Equivalent and absolutely continuous measure changes for jump-diffusion processes

On the Coupling Property of Lévy Processes

We give necessary and sufficient conditions guaranteeing that the coupling for Lévy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall-Rogers on couplings of random walks. In particular, we obtain that a Lévy process admits a successful coupling, if it is a strong Feller process or if the Lévy (jump) measure has an absolutely continuous component.

On a transformation between distributions obeying the principle of a single big jump

Beck et al. (2013) introduced a new distribution class J which contains many heavy-tailed and light-tailed distributions obeying the principle of a single big jump. Using a simple transformation which maps heavy-tailed distributions to light-tailed ones, we find some light-tailed distributions, which belong to the class J but do not belong to the convolution equivalent distribution class and which are not even weakly tail equivalent to any convolution equivalent distribution. This fact helps to understand the structure of the light-tailed distributions in the class J and leads to a negative an…

On the absolute continuity of Lévy processes with drift

We consider the problem of absolute continuity for the one-dimensional SDE \[X_t=x+\int_0^ta(X_s) ds+Z_t,\] where $Z$ is a real Lévy process without Brownian part and $a$ a function of class $\mathcal{C}^1$ with bounded derivative. Using an elementary stratification method, we show that if the drift $a$ is monotonous at the initial point $x$, then $X_t$ is absolutely continuous for every $t>0$ if and only if $Z$ jumps infinitely often. This means that the drift term has a regularizing effect, since $Z_t$ itself may not have a density. We also prove that when $Z_t$ is absolutely continuous, the…

Existence of invariant densities for semiflows with jumps

The problem of existence and uniqueness of absolutely continuous invariant measures for a class of piecewise deterministic Markov processes is investigated using the theory of substochastic semigroups obtained through the Kato--Voigt perturbation theorem on the $L^1$-space. We provide a new criterion for the existence of a strictly positive and unique invariant density for such processes. The long time qualitative behavior of the corresponding semigroups is also considered. To illustrate our general results we give a detailed study of a two dimensional model of gene expression with bursting.

No-arbitrage conditions and absolutely continuous changes of measure

We study the stability of several no-arbitrage conditions with respect to absolutely continuous, but not necessarily equivalent, changes of measure. We first consider models based on continuous semimartingales and show that no-arbitrage conditions weaker than NA and NFLVR are always stable. Then, in the context of general semimartingale models, we show that an absolutely continuous change of measure does never introduce arbitrages of the first kind as long as the change of measure density process can reach zero only continuously.

Optimal control with absolutely continuous strategies for spectrally negative Levy processes

In the last few years there has been renewed interest in the classical control problem of de Finetti for the case that underlying source of randomness is a spectrally negative Levy process. In particular a significant step forward is made in an article of Loeffen where it is shown that a natural and very general condition on the underlying Levy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Levy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem but no…

Stochastic Fokker-Planck PIDE for conditional McKean-Vlasov jump diffusions and applications to optimal control

The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate an Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of conditional McKean-Vlasov jump diffusions. Th…

Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients

We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of R^d with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.

On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes

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…

On jump-diffusion processes with regime switching: martingale approach

We study jump-diffusion processes with parameters switching at random times. Being motivated by possible applications, we characterise equivalent martingale measures for these processes by means of the relative entropy. The minimal entropy approach is also developed. It is shown that in contrast to the case of Lévy processes, for this model an Esscher transformation does not produce the minimal relative entropy.

Uniform Large deviations for infinite dimensional stochastic systems with jumps

Uniform large deviation principles for positive functionals of all equivalent types of infinite dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational representation formula which for an infinite sequence of i.i.d real Brownian motions and a Poisson random measure was shown in [5].

A stochastic volatility model with jumps

We consider a stochastic volatility model with jumps where the underlying asset price is driven by the process sum of a 2-dimensional Brownian motion and a 2-dimensional compensated Poisson process. The market is incomplete, resulting in infinitely many equivalent martingale measures. We find the set equivalent martingale measures, and we hedge by minimizing the variance using Malliavin calculus.

