On strong solutions for positive definite jump diffusions

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics.Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the αth positive semidefinite power of the process itself with 0.5<α<1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions.     

Self-normalization: Taming a wild population in a heavy-tailed world

The past two decades have witnessed the active development of a rich probability theory of Studentized statistics or self-normalized processes, typified by Student's t-statistic as introduced by W. S. Gosset more than a century ago, and their applications to statistical prob-lems in high dimensions, including feature selection and ranking, large-scale multiple testing and sparse, high dimensional signal detection. Many of these applications rely on the robust-ness property of Studentization/self-normalization against heavy-tailed sampling distributions. This paper gives an overview of the salient progress of self-normalized limit theory, from Studen-t's t-statistic to more general Studentized nonlinear statistics. Prototypical examples include Studentized one- and two-sample U-statistics. Furthermore, we go beyond independence and glimpse some very recent advances in self-normalized moderate deviations under dependence.     

Functional central limit theorems for self-normalized least squares processes in regression with possibly infinite variance data

Based on an R²-valued random sample {(y_i,x_i),1≤i≤n} on the simple linear regression model y_i=x_iβ+α+ε_i with unknown error variables ε_i, least squares processes (LSPs) are introduced in D[0,1] for the unknown slope β and intercept α, as well as for the unknown β when α=0. These LSPs contain, in both cases, the classical least squares estimators (LSEs) for these parameters. It is assumed throughout that {(x,ε),(x_i,ε_i),i≥1} are i.i.d. random vectors with independent components x and ε that both belong to the domain of attraction of the normal law, possibly both with infinite variances. Functional central limit theorems (FCLTs) are established for self-normalized type versions of the vector of the introduced LSPs for (β,α), as well as for their various marginal counterparts for each of the LSPs alone, respectively via uniform Euclidean norm and sup–norm approximations in probability. As consequences of the obtained FCLTs, joint and marginal central limit theorems (CLTs) are also discussed for Studentized and self-normalized type LSEs for the slope and intercept. Our FCLTs and CLTs provide a source for completely data-based asymptotic confidence intervals for β and α.     

Self-normalized limit theorems for linear processes generated by <Emphasis Type="Italic">ρ</Emphasis>-mixing innovations<Superscript>*</Superscript>

In this paper, we study the asymptotic behavior of the self-normalizer V _n ² for partial sums of linear processes generated by strictly stationary ρ-mixing innovations with infinite variance. Further, by using this we derive self-normalized versions of the CLT, the functional CLT, and the almost sure CLT for partial sums of the processes.     

Exit times for a class of piecewise exponential Markov processes with two-sided jumps

We consider first passage times for piecewise exponential Markov processes that may be viewed as Ornstein–Uhlenbeck processes driven by compound Poisson processes. We allow for two-sided jumps and as a main result we derive the joint Laplace transform of the first passage time of a lower level and the resulting undershoot when passage happens as a consequence of a downward (negative) jump. The Laplace transform is determined using complex contour integrals and we illustrate how the choice of contours depends in a crucial manner on the particular form of the negative jump part, which is allowed to belong to a dense class of probabilities. We give extensions of the main result to two-sided exit problems where the negative jumps are as before but now it is also required that the positive jumps have a distribution of the same type. Further, extensions are given for the case where the driving Lévy process is the sum of a compound Poisson process and an independent Brownian motion. Examples are used to illustrate the theoretical results and include the numerical evaluation of some concrete exit probabilities. Also, some of the examples show that for specific values of the model parameters it is possible to obtain closed form expressions for the Laplace transform, as is the case when residue calculus may be used for evaluating the relevant contour integrals.     

Asymptotic properties of nonlinear autoregressive Markov processes with state-dependent switching

In this paper, we consider a class of nonlinear autoregressive (AR) processes with state-dependent switching, which are two-component Markov processes. The state-dependent switching model is a nontrivial generalization of Markovian switching formulation and it includes the Markovian switching as a special case. We prove the Feller and strong Feller continuity by means of introducing auxiliary processes and making use of the Radon-Nikodym derivatives. Then, we investigate the geometric ergodicity by the Foster-Lyapunov inequality. Moreover, we establish the V-uniform ergodicity by means of introducing additional auxiliary processes and by virtue of constructing certain order-preserving couplings of the original as well as the auxiliary processes. In addition, illustrative examples are provided for demonstration.     

Local times of stochastic processes with positive definite bivariate densities

A new class of stochastic processes, called processes of positive bivariate type, is defined. Such a process is typically one whose bivariate density functions are positive definite, at least for pairs of time points which are sufficiently mutually close. The class includes stationary Gaussian processes and stationary reversible Markov processes, and is closed under the operations of composition and convolution. The purpose of this work is to show that the local times of such processes can be investigated in a natural way. One of the main contributions is an orthogonal expansion of the local time which is new even in the well-studied stationary Gaussian case. The basic tool in its construction is the Lancaster-Sarmanov expansion of a bivariate density in a series of canonical correlations and canonical variables.     

