Some Theorems concerning Extrema of Brownian Motion With

We often use the notation ( ) to denote either ( ) or ( ). For example, ( )− ( ) denotes any one of ( )− ( ), ( )− ( ), ( )− ( ) and ( )− ( ). A point in R is called a point of local minimum (resp. local maximum) of a sample function if there exists a neighborhood of such that ( ) = ( ) (resp. ( ) = ( )). A point of either local minimum or local maximum is called an extreme-point. The following are typical of those problems and theorems we discuss in this paper. (I) Under what condition on does the probability distribution of ( ) admit a strictly positive -density? (II) Under what condition on and does the joint probability distribution of ( ) and ( ) admit a strictly positive -density? (III) Almost all sample functions have the following property: There are no distinct extreme-points and with ( ) = ( ). We give some sufficient conditions that will give positive answers to the problems (I) and (II) and then give a proof of (III). Formulating the problems somewhat generally we state our main results in the following theorems.