The Impact of Pinning Points on Memorylessness in Lévy Random Bridges
Mohammed Louriki
TL;DR
This work analyzes Lévy bridges with random length $\tau$ and random pinning point $Z$ to understand how memorylessness depends on the pinning-point distribution. By conditioning on $(\tau,Z)$ and employing Lebesgue's decomposition of $\mathbb{P}_Z$, it proves a sharp criterion: the bridge is Markov if and only if the absolutely continuous part $a_{ac}$ of $Z$'s law is zero; when $a_{ac}>0$, the memoryless property can fail for some times. The paper also shows that the two-dimensional process $Y_t=(Z,\zeta_t)$ is Markov, offering a tractable joint dynamics perspective. The results extend Brownian and Levy-bridge analyses to include singular-continuous pinning points and provide practical insights for information-based asset pricing models where pinning-point distributions govern memory effects and filtration structure.
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 Lévy bridges with random length and random pinning points and analyze their Markov property. Our study demonstrates that the Markov property of Lévy 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 Lévy bridge fails to exhibit Markovian behavior when the pinning point has an absolutely continuous part.