space $R^{d}$ there are not too many directions, we derive a theorem stating that exit time of any (non-constant) Lévy process on $R^{d}$ from a ball has exponentially light tails.
The main tool for stochastic calculus with respect to a multidimensional process $B$ with small Hölder regularity index is rough path theory. Once $B$ has been lifted to a rough path, a stochastic calculus -- as well as solutions to stochastic differential equations driven by $B$ -- follow by standard arguments. Although such a lift has been proved to exist by abstract arguments \cite{LyoVic07}, a first general, explicit construction has been proposed in \cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The purpose of this short note is to convey the main ideas of the Fourier no…
We study stochastic tree fluid networks driven by a multidimensional Levy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Levy process by solving a multidimensional Skorokhod reflection problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion …
Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with Lévy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced.
Risk diversification for infinitely divisible distributions
School of Mathematics, Statistics and Actuarial Science, University of Essex（埃塞克斯大学数学、统计与精算科学学院）；Department of Statistics and Finance, School of Management, University of Science and Technology of China（中国科学技术大学管理学院统计与金融系）
Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck process, solution of the S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a real $n\times n$ matrix and $B$ a Lévy process without Gaussian part. We show that when $A$ is non-singular, the law of $X_1$ is absolutely continuous in $\r^n$ if and only if the jumping measure of $B$ fulfils a certain geometric condition with respect to $A,$ which we call the exhaustion property. This optimal criterion is much weaker than for the background driving Lévy process $B$, which might be very singular and sometimes even have a one-dimensional discrete jump…
This paper considers a general stochastic SIR epidemic model driven by a multidimensional Levy jump process with heavy tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita's inequality instead of the Burkholder-Davis-Gundy inequality for continuous processes, and allows for the treatment of infinite Levy measures by the definition of new threshold values. An SIR model driven by a tempered stable process is pr…
In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a pure-jump Lévy process with finite Lévy measure, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.
Motivated by the stochastic Lotka-Volterra model, we introduce discrete-state interacting multitype branching processes. We show that they can be obtained as the sum of a multidimensional random walk with a Lamperti-type change proportional to the population size; and a multidimensional Poisson process with a time-change proportional to the pairwise interactions. We define the analogous continuous-state process as the unique strong solution of a multidimensional SDE. We prove that the scaling limits of the discrete-state process correspond to its continuous counterpart. In addition, we show th…
When is it possible to interpret a given Markov process as a Lévy-like process? Since the class of Lévy processes can be defined by the relation between transition probabilities and convolutions, the answer to this question lies in the existence of a convolution-like operator satisfying the same relation with the transition probabilities of the process. It is known that the so-called Sturm-Liouville convolutions have the desired properties and therefore the question above has a positive answer for a certain class of one-dimensional diffusions. However, more general processes have never been sy…
We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a care…
Let $(U_t,V_t)$ be a bivariate Lévy process, where $V_t$ is a subordinator and $U_t$ is a Lévy process formed by randomly weighting each jump of $V_t$ by an independent random variable $X_t$ having cdf $F$. We investigate the asymptotic distribution of the self-normalized Lévy process $U_t/V_t$ at 0 and at $\infty$. We show that all subsequential limits of this ratio at 0 ($\infty$) are continuous for any nondegenerate $F$ with finite expectation if and only if $V_t$ belongs to the centered Feller class at 0 ($\infty$). We also characterize when $U_t/V_t$ has a non-degenerate limit distributio…
The study of multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined through the theory of stochastic integration. Here, inspired by the Lévy-Cieselski construction of the Wiener process, we propose an alternative representation of multidimensional Gauss-Markov processes as expansions on well-chosen Schauder bases, with independent random coefficients of normal law with zero mean and unitary variance. We thereby offer a natur…
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
From the point of view of stochastic analysis the Caputo and Riemann-Liouville derivatives of order $\al \in (0,2)$ can be viewed as (regularized) generators of stable Lévy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. These extensions are introduced and some well-posedness results are obtained that generalize, simplify and unify lots of known facts. This probabilistic analysis leads one to study…
We generalise multivariate subordination of Lévy processes as introduced by Barndorff-Nielsen, Pedersen, and Sato to Hilbert space valued Lévy processes. The processes are explicitly characterised and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space as a subordination …
