Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.

We give a representation of the solution for a stochastic linear equation of the form Xt = Yt+ ∫ (0,t] Xs−dZs where Z is a càdlàg semimartingale and Y is a càdlàg adapted process with bounded variation on finite intervals. As an application we study the case where Y and −Z are nondecreasing, jointly have stationary increments and the jumps of −Z are bounded by 1. Special cases of this process are shot-noise processes, growth collapse (additive increase, multiplicative decrease) processes and clearing processes. When Y and Z are in addition independent Lévy processes the resulting X is called a generalized Ornstein-Uhlenbeck process.

The paper investigates how buyer-supplier firm-specific relationships affect security prices. Starting from the empirical inconsistencies in some standard structural models we propose a structural model of firm dependence in a vertically connected network of firms based on cash flow transfers between the buyers and the suppliers. A closed econ-omy network completeness depends only on the topology of the network. We develop analytical formulas for corporate debt, credit default swaps and collateralized debt obli-gations. We prove that network disintegration does not necessarily reduce corporate and portfolio yields. In fact, it can raise them if the externally generated cash flows and internal network flows are positively correlated. 2 1

We provide the increasing q-harmonic functions associated to the following family of integro-differential operators, for any α> 0, γ ≥ 0 and f ∈ D(L (α,ψ,γ)), (0.1) L (α,ψ,γ) f(x) = x −α

Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the Ewens-Pitman two-parameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of categories, • extending the class of tractable models by exploring the idea of regeneration, • analysing regenerative properties of the Ewens-Pitman partitions, • studying asymptotic features of the regenerative compositions. 1

Abstract. Motivated by an original on-line page-ranking algorithm, starting from an arbitrary Markov chain (Cn) on a discrete state space S, a Markov chain (Cn, Mn) on the product space S 2, the cat and mouse Markov chain, is constructed. The first coordinate of this Markov chain behaves like the original Markov chain and the second component changes only when both coordinates are equal. The asymptotic properties of this Markov chain are investigated. A representation of its invariant measure is in particular obtained. When the state space is infinite it is shown that this Markov chain is in fact null recurrent if the initial Markov chain (Cn) is positive recurrent and reversible. In this context, the scaling properties of the location of the second component, the mouse, are investigated in various situations: simple random walks in Z and Z 2, reflected simple random walk in N and also in a continuous time setting. For several of these processes, a time scaling with rapid growth gives an interesting asymptotic behavior related to limit results for occupation times