Bounding the Maximal Height of a Diffusion by the Time Elapsed

Let X=(X _t ) _t0 be a one-dimensional time-homogeneous diffusion process associated with the infinitesimal generator

Reverse time diffusions

The paper considers a diffusion evolving in $\text{R}$ⁿ. The stochastic differential equations giving the same process, but with the time parameter evolving in the negative direction, are obtained under a certain integrability hypothesis when the diffusion has a density function on a time varying submanifold of $\text{R}$ⁿ.     

Ergodicity and Parameter Estimates for Infinite-Dimensional Fractional Ornstein-Uhlenbeck Process

Existence and ergodicity of a strictly stationary solution for linear stochastic evolution equations driven by cylindrical fractional Brownian motion are proved. Ergodic behavior of non-stationary infinite-dimensional fractional Ornstein-Uhlenbeck processes is also studied. Based on these results, strong consistency of suitably defined families of parameter estimators is shown. The general results are applied to linear parabolic and hyperbolic equations perturbed by a fractional noise. This work was partially supported by the GACR Grant 201/04/0750 and by the MSMT Research Plan MSM 4977751301.     

Transportation inequalities for stochastic delay evolution equations driven by fractional Brownian motion

We discuss stochastic functional partial differential equations and neutral partial differential equations of retarded type driven by fractional Brownian motion with Hurst parameter H>1/2. Using the Girsanov transformation argument, we establish the quadratic transportation inequalities for the law of the mild solution of those equations driven by fractional Brownian motion under the L² metric and the uniform metric.     

Harnack inequalities for SDEs driven by subordinate Brownian motions

By using regularization approximations of the underlying subordinator and a gradient estimate approach, the dimension-independent Harnack inequalities are established for the inhomogeneous semigroup associated with a class of SDEs with Lévy noise containing a subordinate Brownian motion. Our estimates in Harnack type inequalities improve the corresponding ones in the recent paper by Wang and Wang (2014) [10].     

Peng’s maximum principle for a stochastic control problem driven by a fractional and a standard Brownian motion

We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2. We apply an anticipative Girsanov transformation to transform the system into another one, driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion. We derive a maximum principle and the associated stochastic variational inequality, which both are generalizations of the classical case.     

A time fractional functional differential equation driven by the fractional Brownian motion

Let $B^H$ be a fractional Brownian motion with Hurst index $H>\frac12$. In this paper, we prove the global existence and uniqueness of the equation $$ \begin{cases} ^CD_t^{\gamma}x(t)=f(x_t)+G(x_t)\frac{d}{dt}B^H(t),\ \ \ \ &t\in(0,T], \x(t)=\eta(t), \ \ \ \ \ &t\in[-r,0], \end{cases} $$ where $\max\{H,2-2H\}<\gamma<1$, $^CD_t^{\gamma}$ is the Caputo derivative, and $x_t\in \mathcal{C}_r=\mathcal{C}([-r,0],\mathbb{R})$ with $x_t(u)=x(t+u),u\in[-r,0]$. We also study the dependence of the solution on the initial condition.     

Necessary Maximum Principle of Stochastic Optimal Control with Delay and Jump Diffusion

In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.     

三维Navier-Stokes方程的随机变分准则

我们将文献(Cipriano F,Cruzeiro A B.Navier-Stokes equation and diffusions on the group of homeomorphisms of the Torus[J].Commun.Math.Phys.,2007,275:255-269)推广到三维情形,即给出三维环面上的Navier-Stokes方程的随机变分准则.     

Sharp maximal inequalities for stochastic integrals in which the integrator is a submartingale

We obtain sharp maximal inequalities for strong subordinates of real-valued submartingales. Analogous inequalities also hold for stochastic integrals in which the integrator is a submartingale. The impossibility of general moment inequalities is also demonstrated.

     

Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Let X(t), t[0,1], be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique fixed point t₀ in the interval [0,1], the behavior of P{sup_t[0,1] X(t)>u} is known for u. We investigate the case where the unique point t₀ = t_u depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes X_u(t) depending on u, which have for each u such a unique point t_u tending to the boundary as u. We derive the asymptotic behavior of P{sup_t[0,1] X(t)>u}, depending on the rate as t_u tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the difficulties to simulate such rare events, when u is large.     

Estimation for Translation of a Process Driven by Fractional Brownian Motion

Abstract

We investigate the general problem of estimating the translation of a stochastic process governed by a stochastic differential equation driven by a fractional Brownian motion. The special case of the Ornstein-Uhlenbeck process is discussed in particular.     

Stochastic Maximum Principle for Forward-Backward Regime Switching Jump Diffusion Systems and Applications to Finance

The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.     

A Note on the Local Time of Fractional Brownian Motion

Asymptotic behavior of the local time at the origin of q-dimensional fractional Brownian motion is considered when the index approaches the critical value 1/q. It is proved that, under a suitable (temporally inhomogeneous) normalization, it converges in law to the inverse of an extremal process which appears in the extreme value theory.     

Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations

In this paper, a blow-up problem to nonlinear stochastic partial differential equations driven by Brownian motions is investigated. In particular, the impact of noises on the life span of solutions is studied. It is interesting to know that the noise can be used to postpone the blow-up time of a stochastic nonlinear system.     

带跳的分数倒向重随机微分方程及相应的随机积分偏微分方程

本文首次把Poisson随机测度引入分数倒向重随机微分方程,基于可料的Girsanov变换证明由Brown运动、Poisson随机测度和Hurst参数在(1/2,1)范围内的分数Brown运动共同驱动的半线性倒向重随机微分方程解的存在唯一性.在此基础上,本文定义一类半线性随机积分偏微分方程的随机黏性解,并证明该黏性解由带跳分数倒向重随机微分方程的解唯一地给出,对经典的黏性解理论作出有益的补充.     

二阶扩散模型的经验似然推断

本文利用经验似然方法得到了二阶扩散模型的漂移系数和扩散系数的经验似然估计量, 并研究这些估计量的相合性和渐近正态性. 进一步在经验似然方法的基础上给出了漂移系数和扩散系数的非对称的置信区间, 并且在一定的条件下证明了调整的对数似然比是渐近卡方分布的.