Existence Of Exponential Moments Research Articles

The Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal Inequalities

The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.

On a Limit Theorem Related to Probabilistic Representation of Solution to the Cauchy Problem for the Schrödinger Equation

A new method of probabilistic approximation of solution to the Cauchy problem for the unperturbed Schrodinger equation by expectations of functionals of a random walk is suggested. In contrast to earlier papers of the authors, the existence of exponential moment for each step of the random walk is not assumed.

From uniform renewal theorem to uniform large and moderate deviations for renewal-reward processes

A uniform key renewal theorem is deduced from the uniform Blackwell's renewal theorem. A uniform LDP (large deviation principle) for renewal-reward processes is obtained, and MDP (moderate deviation principle) is deduced under conditions much weaker than existence of exponential moments.

Affine Diffusion Processes: Theory and Applications

We revisit affine diffusion processes on general and on the canonical state space in particular. A detailed study of theoretic and applied aspects of this class of Markov processes is given. In particular, we derive admissibility conditions and provide a full proof of existence and uniqueness through stochastic invariance of the canonical state space. Existence of exponential moments and the full range of validity of the affine transform formula are established. This is applied to the pricing of bond and stock options, which is illustrated for the Vasicek Cox-Ingersoll-Ross and Heston Models.

Upper Bounds for Exponential Moments of Hitting Times for Semi-Markov Processes

Necessary and sufficient conditions for the existence of exponential moments for hitting times for semi-Markov processes are found. These conditions and the corresponding upper bounds for exponential moments are given in terms of test-functions. Applications to hitting times for semi-Markov random walks and queuing systems illustrate the results.

Non-Gaussian surface pinned by a weak potential

We consider a model of a two-dimensional interface of the SOS type, with finite-range, even, strictly convex, twice continuously differentiable interactions. We prove that, under an arbitrarily weak potential favouring zero-height, the surface has finite mean square heights. We consider the cases of both square well and $\delta$ potentials. These results extend previous results for the case of nearest-neighbours Gaussian interactions in \cite{DMRR} and \cite{BB}. We also obtain estimates on the tail of the height distribution implying, for example, existence of exponential moments. In the case of the $\delta$ potential, we prove a spectral gap estimate for linear functionals. We finally prove exponential decay of the two-point function (1) for strong $\delta$-pinning and the above interactions, and (2) for arbitrarily weak $\delta$-pinning, but with finite-range Gaussian interactions.