关于O-U型马氏过程维数和碰撞的一点结果

本文给出了维O－U型马氏过程像集的Housdorff维数上下界的一个估计，并研究了两个独立的一维O－U型马氏过程的碰撞问题。     

算子自相似马氏过程的Hausdorff维数定理

对各向同性,算子自相似马氏过程,本文在恰当的条件下得到了其象集与图集的Ｈａｕｓｄｏｒｆｆ维数。其证明表明该维数的估计与其算子特征值的实部有关。     

d维平稳高斯过程极集的充分条件及维数

本文研究了d维平稳高斯过程极集的性质,给出了d维平稳高斯过程广义极性的充分条件,并通过一个特殊的Cantor型集的构造将极集的维数与容度巧妙地结合起来,得到了d维平稳高斯过程非极集的Hausdorff维数的下确界.     

ON THE HITTING PROBABILITY AND POLARITY FOR A CLASS OF SELF-SIMILAR MARKOV PROCESSES

1IntroductionSelf-similarMarkoyprocesseson(0,co)wereintroducedbyLamperti[ZI.AnormalMarkovprocessX=(fi,F,R,Xt,ot,p")withstatespace(0,co)iscalledself-similarMarkovprocessoforderorjor>0,ifitstransitionfunctionPt(x,A)satisfiesPt(x,A)=Pa,(a"x,a"A)Vt>9,a>0,x>0,AEB(0,co).Itwasshownin[2]and[11]thateveryself-similarMarkovprocesson(0,co)automaticllyisstronglyMarkovianandhasanicepath",soitisaHuntprocessinthesenseof[1].Thefractalpropertiesoftheprocesseswerestudiedin[6]and[8].TheitoExcursiontheor…     

自相似分量过程Hausdorff维数结果

设Ｘ（ｔ）（ｔ∈Ｒ＋）是一个具有独立自相似分量过程。我们在一些较弱的条件下得到了它的像集和图订的Ｈａｕｓｄｏｒｆｆ维数。     

N指标d维广义Wiener过程像集的一致维数

研究了N指标d维广义Wiener过程像集的一致维数和测度，得到了其像集的致Hausdorff维数和一致Packing维数。     

N指标d维广义Wiener过程象集代数和性质

令Ｗ↑￣（ｔ）：Ｒ＋＾Ｎ→Ｒ＾ｄ是Ｎ指标ｄ维广义Ｗｉｅｎｅｒ过程，对任意紧集Ｅ，Ｆ∩→Ｒ＋＾Ｎ／｛０｝，本文研究了代数和Ｗ↑￣（Ｅ）－Ｗ↑￣（Ｆ）的Ｈａｕｓｄｏｒｆｆ维数及内点存在性，此结果包含并推广了Ｂｒｏｗｎｉａｎ Ｓｈｅｅｔ的结果。     

Countably many simultaneous random changes of time scales

Simultaneous changes of time scales of the components of a vector Markov process are defined and developed. Measurability properties, Dynkin's lemma, and the strong Markov property are established for the transformed process.     

Ornstein－Uhlenbeck型马氏过程局部时与占位时的存在性

本文考虑Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ型马氏过程的局部时，证明了在一定情形下局部时的存在性，并给出了不存在的反例，同时讨论了这类过程的占位时，指出了在某些限制性条件下，占位时密度的平方可积性．     

Lévy Approximation of Increment Processes with Markov Switching

We study weak convergence of increment processes with embedded Markov chain switching in a series scheme. The limit process is a Lévy process where the jump part is a compound Poisson process. A result concerning the rate of convergence is also given. This study is motivated by risk theory and its applications.     

Affine representations of fractional processes with applications in mathematical finance

Fractional processes have gained popularity in financial modeling due to the dependence structure of their increments and the roughness of their sample paths. The non-Markovianity of these processes gives, however, rise to conceptual and practical difficulties in computation and calibration. To address these issues, we show that a certain class of fractional processes can be represented as linear functionals of an infinite dimensional affine process. This can be derived from integral representations similar to those of Carmona, Coutin, Montseny, and Muravlev. We demonstrate by means of several examples that this allows one to construct tractable financial models with fractional features.     

Time change equations for Lévy-type processes

We consider time change equations for Lévy-type processes. In this context we generalize the results of Böttcher et al. (2013) significantly. Namely, we are able to incorporate measurable instead of continuous multipliers. This opens a gate to find whole classes of symbols for which corresponding processes do exist. In order to establish our results we carefully analyze the connection between time change equations and classical initial value problems. This relationship allows us to transfer well-known results from this classical subject of pure mathematics into the theory of stochastic processes. On the way to prove our main theorem we establish generalizations of results on paths of Lévy-type processes.     

Extremes of q-Ornstein–Uhlenbeck processes

Two limit theorems are established on the extremes of a family of stationary Markov processes, known as $q$-Ornstein–Uhlenbeck processes with $q\in (?1,1)$. Both results are crucially based on the weak convergence of the tangent process at the lower boundary of the domain of the process, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown–Resnick-type limit theorem on the minimum process of i.i.d. copies of the $q$-Ornstein–Uhlenbeck process: with appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.     

State estimation for cox processes on general spaces

Let N be an observable Cox process on a locally compact space E directed by an unobservable random measure M. Techniques are presented for estimation of M, using the observations of N to calculate conditional expectations of the form E [M]|$\text{F}$_A], where $\text{F}$_A is the σ–algebra generated by the restriction of N to A. We introduce a random measure whose distribution depends on N_A, from which we obtain both exact estimates and a recursive method for updating them as further observations become available. Application is made to the specific cases of estimation of an unknown, random scalar multiplier of a known measure, of a symmetrically distributed directing measure M and of a Markov–directed Cox process on $\text{R}$. By means of a Poisson cluster representation, the results are extended to treat the situation where N is conditionally additive and infinitely divisible given M.     

Large time asymptotics of Feynman–Kac functionals for symmetric stable processes

Let μ be a positive Kato measure on associated with the Green kernel of the transient symmetric α‐stable process, the Markov process with generator (). Let be the positive continuous additive functional in the Revuz correspondence with μ. If, in addition, μ is of compact support, we give exact large time asymptotics of the expectation of the Feynman–Kac functional, .     

Volterra-type Ornstein–Uhlenbeck processes in space and time

We propose a novel class of temporo-spatial Ornstein–Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.     

Process convergence for the complexity of Radix Selection on Markov sources

A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered.The complexity is studied as a stochastic process indexed by the set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science.For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations.     

Two-mode control of Brownian Motion with quadratic loss and switching costs

We consider a simple problem in the optimal control of Brownian Motion. There are two modes of control available, each with its own drift and diffusion coefficients, and switching costs are incurred whenever the control mode is changed. Finally, holding costs are incurred according to a quadratic function of the state of the system, and all costs are continuously discounted. It is shown that there exists an optimal policy involving just two critical numbers, and formulas are given for computation of the critical numbers.