We consider the optimal stopping problem consisting in, given a strong Markov process, a reward function and a discount rate, finding the stopping time such that the expected reward at the stopping time is maximum. The approach we follow, has two main components: the Dynkin's characterization of the value function as the smallest excessive function dominating the reward; and the Riesz representation of excessive functions in terms of the Green kernel, the main reference being Salminen 85. In the context of one-dimensional diffusions we give a complete characterization of the solution under som…
This paper considers the classical SIR epidemic model driven by a multidimensional Lévy jump process. We consecrate to develop a mathematical method to obtain the asymptotic properties of the perturbed model. Our method differs from previous approaches by the use of the comparison theorem, mutually exclusive possibilities lemma, and some new techniques of the stochastic differential systems. In this framework, we derive the threshold which can determine the existence of a unique ergodic stationary distribution or the extinction of the epidemic. Numerical simulations about different perturbatio…
Consider a multidimensional diffusion process $X=\{X\left(t\right) :t\in\lbrack0,1]\}$. Let $\varepsilon>0$ be a \textit{deterministic}, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of $X$, we construct a probability space, supporting both $X$ and an explicit, piecewise constant, fully simulatable process $X_{\varepsilon}$ such that \[ \sup_{0\leq t\leq1}\left\Vert X_{\varepsilon}\left(t\right) -X\left(t\right) \right\Vert_{\infty}<\varepsilon \] with probability one. Moreover, the user can adaptively choose $\varepsilon^…
Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric Lévy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the Lévy exponent $ψ(\la)$ is regularly varying at zero with index $1<β\leq 2$, and satisfies some additional regularity conditions, \begin{eqnarray*} && {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- E(\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx)\over t\sqrt{ψ^{-1}(1/t)}}\label{r5.0tweaksabs} && \stackrel{\mathcal{L}}{\Longrightarrow}(8c_{ψ,1 })^{1/2}(\int_{-\infty}^{\finfty} (L_{β,1}^{x})^{2} dx)^{1/2} η\end{eqnarray*} as $t\rar\infty$, where …
In this article we consider Lévy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional Lévy processes is also deduced from these res…
We show some Chung-type $\liminf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.
Let $^{(r,s)}X_t$ be the Lévy process $X_t$ with the $r$ largest positive jumps and $s$ smallest negative jumps up till time $t$ deleted and let $^{(r)}\widetilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till time $t$ deleted. Let $a_t \in \mathbb{R}$ and $b_t>0$ be non-stochastic functions in $t$. We show that the tightness of $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}\widetilde X_t - a_t)/b_t$ at $0$ implies the tightness of all normed ordered jumps, hence the tightness of the untrimmed process $(X_t -a_t)/b_t$ at $0$. We use this to deduce that the trimmed process $({}^{(r,s)}X…
Motivated by the multilevel Monte Carlo method introduced by Giles [5], we study the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are respectively Euler approximations with time steps $1/n$ and $1/nm$ of a given stochastic differential equation $X$ driven by a pure jump Lévy process. In this paper, we prove that this normalized multilevel error converges to different non-trivial limiting processes with various sharp rates $u_{n,m}$ depending on the behavior of the Lévy measure around zero. Our results are consistent with those of Jacod [9] …
We consider Kallenberg's hypothesis on the characteristic function of a Lévy process and show that it allows the construction of weakly continuous bridges of the Lévy process conditioned to stay positive. We therefore provide a notion of normalized excursions Lévy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Lévy process conditioned to stay positive by using the weakly continuous bridges of the Lévy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper …
We consider rough paths with jumps. In particular, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against cadlag processes. A class of Levy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and and a certain integrability property of the Carnot--Caratheodory norm with respect to the Levy measure on the group, using Hunt's framework of Lie group valued Levy processes. Examples of Levy rough paths include standard multi-dimen…
This paper presents a set of results relating to the occupation time $α(t)$ of a process $X(\cdot)$. The first set of results concerns exact characterizations of $α(t)$ for $t\geq0$, e.g., in terms of its transform up to an exponentially distributed epoch. In addition we establish a central limit theorem (entailing that a centered and normalized version of $α(t)/t$ converges to a zero-mean Normal random variable as $t\rightarrow\infty$) and the tail asymptotics of $P(α(t)/t\geq q)$. We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new o…
We introduce and investigate in this paper a new convenient method of introduction of a norm in the multidimensional rearrangement probability invariant space.
Gonzalez Cazares and Ivanovs (2021) suggested a new method for "recovering" the Brownian motion component from the trajectory of a Levy process that required sampling from an independent Brownian motion process. We show that such a procedure works equally well without any additional source of randomness if one uses normal quantiles instead of the ordered increments of the auxiliary Brownian motion process.
We study the almost sure behaviour of suitably normalised multivariate Levy processes as t goes to zero. Among other results we find necessary and sufficient conditions for a law of a very slowly varying function which includes a general law of the iterated logarithm in this setting. We also look at the corresponding cluster set problem.
