A trivariate version of ''Lévy's equivalence'' (Q1067685)

It is shown that the trivariate stochastic processes \(\{(M_ t-W_ t,M_ t,\Theta_ t),t\geq 0\}\) and \(\{(| W_ t|,L_ t,T_ t),t\geq 0\}\) have the same distributions when: \(W=\{W_ t,t\geq 0\}\) is a Wiener process, \(M_ t\) is the maximum value attained by W over the time interval [0,t], \(\Theta_ t\) is the time the maximum value is attained, \(L_ t\) is the local time of W at level zero and time t, and \(T_ t\) is the last time W is zero in the time interval [0,t]. A straightforward proof, based on Tanaka's formula, establishes this result by deriving an almost sure version of the equivalence.