Optimally stopping a Brownian bridge with an unknown pinning time: A Bayesian approach

Abstract

We consider the problem of optimally stopping a Brownian bridge with an unknown pinning time so as to maximise the value of the process upon stopping. Adopting a Bayesian approach, we allow the stopper to update their belief about the value of the pinning time through sequential observations of the process. Uncertainty in the pinning time influences both the conditional dynamics of the process and the expected (random) horizon of the optimal stopping problem. Structural properties of the optimal stopping region are shown to be qualitatively different under different prior distributions, however we provide a sufficient condition for the existence of a one-sided stopping region. Certain gamma and beta distributed priors are shown to satisfy this condition and these cases are subsequently considered in detail. Remarkably, in the gamma case the optimal stopping problem becomes time homogeneous and is completely solvable in closed form. In the beta case we find that the optimal stopping boundary takes on a square-root form, similar to the classical solution with a known pinning time. A two-point prior distribution is also considered in which a richer structure emerges (with multiple optimal stopping boundaries). Finally, when one of the values of the two-point prior is set to infinity (such that the process may never pin) we observe that the optimal stopping problem is also solvable in closed form.

Optimal stopping of an Ornstein-Uhlenbeck bridge

“…The first result in OSPs with Markov bridges was given by Shepp (1969), who circumvented the complexity of dealing with a Brownian Bridge (BB) by using a time-space transformation that allowed to reformulate the problem into a more tractable one with a Brownian motion underneath. Since then, more than fifty years ago, the use of Markov bridges in the context of OSPs has been narrowed to extending the result of Shepp (1969): Ekström and Wanntorp (2009) and Ernst and Shepp (2015) studied alternative methods of solutions; Ekström and Wanntorp (2009) and de Angelis and Milazzo (2020) looked at a broader class of gain functions, Glover (2020) randomized the horizon while Föllmer (1972), Leung et al (2018), and Ekström and Vaicenavicius (2020) analyzed the randomization of the bridge's terminal point.…”

Optimal stopping for the exponential of a Brownian bridge

“…Optimal stopping of a Brownian bridge with random pinning point or random pinning time were also studied in [10] and [15], respectively. In [10], the authors consider more general versions of the problem addressed in [13] and, among other things, they give general sufficient conditions for optimal stopping rules in the form of a hitting time to a one-sided stopping region.…”

“…In [10], the authors consider more general versions of the problem addressed in [13] and, among other things, they give general sufficient conditions for optimal stopping rules in the form of a hitting time to a one-sided stopping region. In [15], the author provides sufficient conditions for a one-sided stopping set and is able to solve the problem in closed form for some choices of the pinning time's distribution.…”

“…We extend the existing literature in several directions. The exponential structure of the gain function makes it impossible to use scaling properties that are central in all the papers where explicit solutions were obtained (see, e.g., [24], [11], [2], [15]). For this reason we must deal directly with a stopping problem for a time-inhomogeneous diffusion.…”

Discounted Optimal Stopping of a Brownian Bridge, with Application to American Options under Pinning

“…The recent paper [19] solves the non-discounted problem using the exponential of a Brownian bridge to model the stock prices. A Brownian bridge with unknown pinning random distribution and a Bayesian approach is advocated by [23]. The analytical results in [13] are extended in [24] by looking at a class of Gaussian bridges that share the same optimal stopping boundary.…”