Goodness-of-fit test for homogeneous Markov processes

We give chi-squared goodness-of-fit tests for homogeneous Markov processes with unknown transition intensities or with transition intensities of known form depending on a finite-dimensional parameter.     

Factorial moments of point processes

We derive joint factorial moment identities for point processes with Papangelou intensities. Our proof simplifies previous combinatorial approaches to the computation of moments for point processes. We also obtain new explicit sufficient conditions for the distributional invariance of point processes with Papangelou intensities under random transformations.     

Renewal equations and moments of branching processes

Some theorems, which strengthen the asymptotic properties of the solutions of renewal equations, are proved. These results are applied in a study of the first two moments of age-dependent branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 3–14, January, 1968.     

Microfoundations for diffusion price processes

We study microeconomic foundations of diffusion processes as models of stock price dynamics. To this end, we develop a microscopic model of a stock market with finitely many heterogeneous economic agents, who trade in continuous time, giving rise to an endogeneous pure-jump process describing the evolution of stock prices over time. When the number of agents in the market is large, we show that the price process can be approximated by a diffusion, with price-dependent drift and volatility coefficients that are determined by small excess demands and trading volume in the microscopic model. We extend the microscopic model further by allowing for non-market interactions between agents, to model herd behavior in the market. In this case, price dynamics can be approximated by a process with stochastic volatility. Finally, we demonstrate how heavy-tailed stock returns emerge when agents have a strong tendency towards herd behavior.     

Aging for interacting diffusion processes

We study the aging phenomenon for a class of interacting diffusion processes {Xt(i),i∈Zd}. In this framework we see the effect of the lattice dimension d on aging, as well as that of the class of test functions f(Xt) considered. We further note the sensitivity of aging to specific details, when degenerate diffusions (such as super random walk, or parabolic Anderson model), are considered. We complement our study of systems on the infinite lattice, with that of their restriction to finite boxes. In the latter setting we consider different regimes in terms of box size scaling with time, as well as the effect that the choice of boundary conditions has on aging. The key tool for our analysis is the random walk representation for such diffusions.     

Comparison theorems for diffusion processes

We prove comparison theorems for diffusion processes onR ^d. From these theorems we derive lower and upper bounds for the transition probabilities of a diffusion process. In contrast to the known estimates for fundamental solutions of parabolic equations our bounds do not depend on the moduli of continuity of the coefficients of the differential operator.     

Testing diffusion processes for non-stationarity

Financial data are often assumed to be generated by diffusions. Using recent results of Fan et al. (J Am Stat Assoc, 102:618–631, 2007; J Financ Econometer, 5:321–357, 2007) and a multiple comparisons procedure created by Benjamini and Hochberg (J R Stat Soc Ser B, 59:289–300, 1995), we develop a test for non-stationarity of a one-dimensional diffusion based on the time inhomogeneity of the diffusion function. The procedure uses a single sample path of the diffusion and involves two estimators, one temporal and one spatial. We first apply the test to simulated data generated from a variety of one-dimensional diffusions. We then apply our test to interest rate data and real exchange rate data. The application to real exchange rate data is of particular interest, since a consequence of the law of one price (or the theory of purchasing power parity) is that real exchange rates should be stationary. With the exception of the GBP/USD real exchange rate, we find evidence that interest rates and real exchange rates are generally non-stationary. The software used to implement the estimation and testing procedure is available on demand and we describe its use in the paper.     

Combinatorial decompositions and homogeneous geometrical processes

This paper considers line processes and random mosaics. The processes are assumed invariant with respect to the group of translations ofR ². An expression for the probabilities ,k=0, 1, 2,... to havek hits on an interval of lengtht taken on a typical line of direction (the hits are produced by other lines of the process) is obtained. Also, the distribution of a length of a typical edge having direction in terms of the process {P _i , _i } is found, hereP _i is the point process of intersections of edges of the mosaic with a fixed line of direction and the mark _i is the intersection angle atP _i . The method is based on the results of combinatorial integral geometry.     

Davis-type inequalities for some diffusion processes

In this paper, we obtain Davis-type maximal inequalities for the Bessel process of dimension δ > 1 and for the radial Ornstein—Uhlenbeck process. The proofs are based on a generalization of Lenglart’s domination principle. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 30–43.     

Interpolation for partly hidden diffusion processes

Let X_t be n-dimensional diffusion process and S_t be a smooth set-valued function. Suppose X_t is invisible when X_t∈S_t, but we can see the process exactly otherwise. Let X_t0∈S_t0 and we observe the process from the beginning till the signal reappears out of the obstacle after t₀. With this information, we evaluate the estimators for the functionals of X_t on a time interval containing t₀ where the signal is hidden. We solve related 3 PDEs in general cases. We give a generalized last exit decomposition for n-dimensional Brownian motion to evaluate its estimators. An alternative Monte Carlo method is also proposed for Brownian motion. We illustrate several examples and compare the solutions between those by the closed form result, finite difference method, and Monte Carlo simulations.     

Another characterization of homogeneous Poisson processes

For a general renewal process N (allowing delay, defect and multiple simultaneous arrivals) the independence of the first renewal epochs of the marked processes got from N by Bernoulli 0/1 thinning is characterized. This independence is well-known to hold true in the case of homogeneous Poisson processes; by way of corollary one obtains the interesting observation that, when coupled with some minimal extra conditions, it in fact already identifies them. The proof is analytic in character.     

Convergence of diffusion processes

For stochastic diffusion equations with coefficients depending on a parameter, necessary and sufficient conditions of the weak convergence of solutions to the solution of a stochastic diffusion equation are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 284–289, February, 1992.     

Two-parameter diffusion processes and martingales

We introduce a class of two-parameter processes which are diffusions on each coordinate and satisfy a particular Markov property related to the partial ordering in R²₊. These processes can be expressed as solutions of some stochastic integral equations driven by a two-parameter Wiener process and two families of ordinary Brownian motions. This result is based on a characterization of two-parameter martingales with orthogonal increments.     

Large deviations for perturbed reflected diffusion processes

In this article, we establish a large deviation principle for the solutions of perturbed reflected diffusion processes. The key is to prove a uniform Freidlin–Ventzell estimate of perturbed diffusion processes.