An occupation time related potential measure for diffusion processes

In this paper, for homogeneous diffusion processes, the approach of Y. Li and X. Zhou [Statist. Probab. Lett., 2014, 94: 48–55] is adopted to find expressions of potential measures that are discounted by their joint occupation times over semi-infinite intervals (-∞, α) and (α, ∞): The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions for Brownian motion with drift, skew Brownian motion, and Brownian motion with two-valued drift, respectively.     

Approximations for functionals and optimal control problems on jump diffusion processes

The paper treats approximations to stochastic differential equations with both a diffusion and a jump component, and to associated functionals and partial-differential-integral equations of the (degenerate or not) elliptic or parabolic type. Approximations for the optimal control problem on such a model, or for the associated nonlinear partial-differential-integral equation are discussed. The techniques are purely probabilistic and are extensions of those in [3], which dealt with the diffusion case.     

A weighted occupation time for a class of measured-valued branching processes

Summary A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on R ^d. Considered as a process in its own right, the first and second order asymptotics are found as time t. Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.Research supported in part by Natural Sciences and Engineering Research Council of Canada     

Scaling transformation and probability distributions for financial time series

The price of financial assets are, since [Bachelier L. Annales de l'Ecole Normale Supérieure 1900;3:XVII:21–86], considered to be described by a (discrete or continuous) time sequence of random variables, i.e., a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [Mandelbrot BB. J Business 1963;36:394–419; Peters EE. Chaos and order in the capital markets. New York: Wiley, 1991; Mantegna RN, Stanley HE. Nature 1995;376:46–49; Evertsz CJG. Fractals. 1995;3:609–616; Bouchaud JP, Potters M. Théorie des risques financiers. Aléa Saclay, 1997]. In this paper we investigate the question of scaling transformation of price processes by establishing a new connection between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.     

Optimal control problems in diffusion processes with a nonsmooth nonlinear boundary condition

We investigate a one-dimensional diffusion process controlled by a nonsmooth nonlinear boundary condition. Existence and estimates of positive solutions of the differential equation are derived. We prove necessary conditions for optimal controls and apply them to two examples.     

Estimation error for occupation time functionals of stationary Markov processes

The approximation of integral functionals with respect to a stationary Markov process by a Riemann sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to explicitly calculate the estimation error and to prove a general finite sample error bound. The presented approach admits general integrands and gives a unifying explanation for different rates obtained in the literature. Several examples demonstrate how the general bound can be related to well-known function spaces.     

A property of distributions of diffusion processes

Translated from Matematicheskie Zametki, Vol. 54, No. 3, pp. 106–113, September, 1993.     

Occupation time theorems for a class of one-dimensional diffusion processes

Summary The long time asymptotic behavior of the occupation times on a half line is studied for a class of one-dimensional diffusion processes whose excursion intervals have very heavy tail probability     

Comparison theorem and estimates for transition probability densities of diffusion processes

We establish several comparison theorems for the transition probability density p _b (x,t,y) of Brownian motion with drift b, and deduce explicit, sharp lower and upper bounds for p _b (x,t,y) in terms of the norms of the vector field b. The main results are obtained through carefully estimating the mixed moments of Bessel processes. All constants are explicit in our lower and upper bounds, which is different from most of the previous estimates, and is important in many applications for example in statistical inferences for diffusion processes.Research partially supported by N.S.F. Grants DMS-0203823, and by Doctoral Program Fundation of the Ministry of Education of China, Grant No. 20020269015. Mathematics Subject Classification (2000):Primary: 60H10, 60H30; Secondary: 35K05     

Limiting distributions of time averages for processes with a semi-Markov interference

We obtain the limiting distribution of the time averages of processes with a semi-Markov interference without assuming the finiteness of the moments of the interference time.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 281–284, February, 1990.     

Nonfickian effect in time and space for diffusion processes

Diffusion processes are usually simulated using the classical diffusion equation. In certain scenarios, such equation induces anomalous behavior and consequently several improvements were introduced in the literature to overcome them. One of the most popular was the replacement of the diffusion equation by an integro‐differential equation. Such equation can be established considering a modification of Fick's mass flux where a delay in time is introduced. In this article, we consider mathematical models for diffusion processes that take into account a memory effect in time and space. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1589–1602, 2015     

Small time asymptotics of diffusion processes

We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity.     

On the occupation time of an iterated process having no local time

We study the asymptotic behaviour of the occupation time process ∫^t₀ I_A(W₁(L₂(s)))ds, t 0, where W₁ is a standard Wiener process and L₂ is a Wiener local time process at zero that is independent from W₁. We prove limit laws, as well as almost sure upper and lower class theorems. Possible extensions of the obtained results are also discussed.     

Long-term average cost control problems for continuous time Markov processes: A survey

This paper addresses the long-term average cost control of continuous time Markov processes. A survey of problems and methods contained in various works is given for continuous control, optimal stopping, and impulse control.     

Density of the transition probability for one class of generalized diffusion processes

The existence of the density of the transition probability is investigated for a generalized diffusion process with transport that satisfies a certain condition of integrability with respect to the Gaussian measure. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 10, pp. 1433–1437, October, 1998.