ON DE FINETTI'S OPTIMAL IMPULSE DIVIDEND CONTROL PROBLEM UNDER CHAPTER 11 BANKRUPTCY

Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow[44]to revisit the De Finetti dividend control problem under the reorganization process and the regulator's intervention documented in U.S.Chapter 11 bankruptcy.We do this by further accommodating the fixed transaction costs on dividends to imitate the real-world procedure of dividend payments.Incorporating the fixed transaction costs transforms the targeting optimal dividend problem into an impulse control problem rather than a singular control problem,and hence computations and proofs that are distinct from[44]are needed.To account for the financial stress that is due to the more subtle concept of Chapter 11 bankruptcy,the surplus process after dividends is driven by a piece-wise spectrally negative Lévy process with endogenous regime switching.Some explicit expressions of the expected net present values under a double barrier dividend strategy,new to the literature,are established in terms of scale functions.With the help of these expressions,we are able to characterize the optimal strategy among the set of admissible double barrier dividend strategies.When the tail of the Lévy measure is log-convex,this optimal double barrier dividend strategy is then verified as the optimal dividend strategy,solving our optimal impulse control problem.

Maximal speed of particles in super-Lévy process

We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of paths. We study the maximal speed of all particles during a given time period, which turns out to be a function of the packing dimension of the time period. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of the historical super-Lévy process.

Existence and joint continuity of local time of multi-parameter fractional Lévy processes

In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.

Reflected BSDEs Driven by Lévy Processes and Countable Brownian Motions

A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.

Lvy过程驱动的倒向重随机Volterra积分方程

考虑一类由Teugels鞅和2个相互独立的布朗运动共同驱动的倒向重随机Volterra积分方程,在系数满足Lipschitz假设条件下,利用不动点定理证明了适应解的存在唯一性.

Lévy过程驱动的随机二维Navier-Stokes方程解的指数性态(英文)

本文研究了Lévy过程驱动的随机二维Navier-Stokes方程弱解的指数性态.给出了不同条件下解的长时间形态,获得了一些特殊情形下解的样本轨道的指数稳定性.

THE ERGODICITY OF STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS WITH LVY JUMP

In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of（pt）t≥0 to equilibrium uniform on any bounded subset in H.

Dynamics of a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise

With the development of information technology,rumors propagate faster and more widely than in the past.In this paper,a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise is proposed.The global positivity of the solution process is proved,and further the basic reproductive number R_(0) is obtained.When R_(0)<1,the dynamical process of system with Lévy jump tends to the rumor-free equilibrium point of the deterministic system,and the rumor tends to extinction;when R_(0)>1,the rumor will keep spreading and the system will oscillate randomly near the rumor equilibrium point of the deterministic system.The results show that the oscillation amplitude is related to the disturbance of the system.In addition,increasing media coverage can effectively reduce the final spread of rumors.Finally,the above results are verified by numerical simulation.

马尔可夫交换Lévy过程模型下的期权定价及其对冲（英文）

In this paper, we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging. In this model, the market interest rate, the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process. We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure. The option price using this model is obtained by the Fourier transform method. We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.

A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation

This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy triplets.This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)with.To construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space.We further prove a new type of Lévy-Khintchine representation formula to characterize.As a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

G-Lévy processes under sublinear expectations

We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations.We then obtain the Lévy-Khintchine formula and the existence for G-Lévy processes.We also introduce G-Poisson processes.

Exact joint laws associated with spectrally negative Levy processes and applications to insurance risk theory

We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson＇s formula is provided.

A Mean-Field Optimal Control for Fully Coupled Forward-Backward Stochastic Control Systems with Lévy Processes

This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes.In these systems,the coefficients contain not only the state processes but also their marginal distribution,and the cost function is of mean-field type as well.The necessary and sufficient conditions for such optimal problems are obtained.Furthermore,the applications to the linear quadratic stochastic optimization control problem are investigated.

Generalized Backward Doubly Stochastic Differential Equations Driven by Lévy Processes with Continuous Coefficients

A new class of generalized backward doubly stochastic differential equations （GBDSDEs in short） driven by Teugels martingales associated with Levy process are investigated. We establish a comparison theorem which allows us to derive an existence result of solutions under continuous and linear growth conditions.

Precise Asymptotics for Lévy Processes

Let {X（t）, t ≥ 0} be a Lévy process with EX（1） = 0 and EX^2（1） 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X（t）, t 〉 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.

A Wiener-Hopf factorization related potential measure for spectrally negative Lévy process

For spectrally negative Lévy process (SNLP), we find an expression, in terms of scale functions, for a potential measure involving the maximum and the last time of reaching the maximum up to a draw-down time. As applications, we obtain a potential measure for the reflected SNLP and recover a joint Laplace transform for the Wiener-Hopf factorization for SNLP.

Mean-Field, Infinite Horizon, Optimal Control of Nonlinear Stochastic Delay System Governed by Teugels Martingales Associated with Lévy Processes

This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.

Exponential Convergence in Probability for Empirical Means of Lévy Processes

Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.

Reflected and Doubly Reflected BSDEs for L'evy Processes:Solutions and Comparison

In this paper we study reflected and doubly reflected backward stochastic differential equations （BSDEs, for short） driven by Teugels martingales associated with L^vy process satisfying some moment condi- tions and by an independent Brownian motion. For BSDEs with one reflecting barrier, we obtain a comparison theorem using the Tanaka-Meyer formula. For BSDEs with two reflecting barriers, we first prove the existence and uniqueness of the solutions under the Mokobodski＇s condition by using the Snell envelope theory and then we obtain a comparison result.