Optimal Stopping and Maximal Inequalities for Linear Diffusions

Given a linear diffusion the solution is found to the optimal stopping problem where the gain is given by the maximum of the process and the cost is proportional to the duration of time. The optimal stopping boundary is shown to be the maximal solution of a nonlinear differential equation expressed in terms of the scale function and the speed measure. Applications to maximal inequalities are indicated.     

Optimal Stopping in the L log L-Inequality of Hardy and Littlewood

Let B = (B_t)_t0 be standard Brownian motion started at zero.We prove for all c > 1and all stopping times for B satisfying E(^r) < for somer > 1/2. This inequality is sharp, and equality is attainedat the stopping time whereu_* = 1 + 1/e^c(c – 1) and = (c – 1)/c for c >1, with X_t = |B_t| and S_t = max_0rt|B_r|. Likewise, we prove for all c > 1 and all stopping times for B satisfying E(^r < for some r > 1/2. This inequalityis sharp, and equality is attained at the stopping time where v_* = c/e(c – 1) and =(c – 1)/c for c > 1. These results contain and refinethe results on the L log L-inequality of Gilat [6] which areobtained by analytic methods. The method of proof used hereis probabilistic and is based upon solving the optimal stoppingproblem with the payoff whereF(x) equals either xlog⁺ x or x log x. This optimal stoppingproblem has some new interesting features, but in essence issolved by applying the principle of smooth fit and the maximalityprinciple. The results extend to the case when B starts at anygiven point (as well as to all non-negative submartingales).1991 Mathematics Subject Classification 60G40, 60J65, 60E15.     

Accurate switching intensities and length scales in quasi-phase-matched materials

We consider unseeded type I second-harmonic generation in quasi-phase-matched quadratic nonlinear materials and derive an accurate analytical expression for the evolution of the average intensity. The intensity-dependent nonlinear phase mismatch that is due to the cubic nonlinearity induced by quasi phase matching is found. The equivalent formula for the intensity of maximum conversion, the crossing of which changes the one-period nonlinear phase shift of the fundamental abruptly by pi , corrects earlier estimates [Opt. Lett. 23, 506 (1998)] by a factor of 5.3. We find the crystal lengths that are necessary to obtain an optimal flat phase versus intensity response on either side of this separatrix intensity.     

Maximal inequalities for the Ornstein-Uhlenbeck process

Let be the Ornstein-Uhlenbeck velocity process solving

with , where 0$"> and is a standard Brownian motion. Then there exist universal constants 0$">and 0$"> such that

for all stopping times of . In particular, this yields the existence of universal constants 0$"> and 0$"> such that

for all stopping times of . This inequality may be viewed as a stopped law of iterated logarithm. The method of proof relies upon a variant of Lenglart's domination principle and makes use of Itô calculus.

     

Path and semimartingale properties of chaos processes

The present paper characterizes various properties of chaos processes which in particular include processes where all time variables admit a Wiener chaos expansion of a fixed finite order. The main focus is on the semimartingale property, p$p$-variation and continuity. The general results obtained are finally used to characterize when a moving average is a semimartingale.     

α-self-similar Markov processes

Summary Let ((X(t)), P ^x) be an -self-similar isotropic Markov process on R ^d {0}. A representation of (X(t)), in terms of the radial and angular process which generalizes the skew product representation for Brownian motion is given.