Exponential functionals of Levy processes

Exponential Convergence for Semilinear SDEs Driven by Lévy Processes on Hilbert Spaces

In this paper, an integration by parts formula was derived for jump processes on Hilbert spaces. Using this formula, we investigated derivative formula and exponential ergodicity for nonlinear SPDEs driven by purely jump processes.

On the distribution of exponential functionals for Levy processes with jumps of rational transform

We derive explicit formulas for the Mellin transform and the distribution of the exponential functional for Levy processes with rational Laplace exponent. This extends recent results by Cai and Kou on the processes with hyper-exponential jumps [N. Cai and S. Kou "Prising Asian options under a general jump diffusion model", (2010)].

Exponential Stability of Solutions to Stochastic Differential Equations Driven by G-Levy Process

In this paper, BDG-type inequality for G-stochastic calculus with respect to G-Levy process is obtained and solutions of stochastic differential equations driven by G-Levy process under non-Lipschitz condition are constructed. Moreover, we establish the mean square exponential stability and quasi sure exponential stability of the solutions be means of G-Lyapunov function method. An example is presented to illustrate the efficiency of the obtained results.

Exponential convergence rate of ruin probabilities for level-dependent Lévy-driven risk processes

We explicitly find the rate of exponential long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows to reduce the pro blem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.

Distribution of some functionals for a Lévy process with matrix-exponential jumps of the same sign

This paper provides a framework for investigations in fluctuation theory for Lévy processes with matrix-exponential jumps. We present a matrix form of the components of the infinitely divisible factorization. Using this representation we establish generalizations of some results known for compound Poisson processes with exponential jumps in one direction and generally distributed jumps in the other direction.

Exponential moments of first passage times and related quantities for Lévy processes

For a Lévy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as $r \to \infty$.

On the range of exponential functionals of Lévy processes

We characterize the support of the law of the exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$ of two one-dimensional independent Lévy processes $ξ$ and $η$. Further, we study the range of the mapping $Φ_ξ$ for a fixed Lévy process $ξ$, which maps the law of $η_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$. It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.

On continuity properties for option prices in exponential Lévy models

For a converging sequence of exponential Lévy models, we give conditions under which the associated sequence of option prices converges. We also study the behaviour of the prices when no such convergence holds. We then consider two special cases, first when the martingale measure is chosen by minimisation of entropy and then when it minimises Hellinger integrals.

Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results th…

The valuation of American options in a multidimensional exponential Levy model

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.

Hermitian Functional Representation of Free Lévy Processes

A functional representation of free Lévy processes is established via an ensemble of unitarily invariant Hermitian matrix-valued Lévy processes. This is accomplished by proving functional asymptotics of their empirical spectral processes towards the law of a free Lévy processes. This result recovers a functional version of Wigner's theorem and introduces a functional version of Marchenko-Pastur's theorem providing the free Poisson process as the noncommutative limit process.

On the existence-uniqueness and exponential estimate for solutions to stochastic functional differential equations driven by G-Lévy process

The existence-uniqueness theory for solutions to stochastic dynamic systems is always a significant theme and has received a huge attention. The objective of this article is to study the mentioned theory for stochastic functional differential equations (SFDEs) driven by G-Lévy process. The existence-uniqueness theorem for solutions to SFDEs driven by G-Lévy process has been determined. The error estimation between the exact solution and Picard approximate solutions has been shown. In addition, the exponential estimate has been derived.

Exponential functionals of Lévy processes with jumps

We study the exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$ of two one-dimensional independent Lévy processes $ξ$ and $η$, where $η$ is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping $Φ_ξ$ for a fixed Lévy process $ξ$, which maps the law of $η_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$, and study the behaviour of the range of $Φ_ξ$ for varying characteristics of $ξ$. Moreover, we derive conditions for selfde…

An $f$-divergence approach for optimal portfolios in exponential Levy models

We present a unified approach to get explicit formulas for utility maximising strategies in Exponential Levy models. This approach is related to $f$-divergence minimal martingale measures and based on a new concept of preservation of the Levy property by $f$-divergence minimal martingale measures. For common $f$-divergences, i.e. functions which satisfy $f"(x)= ax^ γ,\, a>0, \, γ\in \mathbb R$, we give the conditions for the existence of corresponding $u_f$- maximising strategies, as well as explicit formulas.

