This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Path Decomposition and Continuity of Local Time for One‐Dimensional Diffusions, I

Diffusions, Markov processes, and martingales

Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

background or methods

/pdf/combinatorial-stochastic-processes-4nf5lxpd0t.pdf

Combinatorial Stochastic Processes

Using Bessel processes, one can solve several open problems involving the integral of an exponential of Brownian motion. This point will be illustrated with three examples. The first one is a formula for the Laplace transform of an Asian option which is “out of the money.” The second example concerns volatility misspecification in portfolio insurance strategies, when the stochastic volatility is represented by the Hull and White model. The third one is the valuation of perpetuities or annuities under stochastic interest rates within the Cox-Ingersoll-Ross framework. Moreover, without using time changes or Bessel processes, but only simple probabilistic methods, we obtain further results about Asian options: the computation of the moments of all orders of an arithmetic average of geometric Brownian motion; the property that, in contrast with most of what has been written so far, the Asian option may be more expensive than the standard option (e.g., options on currencies or oil spreads); and a simple, closed-form expression of the Asian option price when the option is “in the money,” thereby illuminating the impact on the Asian option price of the revealed underlying asset price as time goes by. This formula has an interesting resemblance with the Black-Scholes formula, even though the comparison cannot be carried too far.

Bessel Processes, Asian Options, and Perpetuities

background or result

A decomposition of Bessel Bridges

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T]of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit time distributions and the fixed time case is recovered by inverting Laplace transforms.

background

https://digitalassets.lib.berkeley.edu/sdtr/ucb/text/310.pdf

Path Decomposition and Continuity of Local Time for One‐Dimensional Diffusions, I

Citations

Diffusions, Markov processes, and martingales

Combinatorial Stochastic Processes

Bessel Processes, Asian Options, and Perpetuities

A decomposition of Bessel Bridges

On Some Exponential Functionals of Brownian Motion

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