Regular Stochastic Process Research Articles

On the random wave equation within the mean square context

<p style='text-indent:20px;'>This paper deals with the random wave equation on a bounded domain with Dirichlet boundary conditions. Randomness arises from the velocity wave, which is a positive random variable, and the two initial conditions, which are regular stochastic processes. The aleatory nature of the inputs is mainly justified from data errors when modeling the motion of a vibrating string. Uncertainty is propagated from these inputs to the output, so that the solution becomes a smooth random field. We focus on the mean square contextualization of the problem. Existence and uniqueness of the exact series solution, based upon the classical method of separation of variables, are rigorously established. Exact series for the mean and the variance of the solution process are obtained, which converge at polynomial rate. Some numerical examples illustrate these facts.</p>

Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

Pricing American Options in a Mild Stochastic Environment

The problem of pricing derivative financial products is central to the theory of capital markets. An option is a financial contract conveying its owner the right of buying or selling a financial asset (underlying asset) at a preset strike price K, at a fixed expiration date T (maturity). Unlike European options, which can be exercised only at maturity date, an American option can be exercised at any time t prior to the maturity date. Most of the option pricing methods, starting with the well-known Black-Scholes model (1973), are based on the assumption that the market uncertainty can be modeled by a Wiener process. In this context, while it is possible to obtain convenient analytical option pricing formulae for European options, it is very difficult to obtain exact results for American options. In the present paper, we assume that the market uncertainty is modeled by a more regular stochastic process, which was called, by A. Halanay, a mild stochastic environment. In this context, we are able to obtain precise stopping rules, determining the exact exercise time and the exact price of an American option.

Central Limit Theorem of the Smoothed Empirical Distribution Functions for Asymptotically Stationary Absolutely Regular Stochastic Processes

Let be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n. A central limit theorem is established for the target statistic , where the underlying random vector forms an asymptotically stationary absolutely regular stochastic process, and is an estimator of a multivariate parameter ξ by using a vector of U‐statistics. The results obtained extend or generalize previous results from the stationary univariate case to the asymptotically stationary multivariate case. An example of asymptotically stationary absolutely regular multivariate ARMA process and an example of a useful estimation of F(ξ) are given in the applications.

Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation

In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

Integration par parties dans l'espace de Wiener et approximation du temps local

Let be a real-valued stochastic process having a continuous local timeL(u,t),u∈ —, 0≦t≦T andXe(t) = (Ψe *X)(t),t ⪴ 0, the regularization ofX by means of the convolution with the approximation of unityΨe. The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, •) by means of a convenient normalization of the numberNXe(u;•) of crossings of the processXe with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.

Canonical commutation relations of quantum mechanics and stochastic regularity

In a previous publication (Boletin de la Sociedad Matematica Mexicana 1975) it was established that any weakly stationary linearly regular stochastic process is unitarily equivalent to a quantum mechanical momentum evolution. The object of the present note is, as promised in the previous publication, to amplify some of the details concerning the just mentioned interesting connection, giving in particular a direct proof of the Szego-Kolmolgorov-Krein characterization of regular stationary processes. We also show that although the so-called decaying states without regeneration do not exist for unstable quantum systems, they are natural for regular stationary processes.