The Wiener disorder problem with finite horizon

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ‘disorder’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation.     

An application of prophet regions to optimal stopping with a random number of observations

Let X ₁,X ₂ ,?…?be any sequence of nonnegative integrable random variables, and let N∈{1,2 , …} be a random variable with known distribution, independent of X ₁,X ₂ , …. The optimal stopping value sup _t E(X_t I(N≥ t)) is considered for two players: one who has advance knowledge of the value of N, and another who does not. Sharp ratio and difference inequalities relating the two players' optimal values are given in a number of settings. The key to the proofs is an application of a prophet region for arbitrarily dependent random variables by Hill and Kertz [T.P. Hill and R.P. Kertz (1983). Stop rule inequalities for uniformly bounded sequences of random variables. Trans. Amer. Math. Soc., 278, 197–207].     

Integral equations for Rost’s reversed barriers: Existence and uniqueness results

We establish that the boundaries of the so-called Rost’s reversed barrier are the unique couple of left-continuous monotonic functions solving a suitable system of nonlinear integral equations of Volterra type. Our result holds for atom-less target distributions $\mu$ of the related Skorokhod embedding problem.The integral equations we obtain here generalise the ones often arising in optimal stopping literature and our proof of the uniqueness of the solution goes beyond the existing results in the field.     

The inverse balayage problem for Markov chains,part II

This paper continues the study of the inverse balayage problem for Markov chains. Let X be a Markov chain with state space $\text{A ? B}{}_2$, let v be a probability measure on B₂ and let M(v) consist of probability measures μ on A whose X-balayage onto B₂ is v. The faces of the compact, convex set M(v) are characterized. For fixed μ?M(v) the set M(μ,v) of the measures ? of the form $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S) ? ·}}$, where S is a randomized stopping time, is analyzed in detail. In particular, its extreme points and edge are explicitly identified. A naturally defined reversed chain X, for which v is an inverse balayage of μ, is introduced and the relation between X and X^ is studied. The question of which $\text{? ? M(μ, v)}$ admit a natural stopping time $\text{S}{}_{\mathrm{?}}$ of X (not involving an independent randomization) such that $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S}{}_{\mathrm{?}}\text{) ? ·}}$, is shown to have rather different answers in discrete and continuous time. Illustrative examples are presented.     

Commutator bounds for eigenvalues,with applications to spectralgeometry

We prove a purely algebraic version of an eigenvalue inequality of Hile and Protter, and derive corollaries bounding differences of eigenvalues of Laplace– Beltrami operators on manifolds. We significantly improve earlier bounds of Yang and Yau, Li, and Harrell.     

Heat Equations with Fractional White Noise Potentials

This paper is concerned with the following stochastic heat equations: where w ^H is a time independent fractional white noise with Hurst parameter H=(h ₁ , h ₂ ,..., h _d ) , or a time dependent fractional white noise with Hurst parameter H=(h ₀ , h ₁ ,..., h _d ) . Denote |H|=h ₁ +h ₂ +...+h _d . When the noise is time independent, it is shown that if ? <h _i <1 for i=1, 2,..., d and if |H|>d-1 , then the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ? <h _i <1 for i=0, 1,..., d and if |H|>d- 2 /( 2h ₀ -1 ) , the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S _ρ , ρ∈ RR , is introduced so that every chaos of an element in S _ρ is in L ₂ . The Lyapunov exponents in S _ρ of the solution are also estimated. Accepted 10 October 2000. Online publication 19 February 2001.