We determine the asymptotic behavior of the realized power variations, or more generally of sums of a given test function evaluated at the successive increments of a Lévy process. One can completely elucidate the first order behavior (convergence in probability, possibly after normalization). As for the associated CLT, one can show some versions of it, but only in a limited number of cases. In some other cases, a CLT just does not exist.
This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $α\in(1,2)$, the i.i.d. sequence \[ \left \{ \left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum _{i=1}^{n}Y_{i},\frac{1}{\sqrt[α]{n}}\sum_{i=1}^{n}Z_{i}\right) \right \} _{n=1}^{\infty} \] converges in distribution to $\tilde{L}_{1}$, where $\tilde{L}_{t}=(\tilde ξ_{t},\tildeη_{t},\tildeζ_{t})$, $t\in [0,1]$, is a multidimensional nonlinear Lévy process with an uncertainty set $Θ$ as a set of Lévy triplets. This nonlinear Lévy p…
Let $X(t)$, $t\geq0$, be a Lévy process in $\mathbb{R}^d$ starting at the origin. We study the closed convex hull $Z_s$ of $\{X(t): 0\leq t\leq s\}$. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set $Z_s$ and find explicit expressions for their means in the case of symmetric $α$-stable Lévy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of $Z_s$ for all $s>0$. Limit theorems for the convex hull of Lévy processes with normal and stable lim…
Suppose we have a high-frequency sample from the Lévy process of the form $X_t^θ=βt+γZ_t+U_t$, where $Z$ is a possibly asymmetric locally $α$-stable Lévy process, and $U$ is a nuisance Lévy process less active than $Z$. We prove the LAN property about the explicit parameter $θ=(β,γ)$ under very mild conditions without specific form of the Lévy measure of $Z$, thereby generalizing the LAN result of A\"ıt-Sahalia and Jacod (2007). In particular, it is clarified that a non-diagonal norming may be necessary in the truly asymmetric case. Due to the special nature of the local $α$-stable property, t…
We prove that the stochastic differential equation $$ Y_{s,t}(x) = Y_{s,s}(x) + \int_0^{t-s} f(Y_{s,s+u}(x)) dX_{s+u}, Y_{s,s}(x)=x\in\R^d. $$ driven by a Lévy process whose paths have finite p-variation almost surely for some $p\in[1,2)$ defines a flow of locally C^1-diffeomorphisms provided the vector field f is $α$-Lipschitz for some $α>p$. Using a path- wise approach we relax the smoothness condition normally required for a class of discontinuous semi-martingales.
We study the upper tail behaviors of the local times of the additive Lévy processes and additive random walks. The limit forms we establish are the moderate deviations and the laws of the iterated logarithm for the L_2-norms of the local times and for the local times at a fixed site.
In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the properties of a wide class of Levy processes (potential, quasi-potential, the probability of the Levy process remaining within the given domain, long time behavior, stable processes). We analyze in detail a number of concrete examples of the Levy processes (stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussia…
We show that the L^p norms of random sequences {s_N} of L^2 normalized holomorphic sections of increasing powers of an ample line bundle on a compact Kahler manifold are almost surely bounded for 2<p< infinity, and are almost surely O((log N)^{1/2}) for p= infinity. This estimate also holds for almost-holomorphic sections of positive line bundles on symplectic manifolds (in the sense of math.SG/0212180) and we give almost sure bounds for the C^k norms. Our methods involve asymptotics of Bergman-Szego kernels and the concentration of measure phenomenon.
We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Lévy noises. We do not impose any moment condition on the driving Lévy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient $\varepsilon \to 0$ and $n \to \infty$ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to t…
Ornstein-Uhlenbeck processes driven by general Lévy process are considered in this paper. We derive strongly consistent estimators for the moments of the underlying Lévy process and for the mean reverting parameter of the Ornstein-Uhlenbeck process. Moreover, we prove that the estimators are asymptotically normal. Finally, we test the empirical performance of our estimators in a simulation study and we fit the model to real data.
We prove Chung-type laws of the iterated logarithm for general Lévy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated logarithm. This reveals laws of the iterated logarithm for Lévy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.
In this paper, we consider a backward in time problem for Ginzburg-Landau equation in multidimensional domain associated with some random data. The problem is ill-posed in the sense of Hadamard. To regularize the instable solution, we develop a new regularized method combined with statistical approach to solve this problem. We prove a upper bound, on the rate of convergence of the mean integrated squared error in $L^2 $ norm and $H^1$ norm.
