The First Exit Time Of The Brownian Motion And Mills' Ratio

In probability theory,the large deviation theory(or the tail probabilities) and the small deviation theory(or the small ball probabilities) are in a sense two complementary directions.The large deviation theory,which is a more classical direction,seeks to control the probability of deviation of a random variable X from its mean M,i.e.one looks for upper bounds on Prob(|X - M|＞t).The small deviation theory seeks to control the probability of X being very small,i.e.it looks for upper bounds on Prob(|X|＜t).In recent years,there has been renewed interest in the topic of large deviations,namely, the asymptotic computation of small probability on an exponential scale.The large deviations estimates have proved to be the crucial tool required to handle many questions in statistics,engineering,statistical mechanics,and applied probability.It has also been found that the small deviation estimates has close connections with the rate of convergence in Strassen's law of the iterated logarithm,and empirical processes.There are a number of excellent texts on large deviation,see e.g.recent books.A recent exposition of the state of the art in small deviation theory can be found in Li.The problem we consider is that replacing random variable X on the left side of the above probability by the Brownian motion,and replacing simple domain on the right side of the above probability by random domains.No matter how difficult,the research on the exit probability of a Brownian motion from various domains is based on the large deviation theory and the small deviation theory.Another part in this paper is the study on the Mills' ratio.Mills' ratio is the standard normal probability beyond a certain point divided by the standard normal density at that point.Bounds on Mills' ratio arise naturally in many areas of probability and statistics. There is also a long history of studying them in various settings.Li have studied the first exit time of a Brownian motion from a single random domain. We develop this problem.In Chapter 2,we study the first exit time of a Brownian motion from the minimum and maximum random domains and obtain the upper and lower estimates of this two probabilities.Lifshits and Shi have studied the first exit time of a Brownian motion from a single parabolic domain.We develop this problem.In Chapter 3,we study the first exit time of a Brownian motion from the minimum and maximum parabolic random domains.Finally, we obtain the upper and lower estimates and also prove that they are still asymptotically equivalent.Lifshits and Shi have studied the first exit time of a Brownian motion from a parabolic domain.We develop this problem.In Chapter 4,we study the first exit time of a Brownian motion with a drift from a random parabolic domain.Finally,the upper and lower estimates of the exit probability are obtained.We also prove that they are still asymptotically.In Chapter 5,we study the first exit time of a Brownian motion from an unbounded convex domain.Li have studied this problem and obtained the upper and lower estimates. However,the upper estimates is not very sharp for some parameter.Based on his research,we construct a Gaussian process.Applying this process to Slepian's inequality and using Gaussian technique,a new upper estimates is obtained.For some parameter, the new upper estimates is more sharper than the old one.In Chapter 6,we study the univariate Mills' ratio and the multivariate Mills' ratio.It is well-known that the univariate Mills' ratio can be estimated by the continued fraction, but the estimate is not simple.We give a formula by polynomial,and improving previous estimates.In addition,an upper bound of multivariate Gaussian probability for a general convex domain is given based on a geometric observation.By using the estimates of the univariate Mills' ratio,we checked that the bound is sharper than known ones on multivariate Mills' ratio in many case.