Asymptotic Results for Heavy-tailed Lévy Processes and their Exponential Functionals

In this paper we first provide several conditional limit theorems for Lévy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of heavy-tailed Lévy processes. As the key point, we observe that the asymptotics mainly depends on the sample paths with early arrival large jump. Both the polynomial decay rate and the exact expression of the limit coefficients are given. As an application, we give an exact description for the extinction speed of continuous-state branching processes in heavy-tail…

Asymptotic results for exponential functionals of Levy processes

In this work we give a complete description to the asymptotic behaviors of exponential functionals of Lévy processes and divide them into five different types according to their convergence rates. Not only their exact convergence speeds are proved, the accurate limit constants are also given. As an application, we study the survival probabilities of continuous-state branching processes in random environment defined in He et al. (2016). Like the discrete case and branching diffusion in random environment, we classify them into five different types according to their extinction speeds.

Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes

We present a novel idea for a coupling of solutions of stochastic differential equations driven by Lévy noise, inspired by some results from the optimal transportation theory. Then we use this coupling to obtain exponential contractivity of the semigroups associated with these solutions with respect to an appropriately chosen Kantorovich distance. As a corollary, we obtain exponential convergence rates in the total variation and standard $L^1$-Wasserstein distances.

Path independence of the additive functionals for stochastic differential equations driven by G-Lévy processes

In the paper, we consider a type of stochastic differential equations driven by G-Lévy processes. We prove that a kind of their additive functionals has path independence and extend some known results.

Extended Factorizations of Exponential Functionals of Lévy Processes

In [16], under mild conditions, a Wiener-Hopf type factorization is derived for the exponential functional of proper Lévy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by considering the exponential functional for killed Lévy processes. As a by-product, we derive some interesting new distributional properties enjoyed by a large class of this random variable, such as the absolute continuity of its distribution and the smoothness, boundedness or complete monotonicity of its density. This type of results is then used to derive similar pro…

On the density of exponential functionals of Lévy processes

In this paper, we study the existence of the density associated to the exponential functional of the Lévy process $ξ$, \[ I_{\ee_q}:=\int_0^{\ee_q} e^{ξ_s} \, \mathrm{d}s, \] where $\ee_q$ is an independent exponential r.v. with parameter $q\geq 0$. In the case when $ξ$ is the negative of a subordinator, we prove that the density of $I_{\ee_q}$, here denoted by $k$, satisfies an integral equation that generalizes the one found by Carmona et al. \cite{Carmona97}. Finally when $q=0$, we describe explicitly the asymptotic behaviour at 0 of the density $k$ when $ξ$ is the negative of a subordinato…

Integral functionals for spectrally positive Levy processes

We find necessary and sufficient conditions for almost sure finiteness of integral functionals of spectrally positive Lévy processes. Via Lamperti type transforms, these results can be applied to obtain new integral tests on extinction and explosion behaviors for a class of continuous-state nonlinear branching processes.

Lévy过程的一个最优泛函Itô公式

An Optimal Functional Itô's Formula For Lévy Processes

本文针对Lévy过程提出了一个最优的局部时-空泛函Itô公式，适用于任意有界连续泛函，且不依赖Hölder连续性条件。

A decomposition for additive functional of Levy processes

Motivated by the recent results of Nualart and Xu \cite{Nualart} concerning limits laws for occupation times of one dimensional symmetric stable processes, this paper proves a decomposition for functionals of one dimensional symmetric Lévy processes under certain conditions on the characteristic exponent and computes the moments of the decomposition.

Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon

There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are establishe…

Exact asymptotic for distribution densities of Levy functionals

A version of the saddle point method is developed, which allows one to describe exactly the asymptotic behavior of distribution densities of Levy driven stochastic integrals with deterministic kernels. Exact asymptotic behavior is established for (a) the transition probability density of a real-valued Levy process; (b) the transition probability density and the invariant distribution density of a Levy driven Ornstein-Uhlenbeck process; (c) the distribution density of the fractional Levy motion.

Malliavin calculus and Clark-Ocone formula for functionals of a square-integrable Lévy process

In this paper, we construct a Malliavin derivative for functionals of square-integrable Lévy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a Lévy process.

Permutation Invariant Functionals of Lévy Processes

We study natural invariance properties of functionals defined on Lévy processes and show that they can be described by a simplified structure of the deterministic chaos kernels in Itô's chaos expansion. These structural properties of the kernels relate intrinsically to a measurability with respect to invariant $σ$-algebras. This makes it possible to apply deterministic functions to invariant functionals on Lévy processes while keeping the simplified structure of the kernels. This stability is crucial for applications. Examples are given as well.

确定性期限上 Lévy 过程指数泛函的矩——恒等式与显式表达式

Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions

本文研究确定性期限上 Lévy 过程指数泛函的复矩，建立两个卷积恒等式并给出对称过程与从属子的显式矩公式。

Functional limit theorems for Levy processes and their almost-sure versions

In this paper we prove a criterion of convergence in distribution in Skorokhod space. We apply this criterion to some special Levy processes and obtain almost-sure versions of limit theorems for these processes.

