The Impact of Pinning Points on Memorylessness in Lévy Random Bridges

Abstract: Random Bridges have gained significant attention in recent years due to their potential applications in various areas, particularly in information-based asset pricing models. This paper aims to explore the potential influence of the pinning point's distribution on the memorylessness and stochastic dynamics of the bridge process. We introduce L\'evy bridges with random length and random pinning points and analyze their Markov property. Our study demonstrates that the Markov property of L\'evy bridges depends on the nature of the distribution of their pinning points. The law of any random variables can be decomposed into singular continuous, discrete, and absolutely continuous parts with respect to the Lebesgue measure (Lebesgue's decomposition theorem). We show that the Markov property holds when the pinning points' law does not have an absolutely continuous part. Conversely, the L\'evy bridge fails to exhibit Markovian behavior when the pinning point has an absolutely continuous part.