Bessel Bridges Research Articles

Random Brownian scaling identities and splicing of Bessel processes

An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of Brownian excursions, due to Williams and Bismut, each involving back-to-back splicing of fragments of two independent three-dimensional Bessel processes. Generalizations of both splicing constructions are described, which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.

The "True" Self-Avoiding Walk with Bond Repulsion on $\mathbb{Z}$: Limit Theorems

The true self-avoiding walk with bond repulsion is a nearest neighbor random walk on $\mathbb{Z}$, for which the probability of jumping along a bond of the lattice is proportional to $\exp(-g \cdot$ number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as $A \rightarrow \infty$, for the distribution of $A^{-2/3}X_{\theta_{s/A}}$, where $\theta_{s/A}$ is a random time distributed geometrically with mean $e^{-s/A}(1 - e^{-s/A})^{-1} = A/s + O(1)$. As a by-product we also obtain an apparently new identity related to Brownian excursions and Bessel bridges.

ANAYTICAL SOLUTIONS FOR THE PRICING OF AMERICAN BOND AND YIELD OPTIONS1

In this paper we use the Cox, Ingersoll, and Ross (1985b) single‐factor, term structure model and extend it to the pricing of American default‐free bond puts. We provide a quasi‐analytical formula for these option prices based on recently established mathematical results for Bessel bridges, coupled with the optimal stopping time method. We extend our results to another interest rate contingent claim and provide a quasi‐analytical solution for American yield option prices which illustrates the flexibility of our framework.