Telegraph Processes with Random Jumps and Complete Market Models

We propose a new generalisation of jump-telegraph process with variable velocities and jumps. Amplitude of the jumps and velocity values are random, and they depend on the time spent by the process in the previous state of the underlying Markov process. This construction is applied to markets modelling. The distribution densities and the moments satisfy some integral equations of the Volterra type. We use them for characterisation of the equivalent risk-neutral measure and for the expression of historical volatility in various settings. The fundamental equation is derived by similar arguments.…

Linear Programming Formulations of Singular Stochastic Control Problems: Time-Homogeneous Problems

Conditions are established under which the optimal control of processes having both absolutely continuous and singular (with respect to time) controls are equivalent to linear programs over a space of measures on the state and control spaces. This paper considers long-term average and discounted criteria and includes budget and resource constraints. The linear programs optimize over measures representing the expected occupation measure of the state and absolutely continuous control processes and a similar expected occupation measure of the state and control when the singular action of the proc…

Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $α$-stable noise.

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

In this paper, we study the transition densities of pure-jump symmetric Markov processes in $ {\mathbb R}^d$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment…

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of Ornstein-Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show, that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous ret…

Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces

We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the generator of the Feller process, we are able to define a notion of absolutely continuous trajectories of measures in terms of some topology on this core. Also, we define a Hamiltonian in terms of the linear generator and a Lagrangian as its Legendre transform. We prove the large deviation principle and show that the rate function can be decomposed as a rate …

Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps

We consider a stochastic volatility model with Lévy jumps for a log-return process $Z=(Z_{t})_{t\geq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{t\geq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{t\geq 0}$ is an independent Lévy process with absolutely continuous Lévy measure $ν$. Small-time expansions, of arbitrary polynomial order, in time-$t$, are obtained for the tails $\bbp(Z_{t}\geq z)$, $z>0$, and for the call-option prices $\bbe(e^{z+Z_{t}}-1)_{+}$, $z\neq 0$, assuming smoothness conditions on the {\PaleGrey density of $ν$} away from the origin and a small-time large…

Ergodicity of stochastic shell models driven by pure jump noise

In the present paper we study a stochastic evolution equation for shell (SABRA \& GOY) models with pure jump \levy noise $L=\sum_{k=1}^\infty l_k(t)e_k$ on a Hilbert space $\h$. Here $\{l_k, k\in \mathbb{N}\}$ is a family of independent and identically distributed (i.i.d.) real-valued pure jump \levy processes and $\{e_k, k\in \mathbb{N}\}$ is an orthonormal basis of $\h$. We mainly prove that the stochastic system has a unique invariant measure. For this aim we show that if the \levy measure of each component $l_k(t)$ of $L$ satisfies a certain order and a non-{degeneracy} condition and is ab…

A double-ended queue with catastrophes and repairs, and a jump-diffusion approximation

Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of th…

带跳跃的半鞅最优传输：一般框架与等价形式

Semimartingale Optimal Transport with Jumps: A General Framework and Equivalent Formulations

研究成本依赖于全微分特征的半鞅最优传输问题，在最小假设下证明最优计划存在并建立Kantorovich型对偶，给出四种等价形式，统一推广了经典最优传输、鞅最优传输、薛定谔桥问题和重心弱最优传输。

Stochastic calculus of variations for general Lévy processes and its applications to jump-type SDE's with non-degenerated drift

We consider an SDE in R^m of the type dX(t)=a(X(t))dt+dU(t) with a Lévy process U and study the problem for the distribution of a solution to be regular in various senses. We do not impose any specific conditions on the Lévy measure of the noise, and this is the main difference between our method and the known methods by J.Bismut or J.Picard. The main tool in our approach is the stochastic calculus of variations for a Lévy process, based on the time-stretching transformations of the trajectories. Three problems are solved in this framework. First, we prove that if the drift coefficient a is no…

Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation

The invariant measure of a one-dimensional Allen-Cahn equation with an additive space-time white noise is studied. This measure is absolutely continuous with respect to a Brownian bridge with a density which can be interpreted as a potential energy term. We consider the sharp interface limit in this setup. In the right scaling this corresponds to a Gibbs type measure on a growing interval with decreasing temperature. Our main result is that in the limit we still see exponential convergence towards a curve of minimizers of the energy if the interval does not grow too fast. In the original scali…

时间不一致奇异控制问题：反射控制与速率爆炸的绝对连续控制

Time-inconsistent singular control problems: Reflection and Absolutely continuous controls with exploding rates

本文研究非指数贴现导致的时间不一致奇异控制问题，提出涵盖传统反射控制与速率爆炸的温和阈值控制的新控制类，建立基于变分不等式的验证定理，并在库存管理案例中首次构造出具有不可达边界的均衡解。

Equivalent martingale measures for Lévy-driven moving averages and related processes

In the present paper we obtain sufficient conditions for the existence of equivalent martingale measures for Lévy-driven moving averages and other non-Markovian jump processes. The conditions that we obtain are, under mild assumptions, also necessary. For instance, this is the case for moving averages driven by an $α$-stable Lévy process with $α\in (1,2]$. Our proofs rely on various techniques for showing the martingale property of stochastic exponentials.

On the rate of convergence of weak Euler approximation for non-degenerate SDEs

The paper estimates the rate of convergence of the weak Euler approximation for the solutions of SDEs with Hoelder continuous coefficients driven by point and martingale measures. The equation considered has a non-degenerate main part whose jump intensity measure is absolutely continuous with respect to the Levy measure of a spherically-symmetric stable process. It includes the nondegenerate diffusions and SDEs driven by Levy processes.

Reflected BSDEs with nonpositive jumps, and controller-and-stopper games

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution is proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman-Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affect both drift …

Markov chain approximations for symmetric jump processes

Markov chain approximations of symmetric jump processes are investigated. Tightness results and a central limit theorem are established. Moreover, given the generator of a symmetric jump process with state space $\mathbbm{R}^d$ the approximating Markov chains are constructed explicitly. As a byproduct we obtain a definition of the Sobolev space $H^{α/2}(\mathbbm{R}^d)$, $α\in (0,2)$, that is equivalent to the standard one.

跳跃扩散过程的Onsager-Machlup泛函的概率流方法

Probability Flow Approach to the Onsager--Machlup Functional for Jump-Diffusion Processes

针对跳跃扩散过程推导Onsager-Machlup泛函的长期挑战，本文提出利用纯扩散与跳跃扩散间的等效概率流方法，首次建立有限跳跃活性跳跃扩散过程该泛函的闭式表达式，并将方法拓展至无限跳跃活性Lévy过程得到时间离散版本。

关于点过程可预测强度的最小值

On minimal predictable intensity of point processes

本文研究了点过程可预测强度的最小值问题，证明在测度绝对连续变换下，标准泊松过程具有最小可预测强度。

次对称马尔可夫过程的刻画

Characterization of subordinate symmetric Markov processes

本文刻画了次对称马尔可夫过程的跳跃核，建立了其遵循Liu-Murugan的等价条件，并阐明了与扩散类型不同的尺度，结果可推广至非次过程。

带加性半鞅噪声的欧拉-丸山格式的渐近误差分布

Asymptotic Error Distributions for Euler-Maruyama Schemes with Additive Semimartingale Noise

该研究建立带加性半鞅噪声的欧拉-丸山格式的渐近误差分布相关联合稳定极限定理，将其应用于对应随机微分方程，得到不同条件下的误差收敛结果。

Finitary isomorphisms of renewal point processes and continuous-time regenerative processes

We show that a large class of stationary continuous-time regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with exponential tails is finitarily isomorphic to a Poisson point process. We further give simple necessary and sufficient conditions for a renewal point process to be finitarily isomorphic to a Poisson point process. This improves results and answers several questions of Soo and of Kosloff and Soo.

