Set-parametric Brownian motion b in a star-shaped set G is considered when the values of b on the boundary of G are given. Under the conditional distribution given these boundary values the process b becomes some set-parametrics Gaussian process and not Brownian motion. We define the transformation of this Gaussian process into another Brownian motion which can be considered as “martingale part...

Consider an N -dimensional Markov chain obtained from N onedimensional random walks by Doob h-transform with the q-Vandermonde determinant. We prove that as N becomes large, these Markov chains converge to an infinite-dimensional Feller Markov process. The dynamical correlation functions of the limit process are determinantal with an explicit correlation kernel. The key idea is to identify rand...

In this paper we study the pricing problem of multiple exercise options in continuous time on a finite time horizon. For the corresponding multiple stopping problem, we prove, under quite general assumptions, the existence of the Snell envelope, a reduction principle as nested single stopping problems, and a Doob-Meyer type decomposition for the Snell envelope. The main technical difficulty ari...

It has been almost sixty years since Kolmogorov introduced a distribution-free omnibus test for the simple null hypothesis that a distribution function coincides with a given distribution function. Doob subsequently observed that Kolmogorov's approach could be simplified by transforming the empirical process to an empirical process based on uniform random variables. Recent use of more sophistic...

For massless vertex-transitive transient graphs, the percolation phase transition for level sets of Gaussian free field on associated continuous cable system is particularly well understood, and in particular critical parameter h˜∗ always equal to zero. On general two weak conditions graph G are given [12], each which implies one inequalities h˜∗≤0 h˜∗≥0. In this article, we give counterexample...

For a distance-regular graph \(\Gamma\) of diameter 3, the \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found parameters \(\Gamma_2\) given intersection array (independently, were by A.A. Makhnev D.V.Paduchikh). In this case, has an eigenvalue \(a_2-c_3\). paper, we study graphs with \(\theta=1\). particular, prove that, \(Q\)-polynomial from series arrays \(\{2...

The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D(m,n) and prove some facts on their structure. We show that the considered tw...

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic “true” stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale pr...

Boston, 1974. 7. J. L. Doob, Stochastic Processes, John Wiley, New York 1953. 8. A. Dvoretzky and Th. Motzkin, A problem of arrangements, Duke Math. J. 14 (1947) 305–313. 9. W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed., John Wiley, New York 1957. 10. I. P. Goulden and L. G. Serrano, Maintaining the spirit of the reflection principle when the boundary line has a...

We construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob h-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and König who studied the analogous conditioning for the Weyl chamber of ty...