A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-noise

We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by strictly stable Levy-processes with stability index bigger than one. The limit process turns out to be a strictly stable Levy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.

On limit theory for functionals of stationary increments Levy driven moving averages

In this paper we obtain new limit theorems for variational functionals of high frequency observations of stationary increments Lévy driven moving averages. We will see that the asymptotic behaviour of such functionals heavily depends on the kernel, the driving Lévy process and the properties of the functional under consideration. We show the "law of large numbers" for our class of statistics, which consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we also prove the associated weak limit theory, which again consists of three differe…

On matrix exponential approximations of the infimum of a spectrally negative Levy process

We recall four open problems concerning constructing high-order matrix-exponential approximations for the infimum of a spectrally negative Levy process (with applications to first-passage/ruin probabilities, the waiting time distribution in the M/G/1 queue, pricing of barrier options, etc). On the way, we provide a new approximation, for the perturbed Cramer-Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taaffe, which fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whi…

Lévy 过程的上确界惩罚

Supremum penalizations for Lévy processes

该研究考察一维 Lévy 过程在由上确界函数加权与归一化后的长时间极限，并分别通过指数时钟和常数时钟给出极限定理。

Onsager-Machlup action functional for stochastic partial differential equations with Levy noise

This work is devoted to deriving the Onsager-Machlup action functional for stochastic partial differential equations with (non-Gaussian) Levy process as well as Gaussian Brownian motion. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. This enables the investigation of the most probable transition path for infinite dimensional stochastic dynamical systems modeled by stochastic partial differential equations, by minimizing the Onsager-Machlup action functional.

Distributional properties of exponential functionals of Levy processes

We study the distribution of the exponential functional $I(ξ,η)=\int_0^{\infty} \exp(ξ_{t-}) \d η_t$, where $ξ$ and $η$ are independent Lévy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in Carmona et al "On the distribution and asymptotic results for exponential functionals of Levy processes". In the special case when $η$ is a Brownian motion with drift we show that this integral equation leads to an important functional e…

Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces

We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric $α$-stable noise and/or cylindrical Wiener noise. We also consider the case of a "singular" Wiener process with unbounded covariance operator. In particular, in the equation with diagonal pure $α$-stable cylindrical noise introd…

Lévy过程指数泛函的隐式更新理论

Implicit renewal theory for exponential functionals of Lévy processes

本文为Lévy过程的指数泛函建立新积分方程，给出其定律的三项Wiener-Hopf型分解，并揭示因子决定局部与渐近行为，补充了现有研究方法。

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Let $ξ=(ξ_t, t\ge 0)$ be a real-valued Lévy process and define its associated exponential functional as follows \[ I_t(ξ):=\int_0^t \exp\{-ξ_s\}{\rm d} s, \qquad t\ge 0. \] Motivated by important applications to stochastic processes in random environment, we study the asymptotic behaviour of \[ \mathbb{E}\Big[F\big(I_t(ξ)\big)\Big] \qquad \textrm{as}\qquad t\to \infty, \] where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $ξ$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $…

Cramer's estimate for the exponential functional of a Levy process

We consider the exponential functional $A_{\infty}=\int_0^{\infty} e^{ξ_s} ds$ associated to a Levy process $(ξ_t)_{t \geq 0}$. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process $ξ$, the main one being Cramer's condition, that asserts the existence of a real $χ>0$ such that ${\Bbb E}(e^{χξ_1})=1$. Then there exists $C>0$ satisfying, when $t \to +\infty$ : $$ {\Bbb P} (A_{\infty}> t) \sim C t^{-χ} \quad . $$ This result can be applied for example to the process $ξ_t = at - S_α(t)$ where $S_α$ stands for the stable subordinator of index $α…

A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes

For a Lévy process $ξ=(ξ_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_ξ=\int_0^{\infty}e^{ξ_t} dt.\] Under mild conditions on $ξ$, we show that the following factorization of exponential functionals \[{\rm{I}}_ξ\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}\] holds, where, $\times $ stands for the product of independent random variables, $H^-$ is the descending ladder height process of $ξ$ and $Y$ is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we gener…

Fluctuations of stable processes and exponential functionals of hypergeometric Levy processes

We study the distribution and various properties of exponential functionals of hypergeometric Levy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained by Kuznetsov "On extrema of stable processes" (2010) and Hubalek and Kuznetsov "A convergent series representation for the density of the supremum of a stable process"…

Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions

We first introduce and derive some basic properties of a two-parameters family of one-sided Levy processes. Their Laplace exponents are given in terms of the Pochhammer symbol. This family includes, in a limit case, the family of Brownian motion with drifts. Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties …

Asian options and meromorphic Levy processes

One method to compute the price of an arithmetic Asian option in a Levy driven model is based on the exponential functional of the underlying Levy process: If we know the distribution of the exponential functional, we can calculate the price of the Asian option via the inverse Laplace transform. In this paper we consider pricing Asian options in a model driven by a general meromorphic Levy process. We prove that the exponential functional is equal in distribution to an infinite product of indepedent beta random variables, and its Mellin transform can be expressed as an infinite product of gamm…

European and Asian Greeks for exponential Lévy processes

In this paper we give easy-to-implement closed-form expressions for European and Asian Greeks for general L2-payoff functions and underlying assets in an exponential Lévy process model with nonvanishing Brownian motion part. The results are based on Hilbert space valued Malliavin Calculus and extend previous results from the literature. Numerical experiments suggest, that in the case of a continuous payoff function, a combination of Malliavin Monte Carlo Greeks and the finite difference method has a better convergence behavior, whereas in the case of discontinuous payoff functions, the Malliav…

Functional limit theorems for Lévy processes satisfying Cramér's condition

We consider a Lévy process that starts from $x<0$ and conditioned on having a positive maximum. When Cramér's condition holds, we provide two weak limit theorems as $x\to -\infty$ for the law of the (two-sided) path shifted at the first instant when it enters $(0,\infty)$, respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.

Some properties of exponential integrals of Lévy processes and examples

The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a Lévy process on $\mathbb R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and $\mathbb R ^d$-valued, respectively. The condition for existence and finiteness of $Z$ is given and then the law $\mathcal L(Z)$ of $Z$ is considered. Some sufficient conditions for $\mathcal L(Z)$ to be selfdecomposable and some sufficient conditions for $\mathcal L(Z)$ to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where $d=1$,…

Gerber-Shiu functionals at Parisian ruin for Lévy insurance risk processes

Inspired by works of Landriault et al. \cite{LRZ-0, LRZ}, we study discounted penalties at ruin for surplus dynamics driven by a spectrally negative Lévy process with Parisian implementation delays. To be specific, we study the so-called Gerber-Shiu functional for a ruin model where at each time the surplus process goes negative, an independent exponential clock with rate $q>0$ is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of the…

一维Lévy过程为避免有界集合的条件化

Conditioning to avoid bounded sets for a one-dimensional Lévy processes

本文研究一维Lévy过程在随机时间趋于无穷前避免各类有界集合的条件化极限，涵盖有限点集、有界$F_σ$-集和整数格等情形。

Functional Laws for Trimmed Levy Processes

Two different ways of trimming the sample path of a stochastic process in D[0, 1]: global ("trim as you go") trimming and record time ("lookback") trimming are analysed to find conditions for the corresponding operators to be continuous with respect to the (strong) J1-topology. A key condition is that there should be no ties among the largest ordered jumps of the limit process. As an application of the theory, via the continuous mapping theorem we prove limit theorems for trimmed Levy processes, using the functional convergence of the underlying process to a stable process. The results are app…

FunctionaL Regular Variation of Lévy-driven Multivariate Mixed Moving Average Processes

We consider the functional regular variation in the space $\mathbb{D}$ of càdlàg functions of multivariate mixed moving average (MMA) processes of the type $X_t = \int\int f(A, t - s) Λ(d A, d s)$. We give sufficient conditions for an MMA process $(X_t)$ to have càdlàg sample paths. As our main result, we prove that $(X_t)$ is regularly varying in $\mathbb{D}$ if the driving Lévy basis is regularly varying and the kernel function $f$ satisfies certain natural (continuity) conditions. Finally, the special case of supOU processes, which are used, e.g., in applications in finance, is considered i…

Law of the exponential functional of one-sided Lévy processes and Asian options

The purpose of this note is to describe, in terms of a power series, the distribution function of the exponential functional, taken at some independent exponential time, of a spectrally negative Lévy process ξwith unbounded variation. We also derive a Geman-Yor type formula for Asian options prices in a financial market driven by e^ξ.

Almost periodic solutions for stochastic differential equations with exponential dichotomy driven by Levy noise

In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by Lévy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and uniqueness of bounded solutions. Furthermore, this unique bounded solution is almost periodic in distribution under slightly stronger conditions. We also give two examples to illustrate our results.

Randomisation and recursion methods for mixed-exponential Levy models, with financial applications

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Levy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Levy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Levy processes, whic…

No-Free-Lunch equivalences for exponential Levy models

We provide equivalence of numerous no-free-lunch type conditions for financial markets where the asset prices are modeled as exponential Levy processes, under possible convex constraints in the use of investment strategies. The general message is the following: if any kind of free lunch exists in these models it has to be of the most egregious type, generating an increasing ealth. Furthermore, we connect the previous to the existence of the numeraire portfolio, both for its particular expositional clarity in exponential Levy models and as a first step in obtaining analogues of the no-free-lunc…