On the structure of a class of distributions obeying the principle of a single big jump

In this paper, we present several heavy-tailed distributions belonging to the new class J of distributions obeying the principle of a single big jump introduced by Beck et al. [1]. We describe the structure of this class from different angles. First, we show that heavy-tailed distributions in the class J are automatically strongly heavy-tailed and thus have tails which are not too irregular. Second, we show that such distributions are not necessarily weakly tail equivalent to a subexponential distribution. We also show that the class of heavy-tailed distributions in J which are neither long-ta…

Location of the Path Supremum for Self-similar Processes with Stationary Increments

In this paper we consider the distribution of the location of the path supremum in a fixed interval for self-similar processes with stationary increments. To this end, a point process is constructed and its relation to the distribution of the location of the path supremum is studied. Using this framework, we show that the distribution has a spectral-type representation, in the sense that it is always a mixture of a special group of absolutely continuous distributions, plus point masses on the two boundaries. Bounds on the value and the derivatives of the density function are established. We fu…

Symmetric Jump Processes and their Heat Kernel Estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Holder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.

跳跃扩散模型中的过滤约简与完备性

Filtration Reduction and Completeness in Jump-Diffusion Models

本文研究不完备跳跃扩散市场中的衍生品定价与对冲，提出通过过滤约简构造虚构完备市场以获得唯一等价鞅测度，并利用一致性概念将其提升回原始经济，从简单设定逐步推广到一般模型。

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

We study some jumping SDE and the corresponding Fokker-Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_t)_{t\in [0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_t)_{t\in[0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli…

一种通用的Run-and-Tumble粒子形状转变方法：一维PDMP框架用于不变测度正则性

Invariant-measure regularity for one-dimensional PDMPs with applications to shape transitions in run-and-tumble particle systems

本文提出了一种通用方法，用于研究受势能作用下Run-and-Tumble粒子的形状转变，通过一维PDMP框架分析不变测度的正则性，揭示了形状转变的普遍性及其在非平衡统计力学中的重要性。

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the f…

Hölder正则性与加权泛函不等式的稳定性

Stability of Hölder regularity and weighted functional inequalities

本文研究度量测度空间上的对称Dirichlet形式，引入跳跃测度尾部条件和加权泛函不等式的新表述，建立了caloric和harmonic函数Hölder正则性的稳定等价刻画，并应用于证明具有边界爆炸行为的对称Markov过程的caloric函数的Hölder连续性。

Fundamental Theorem of Asset Pricing under fixed and proportional transaction costs

We show that the lack of arbitrage in a model with both fixed and proportional transaction costs is equivalent to the existence of a family of absolutely continuous single-step probability measures, together with an adapted process with values between the bid-ask spreads that satisfies the martingale property with respect to each of the measures. This extends Harrison and Pliska's classical Fundamental Theorem of Asset Pricing to the case of combined fixed and proportional transaction costs.

A local limit theorem for densities of the additive component of a finite Markov Additive Process

In this paper, we are concerned with centered Markov Additive Processes $\{(X_t,Y_t)\}_{t\in\T}$ where the driving Markov process $\{X_t\}_{t\in\T}$ has a finite state space. Under suitable conditions, we provide a local limit theorem for the density of the absolutely continuous part of the probability distribution of $t^{-1/2}Y_t$ given $X_0$. The rate of convergence and the moment condition are the expected ones with respect to the i.i.d case. An application to the joint distribution of local times of a finite jump process is sketched.

度量测度空间上无界区域中跳过程的生存概率

Survival probability for jump processes in unbounded domains on metric measure spaces

该研究针对度量测度空间上谱底为正的无界区域中的对称跳过程，推导了带显式常数的生存概率渐近上下界，为α-稳定过程的谱底正性提供概率解释与几何条件，且在喇叭形区域中得到精确衰减速率。

Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps

It is common practice to treat small jumps of Lévy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistica…