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.

Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include Non-Gaussian models that are solutions to OU (Ornstein-Uhlenbeck) equations driven by one sided discontinuous L¶evy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general, mean corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean corrected exponential model is not a martingale in the ¯ltration in which it is originally de¯ned. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered ¯ltrations consistent with the one dimensional marginal distributions of the level of the process at each future date. 1

Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents

ABSTRACT. The behavior of connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost (correlated losses). The stationary behavior of this algorithm is analyzed when the rate of occurrence of clumps becomes arbitrarily small. From a probabilistic point of view, it is shown that exponential functionals associated to compound Poisson processes play a key role. A formula for the fractional moments and some density functions are derived. Analytically, to get the explicit expression of the distributions involved, the natural framework of this study turns out to be the q-calculus. Different loss models are then compared using concave ordering. Quite surprisingly, it is shown that, for a fixed loss rate, the correlated loss model has a higher throughput than an uncorrelated loss model. CONTENTS

Let B be a standard one-dimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solution X of the Ito SDE dXv = n 4 \Gamma X 2 v \Gamma t \Gamma R v 0 Xudu \Delta \Gamma1 o dv + 2 p XvdBv on the interval [0; V t (X)), where V t (X) := inffv : R v 0 Xudu = tg, and Xv = 0 for all v V t (X). This conditioned form of the Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as n !1 and 2k= p n ! ` of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a Galton-Watson branching process conditioned on its total progeny....

. Let (Z t ) t2R+ be a martingale in L 4 having the chaos representation property and angle bracket dhZ t ; Z t i = dt. We show that the positive functionals F of (Z t ) t2R+ satisfy the modied logarithmic Sobolev inequality E[F log F ] E[F ] log E[F ] 1 2 E 1 F Z 1 0 (2 i t )(D t F ) 2 dt ; where D is the gradient operator dened by lowering the degree of multiple stochastic integrals with respect to (Z t ) t2R+ and (i t ) t2R+ f0; 1g is a process given by the structure equation satised by (Z t ) t2R+ . Resume. Soit (Z t ) t2R+ une martingale dans L 4 qui satisfait la propriete de representation chaotique, avec dhZ t ; Z t i = dt. On montre que les fonctionnelles positives F de (Z t ) t2R+ satisfont l'inegalite de Sobolev logarithmique modiee E[F log F ] E[F ] log E[F ] 1 2 E 1 F Z 1 0 (2 i t )(D t F ) 2 dt ; ou D est l'operateur gradient qui abaisse le degre des integrales stochastiques multiples par rapport a (Z t ) t2R+ , et (i t ) t2R+ f0; 1g est un processus donne par l'equation de structure satisfaite par (Z t ) t2R+ . Key words: Logarithmic Sobolev inequalities, normal martingales, Azema martingales, Poisson random measures. Mathematics Subject Classication. 60G44, 60G60, 46E35, 46E39. 1

The so-called intensity-based approach to the modelling and valuation of defaultable securities has attracted a considerable attention of both practitioners and academics in recent years; to mention a few papers in this vein: Duffie [8], Duffie and Lando [9], Duffie et al. [10], Jarrow and Turnbull [13], Jarrow et al.

The aim of these notes is to provide a relatively concise- but still self-contained- overview of mathematical notions and results which underpin the valuation of defaultable claims. Though the default risk modelling was extensively studied in numerous recent papers, it seems nonetheless that some of these papers lack a sound theoretical background. Our goal is to furnish results which cover both the classic value-of-the-firm (or structural) approach, as well as the more recent intensity-based methodology. For a more detailed account of mathematical results

The infinitely divisible distributions on R + of random variables C t , S t and T t with Laplace transforms ` 1 cosh p 2 ' t ; / p 2 sinh p 2 ! t ; and / tanh p 2 p 2 ! t respectively are characterized for various t ? 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their L'evy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of C¹ and S³ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and ...

Let M ffi be the maximum of a standard Bessel bridge of dimension ffi . A series formula for P (M ffi a) due to Gikhman and Kiefer for ffi = 1; 2; : : : is shown to be valid for all real ffi ? 0. Various other characterizations of the distribution of M ffi are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of M ffi as is described both as ffi !1 and as ffi # 0. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, Bessel process, zeros of Bessel functions, Riemann zeta function Contents 1 Introduction 3 2 The maximum of a diffusion bridge 8 3 The Gikhman-Kiefer Formula 9 4 The law of T ffi and the agreement formula 11 5 The first passage transform and its derivatives 13 6 Moments 16 7 Dimensions one and three 20 8 Limits as ffi !1 22 9 Limits as ffi # 0 24 10 Relation to last exit times 27 11 A series involving the zeros of J 30 A Some Useful Formulae 33 A.1 Bessel Functions : : : : : : : : : : :...