Revuz measures under time change

In this paper, we shall study how energy functionals and Revuz measures change under time change of Markov processes and provide an intuitive and direct approach to the computation of the Levy system and jumping measure of time changed process.     

Optimal models for the first arrival time distribution function in continuous time — With a special case

Optimal Models for first arrival time ( _H ) and first arrival target total return (W _H ) distribution functions on MDP in continuous time are presented. Asymptotic expansions of _H andW _H are derived and expressed in simple, explicit forms, and some of their properties are discussed. Two methods to find an optimal policy for distribution function of _H are given. Several necessary and sufficient conditions for the existence of the optimal policy are obtained. This result leads to that the scope of finding the optimal policy is greatly reduced. A special case is also discussed and some deep results are given.     

Quantized space-time and time’s arrow

Motivated by the latest direct detection of time asymmetry in Kaon decay at CERN and Fermilab, we suggest a theoretical rationale for this puzzle, in terms of quantized time.     

Large time behavior of Euler-Poisson system for semiconductor

In this note,we present a framework for the large time behavior of general uniformly bounded weak entropy solutions to the Cauchy problem of Euler-Poisson system of semiconductor devices.It is shown that the solutions converges to the stationary solutions exponentially in time.No smallness and regularity conditions are assumed.     

Ornstein–Uhlenbeck operators with time periodic coefficients

We study the realization of the differential operator in the space of continuous time periodic functions, and in L ² with respect to its (unique) invariant measure. Here L(t) is an Ornstein-Uhlenbeck operator in , such that L(t + T) = L(t) for each .      

Space–time fractional diffusion on bounded domains

Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space–time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.     

Comparison of the performance of a time‐dependent short‐interest rate model with time‐independent models

The coefficients in the stochastic differential equation that the short interest rate follows are of vital importance in the subsequent modelling of bond prices and other interest rate products. Empirical tests have previously been performed by various authors who compare a variety of popular short‐rate models. Most recently, Ahn and Gao compared their model with affine‐drift models and showed that their model with a non‐linear drift function outperforms the others. This paper compares the model developed by Goard, which is a time‐dependent generalization of the Ahn–Gao model, with the Ahn–Gao model itself. It is found that the time‐dependent model using a second‐order Fourier series in time, outperforms the Ahn–Gao model for all data sets considered.     

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

We investigate the long time behavior of the following efficient second-order in time scheme for the 2D Navier–Stokes equations in a periodic box: $$\begin{array}{ll}{\frac{3\omega^{n+1} - 4\omega^n + \omega^{n-1}}{2k} + \nabla^\perp(2\psi^n - \psi^{n-1}) \cdot \nabla(2\omega^n - \omega^{n-1})- \nu\Delta\omega^{n+1} = f^{n+1},} \\ {\quad -{\Delta} {\psi}^{n} = {\omega}^{n}.}\end{array}$$ The scheme is a combination of a 2nd-order in time backward-differentiation and a particular explicit Adams–Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson solver is needed at each time step. We prove uniform in time bounds on this scheme in ${{\dot{L}^2,\, \dot{H}^1_{per}}}$ and ${{\dot{H}^2_{per}}}$ provided that the time-step is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing time-step.     

Hamilton–Jacobi–Bellman equations on time scales

In this paper, we consider a class of optimal control problems on time scales without state constraints, target conditions or the fixed terminal time. We first present and show a time scale version of the Bellman optimality principle. On this basis, using a chain rule of multivariables on time scales, we will derive Hamilton–Jacobi–Bellman equations on a time scale for these kind of optimal control problems. Finally, the quantum time scale is considered as an example to illustrate our results.     

Existence time for the semilinear Schrödinger equation

Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L ^p ? L ^q estimate and the energy estimate.     

On waiting time in the Markov-Pólya scheme

We consider the Markov-Pólya urn scheme. The urn contains a given number of balls of each of N different colors. The balls are sequentially drawn from the urn one at a time, independently of each other. and with equal probabilities to be drawn. After each draw, the ball drawn is returned into the urn together with c balls of the same color, c∈{−1, 0, 1, 2, ...}. The drawing process halts when, for the first time, the frequencies of k unspecified colors attain or exceed the corresponding (random) levels settled before the beginning of the trials. Limit distributions of the stopping time υ_c(N, K) are considered as N→∞, k=k(N); in particular, the dependence of the asymptotic properties of the waiting time on the parameter c is studied. Results concerning υ_c(N, K) are derived as consequences of general limit theorems for decomposable statistics

Bahadur–Kiefer representations for time dependent quantile processes

We define a time dependent empirical process based on n independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur–Kiefer representations.     

Multi‐asset barrier options and occupation time derivatives

A general framework is formulated to price various forms of European style multi‐asset barrier options and occupation time derivatives with one state variable having the barrier feature. Based on the lognormal assumption of asset price processes, the splitting direction technique is developed for deriving the joint density functions of multi‐variate terminal asset prices with provision for single or double barriers on one of the state variables. A systematic procedure is illustrated whereby multi‐asset option price formulas can be deduced in a systematic manner as extensions from those of their one‐asset counterparts. The formulation has been applied successfully to derive the analytic price formulas of multi‐asset options with external two‐sided barriers and sequential barriers, multi‐asset step options and delayed barrier options. The successful numerical implementation of these price formulas is demonstrated.