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The joint distribution of the running maximum and its location of D-valued Markov processes

Tt=min{O~u~t: Y(u)=Mt or Y(u)=Mt}. Since the set {O~u~t: Y(u)=Mt or Y(u)=Mt} is non-empty and closed, the random variable T, is well defined. We give an explicit formula for the joint distribution of (Tt , M t ) . Although the derivation of our result requires only a short argument, the obtained formula is very useful in connection with recent results of Durbin (1985), (1992). To be precise, let