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108
On the pricing of corporate debt: The risk structure of interest rates
 JOURNAL OF FINANCE
, 1974
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Optimal approximation by piecewise smooth functions and associated variational problems
 Commun. Pure Applied Mathematics
, 1989
"... (Article begins on next page) The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Mumford, David Bryant, and Jayant Shah. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. ..."
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Cited by 1200 (13 self)
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(Article begins on next page) The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Mumford, David Bryant, and Jayant Shah. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied
Option pricing when underlying stock returns are discontinuous
 Journal of Financial Economics
, 1976
"... The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying ..."
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Cited by 839 (2 self)
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The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the moregeneral cast when the underlying stock returns are gcncrated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black&holes formula in that it does not dcpcnd on investor prcfcrenccs or knowledge of the expcctsd return on the underlying stock. Morcovcr, the same analysis applied to the options can bc extcndcd to the pricingofcorporatc liabilities. 1. Intruduction In their classic paper on the theory of option pricing, Black and Scholcs (1973) prcscnt a mode of an:llysis that has rcvolutionizcd the theory of corporate liability pricing. In part, their approach was a breakthrough because it leads to pricing formulas using. for the most part, only obscrvablc variables. In particular,
QuasiRandom Sequences and Their Discrepancies
 SIAM J. Sci. Comput
, 1994
"... Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is meas ..."
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Cited by 87 (6 self)
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Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is measured by its discrepancy, which is of size (log N) s N \Gamma1 for large N , as opposed to discrepancy of size (log log N) 1=2 N \Gamma1=2 for a random sequence (i.e. for almost any randomlychosen sequence). Several types of discrepancy, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancy are presented for a wide choice of dimension s, number of points N and different quasirandom sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasirandom sequence is almost exactly the same as that of a randomly chosen sequence...
Brownian Motion in a Weyl Chamber, NonColliding Particles, and Random Matrices
, 1997
"... . Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting posit ..."
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Cited by 82 (2 self)
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. Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time t without having collided by time t. We show that the probability that there will be no collision up to time t is asymptotic to a constant multiple of t \Gamman(n\Gamma1)=4 as t goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point ...
Exact simulation of diffusions
 Annals of Applied Probability
, 2005
"... We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without ..."
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Cited by 50 (16 self)
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We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without further reference to the dynamics of the target process. 1. Introduction. Exact
Diffusionbased motion planning for a nonholonomic flexible needle model
 Proc. IEEE Int. Conf. Robot. Autom. (ICRA); 2005
"... Abstract — Fine needles facilitate diagnosis and therapy because they enable minimally invasive surgical interventions. This paper formulates the problem of steering a very flexible needle through firm tissue as a nonholonomic kinematics problem, and demonstrates how planning can be accomplished us ..."
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Abstract — Fine needles facilitate diagnosis and therapy because they enable minimally invasive surgical interventions. This paper formulates the problem of steering a very flexible needle through firm tissue as a nonholonomic kinematics problem, and demonstrates how planning can be accomplished using diffusionbased motion planning on the Euclidean group, SE(3). In the present formulation, the tissue is treated as isotropic and no obstacles are present. The bevel tip of the needle is treated as a nonholonomic constraint that can be viewed as a 3D extension of the standard kinematic cart or unicycle. A deterministic model is used as the starting point, and reachability criteria are established. A stochastic differential equation and its corresponding FokkerPlanck equation are derived. The EulerMaruyama method is used to generate the ensemble of reachable states of the needle tip. Inverse kinematics methods developed previously for hyperredundant and binary manipulators that use this probability density information are applied to generate needle tip paths that reach the desired targets. Index Terms — needle steering, nonholonomic path planning, probability density function, EulerMaruyama method, medical robotics I.
Free martingale polynomials
 Journal of Functional Analysis
"... ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. ..."
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Cited by 35 (3 self)
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ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota’s finite operator calculus can be modified for the free context.
Equivalent and absolutely continuous measure changes for jumpdiffusion processes” to appear in the Annals of Applied Probability
"... We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential. ..."
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Cited by 33 (3 self)
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We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential.
Brownian gibbs property for airy line ensembles
"... 1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is ad ..."
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1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is added to each of the curves of this scaling limit, an xtranslation invariant process sometimes called the multiline Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and nonintersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers ” and “outliers”. We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property – called the Brownian Gibbs property – of being invariant under the following action. Select an index 1 ≤ k ≤ N and erase Bk on a fixed time interval (a, b) ⊆ (−N, N); then replace this erased curve with a new curve on (a, b) according to the law of a Brownian bridge between the two existing endpoints ( a, Bk(a) ) and ( b, Bk(b) ) , conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edgescaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the longstanding conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multiline Airy process, and readily yields several other interesting properties of this process. 1.