Narrow Results

In this paper, we discuss a new framework for operator-valued Gaussian processes and their covariance kernels. Our emphasis is four-fold: (i) starting with a positive operator-valued measure (POVM) $Q$, we present algorithms for constructing an associated centered, operator-valued, Gaussian process $X$ with $Q$ as its covariance kernel; (ii) we present different classes of POVMs, and we examine the corresponding classes of operator-valued Gaussian processes $X$; (iii) for the operator-valued Gaussian processes $X$ at hand, and the non-commutative framework, we present the corresponding Itô-integrals; and we (iv) outline features of the operator-valued setting which are different from the more familiar case of scalar-valued Gaussian processes.

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏ_t = -∇U(Y_t) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

We consider a finite dimensional deterministic dynamical system with finitely many local attractors K^ι, each of which supports a unique ergodic probability measure P^ι, perturbed by a multiplicative non-Gaussian heavy-tailed Lévy noise of small intensity ε > 0. We show that the random system exhibits a metastable behavior: there exists a unique ε-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures P^ι. In particular our approach covers the case of dynamical systems of Morse–Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative α-stable Lévy noise in the Itô, Stratonovich and Marcus sense. As examples we consider α-stable Lévy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.

In this paper, the numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations have been obtained by second kind Chebyshev wavelets. The second kind Chebyshev wavelets are orthonormal and have compact support on $[0,1]$. The block pulse functions and their relations to second kind Chebyshev wavelets are employed to derive a general procedure for forming stochastic operational matrix of second kind Chebyshev wavelets. The system of integral equations has been reduced to a system of nonlinear algebraic equations and solved for obtaining the numerical solutions. Convergence and error analysis of the proposed method are also discussed. Furthermore, some examples have been discussed to establish the accuracy and efficiency of the proposed scheme.

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs.

The generation and propagation of tsunami caused by a curvilinear stochastic seismic faulting driven by two Gaussian white noise processes in the $x$- and $y$-directions are investigated. This model is used to study the tsunami build up and propagation during and after a realistic curvilinear source model represented by a random spreading slip-fault model. The amplification of tsunami amplitudes builds up progressively as time increases during the generation process due to wave focusing while the maximum wave amplitude decreases with time during the propagation process due to geometric spreading and dispersion. The increase of the normalized noise intensities on the random bottom leads to an increase in oscillations and amplitude of the free surface elevation. Tsunami waveforms using linearized shallow water theory for constant water depth are analyzed analytically by transform methods. The mean and variance of the random tsunami waves are derived and analyzed as a function of the noise intensities, propagated uplift length and the average depth of the ocean along the generation and propagation path.

In this paper we construct and study an integral of operator-valued functions with respect to Hilbert space-valued measures generated by a resolution of identity. Our integral generalizes the Itô stochastic integral with respect to normal martingales and the Itô integral on a Fock space.

In Black-Scholes type security markets, the completeness, arbitrage-free condition, pricing (European) contingent claims, etc. are thought to be well-understood by now. However, some careful studies show that the issue seems not to be as simple as expected. Problems related to the above-mentioned notions lead to several interesting questions for Itô's integrals and (backward) stochastic differential equations. In this paper, some relevant partial results will be presented and several open questions will be posed.

The isometry formula for the Itô integral is generalized to a new stochastic integral involving both adapted and instantly independent stochastic processes. The proof is rather elementary. An example is given to ilustrate the isometry formula.