On the joint distribution of the supremum functional and its last occurrence for subordinated linear Brownian motion

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In this communication, a convenient Laplace transform of the bivariate supremum and the last time the supremum is attained, is established when the underlying Lévy process is subordinate Brownian motion with drift. Explicit integral representations of the Laplace transform of the joint supremum and the last time it occurred are derived in terms of the Lévy–Khintchine exponent of the subordinator Laplace exponent. As an example, a subordinator with exponential Lévy measure is exploited.