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93
Ruin probabilities and penalty functions with stochastic rates of interest
 Stochastic Processes and their Applications 112
, 2004
"... Assume that a compound Poisson surplus process is invested in a stochastic interest process which is assumed to be a Lévy process. We derive recursive and integral equations for ruin probabilities with such an investment. Lower and upper bounds for the ultimate ruin probability are obtained from the ..."
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Cited by 10 (3 self)
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Assume that a compound Poisson surplus process is invested in a stochastic interest process which is assumed to be a Lévy process. We derive recursive and integral equations for ruin probabilities with such an investment. Lower and upper bounds for the ultimate ruin probability are obtained from these equations. When the interest process is a Brownian motion with drift, we give a unified treatment to ruin quantities by studying the expected discounted penalty function associated with the time of ruin. An integral equation for the penalty function is given. Smooth properties of the penalty function are discussed based on the integral equation. Errors in a known result about the smooth properties of the ruin probabilities are corrected. Using a differential argument and moments of exponential functionals of Brownian motions, we derive an integrodifferential equation satisfied by the penalty function. Applications of the integrodifferential equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the amount of claim causing ruin, and so on. Some known results about ruin quantities are recovered from the
Connecting Yule process, bisection and binary search tree via martingales
 J. Iranian Statistical Society
, 2004
"... We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process. Key words. Binary search tree, branching random walk, Yule process, convergence o ..."
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Cited by 9 (3 self)
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We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process. Key words. Binary search tree, branching random walk, Yule process, convergence of martingales, functional equations. A.M.S. Classification. 60J25, 60J80, 60J85, 68W40, 60G42, 60G44. 1
On the valuation of arithmetic–average Asian options: explicit formulas
, 1999
"... In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced in [D] to computing the negative moments of Yor’s accumulation processes for which functional recursion rules are given. Stressing the role of Theta functions, this paper now solves these recursio ..."
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Cited by 8 (3 self)
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In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced in [D] to computing the negative moments of Yor’s accumulation processes for which functional recursion rules are given. Stressing the role of Theta functions, this paper now solves these recursion rules and expresses these negative moments as linear combinations of certain Theta integrals. Using the Jacobi transformation formula, very rapidly and very stably convergent series for them are derived. In this way a computable series for Black–Scholes price of the Asian option results which is numerically illustrated. Moreover, the Laguerre series approach of [D] is made rigorous, and extensions and modifications are discussed. The key for this is the analysis of the integrability and growth properties of the Asia density in [Y], basic problems which seem to be addressed here for the first time. 1. Introduction: Asian
Bessel processes, the integral of geometric Brownian motion, and Asian options
 Theor. Probab. Appl
, 2004
"... Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been stu ..."
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Cited by 7 (0 self)
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Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the HartmanWatson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the HartmanWatson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.
On the integral of geometric Brownian motion
 Adv. Appl. Prob
, 2003
"... Abstract. This paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is t ..."
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Abstract. This paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is the main methodological contribution of the paper. The one of our integrals has a similar structure to that obtained in [Y], while the other is in terms of Hermite functions as those of [Du01]. Performing or not performing a certain Girsanov transformation is identified as the source of these two forms of the laws. For exponents equal to 1 our results specialize to those obtained in [Y], but for exponents equal to minus 1 they give representations for the laws which are markedly different from those obtained in [Du01].
A TRANSFORMATION FOR LÉVY PROCESSES WITH ONESIDED JUMPS AND APPLICATIONS
"... Abstract. The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts, [27, 29, 21, 17], as well as in older work of Urbanik [35]. We show how some specific instances of thi ..."
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Cited by 5 (3 self)
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Abstract. The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts, [27, 29, 21, 17], as well as in older work of Urbanik [35]. We show how some specific instances of this mapping prove to be useful for a variety of applications. 1.
Accurate Approximations for European Asian Options
, 1998
"... In the nperiod binomial tree model, we provide fast algorithms to compute very accurate lower and upper bounds on the value of a Europeanstyle Asian option. These algorithms are inspired by the continuoustime analysis of Rogers and Shi [12]. Specifically we consider lower bounds that are given by ..."
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Cited by 4 (1 self)
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In the nperiod binomial tree model, we provide fast algorithms to compute very accurate lower and upper bounds on the value of a Europeanstyle Asian option. These algorithms are inspired by the continuoustime analysis of Rogers and Shi [12]. Specifically we consider lower bounds that are given by grouping stockprice paths in the tree according to the value of a certain random variable Z, and treating all paths in a group as having the same arithmetic stockprice average, namely the average of the arithmetic average over the group. For a specific Z , Rogers and Shi show analytically that the error in the lower bound is small. However they are only able to estimate the lower and upper bounds numerically as integrals in continuous time. Indeed, for their choice of Z these bounds are difficult to compute exactly in the binomial tree model. We show how to choose Z so that the bounds can be computed exactly in time proportional to n 4 by forward induction. Moreover, our bounds are stric...
Fluctuation theory and exit systems for positive selfsimilar Markov processes
 Preprint. AND ASYMPTOTIC nTUPLE LAWS AT FIRST AND LAST PASSAGE 563
, 2009
"... For a positive selfsimilar Markov process, X, we construct a local time for the random set, �, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) ..."
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Cited by 4 (1 self)
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For a positive selfsimilar Markov process, X, we construct a local time for the random set, �, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) associated to a positive selfsimilar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set � and the process X sampled on the local time scale. The process (R, H) is described in terms of a ladder process linked to the Lévy process associated to X via Lamperti’s transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finitedimensional convergence of (R, H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012–1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Lévy process oscillates. 1. Introduction. In
A Note on the Distribution of Integrals of Geometric Brownian Motion ∗
, 2000
"... The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At: = ∫ t exp{Zs}ds, t ≥ 0, where {Zs: s ≥ 0} is a onedimensional ..."
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Cited by 3 (1 self)
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The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At: = ∫ t exp{Zs}ds, t ≥ 0, where {Zs: s ≥ 0} is a onedimensional