Stopping for two-dimensional stochastic processes

The aim of this paper is to introduce some techniques that can be used in the study of stochastic processes which have as parameter set the positive quadrant of the plane R²₊. We define stopping lines and derive an interesting property of measurability for them. The notion of predictability is developed, and we show the connection between predictable processes, fields associated with stopping lines, and predictable stopping lines. We also give a theorem of section for predictable sets. Extension to processes indexed by any partially ordered set with some regularity assumptions can be carried out quite easily with the same techniques.     

Self-Normalized LIL for Hanson–Russo Type Increments

Let X ₁, X ₂,... be independent, but not necessarily identically distributed random variables in the domain of attraction of a normal law or a stable law with index 0 < α < 2. Using suitable self-normalizing (or Studentizing) factors, laws of the iterated logarithm for self-normalized Hanson–Russo type increments are discussed. Also, some analogous results for self-normalized weighted sums of i.i.d. random variables are given.     

Point processes in the plane

In this survey paper, two-parameter point processes are studied in connection with martingale theory and with respect to the partial-order induced by the Cartesian coordinates of the plane. Point processes are characterized by jump stopping times and by their two-parameter compensators. Properties of the doubly stochastic Poisson process, such as predictability, are discussed. A definition for the Palm measure of a two-parameter stationary point process is proposed.     

On order-preservation and positive correlations for multidimensional diffusion processes

Summary As a continuation of the study by Herbst and Pitt (1991), this note presents two criteria. The first one is on the order-preservation for two (may be different) multidimensional diffusion processes. The second one is on the preservation of positive correlations for a diffusion process.Research supported in part by the Ying-Tung Fok Educational Foundation and the National Natural Science Foundation of China     

Asymptotically one-dimensional diffusions on the Sierpinski gasket and theabc-gaskets

Summary Diffusion processes on the Sierpinski gasket and theabc-gaskets are constructed as limits of random walks. In terms of the associated renormalization group, the present method uses the inverse trajectories which converge to unstable fixed points corresponding to the random walks on one-dimensional chains. In particular, non-degenerate fixed points are unnecessary for the construction. A limit theorem related to the discrete-time multi-type non-stationary branching processes is applied.     

Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions of SDEs. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate the Markov property. To prove uniqueness we solve a general martingale problem for càdlàg processes. This result is of independent interest. Application of our results to generalized exponential Lévy model are present in the last section.     

A weak invariance principle for self-normalized products of sums of mixing sequences

Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of partial sums is obtained.     

Potential Theory of Geometric Stable Processes

In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x|^d log ²|x|), while the Lévy density behaves like 1/|x|^d. We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes. The research of this author is supported in part by MZT grant 0037118 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

Ergodicity for time-changed symmetric stable processes

In this paper we study ergodicity and related semigroup property for a class of symmetric Markov jump processes associated with time-changed symmetric α$\alpha$-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric α$\alpha$-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.     

On tightness and stopping times

This paper studies conditions of tightness for sequences of processes, which conditions are mostly based on the use of ‘dominating’ increasing processes. The results obtained follow in directions initiated by Aldous and Rebolledo and are particularly well-suited for studying sequences of semimartingales. Also obtained are results that extend sufficient conditions of Aldous's type to processes that are not quasi-left-continuous.     

Tempering stable processes

A tempered stable Lévy process combines both the α$\alpha$-stable and Gaussian trends. In a short time frame it is close to an α$\alpha$-stable process while in a long time frame it approximates a Brownian motion. In this paper we consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization. We prove short and long time behavior of tempered stable Lévy processes and investigate their absolute continuity with respect to the underlying α$\alpha$-stable processes. We find probabilistic representations of tempered stable processes which specifically show how such processes are obtained by cutting (tempering) jumps of stable processes. These representations exhibit α$\alpha$-stable and Gaussian tendencies in tempered stable processes and thus give probabilistic intuition for their study. Such representations can also be used for simulation. We also develop the corresponding representations for Ornstein–Uhlenbeck-type processes.     

Asymptotic properties of jump-diffusion processes with state-dependent switching

This work is concerned with a class of jump-diffusion processes with state-dependent switching. First, the existence and uniqueness of the solution of a system of stochastic integro-differential equations are obtained with the aid of successive construction methods. Next, the non-explosiveness is proved by truncation arguments. Then, the Feller continuity is established by means of introducing some auxiliary processes and by making use of the Radon–Nikodym derivatives. Furthermore, the strong Feller continuity is proved by virtue of the relation between the transition probabilities of jump-diffusion processes and the corresponding diffusion processes. Finally, on the basis of the above results, the exponential ergodicity is obtained under the Foster–Lyapunov drift conditions. Some examples are provided for illustration.