Results 1  10
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20
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... 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, expl ..."
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Cited by 97 (10 self)
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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.
Conditioned Brownian trees
"... We consider a Brownian tree consisting of a collection of onedimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus ..."
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Cited by 15 (6 self)
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We consider a Brownian tree consisting of a collection of onedimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the socalled Integrated SuperBrownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive halfline. We also establish a Verwaatlike theorem showing that this conditioned Brownian tree can be obtained by rerooting the unconditioned one at the vertex corresponding to the minimal spatial position. In terms of ISE, this theorem yields the following fact: Conditioning ISE to put no mass on]−∞, −ε [ and letting ε go to 0 is equivalent to shifting the unconditioned ISE to the right so that the leftmost point of its support becomes the origin. We derive a number of explicit estimates and formulas for our conditioned Brownian trees. In particular, the probability that ISE puts no mass on] − ∞, −ε [ is shown to behave like 2ε 4 /21 when ε goes to 0. Finally, for the conditioned Brownian tree with a fixed height h, we obtain a decomposition involving a spine whose distribution is absolutely continuous with respect to that of a ninedimensional Bessel process on the time interval [0,h], and Poisson processes of subtrees originating from this spine. 1
LIMITING LAWS ASSOCIATED WITH BROWNIAN MOTION PERTURBED BY NORMALIZED EXPONENTIAL WEIGHTS, I
, 2008
"... Let (Bt; t ≥ 0) be a one dimensional Brownian motion, with local time process (Lx t; t ≥ 0, x ∈ R). We determine the rate of decay of Z V [ { t (x): = Ex exp − 1 L ..."
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Cited by 14 (6 self)
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Let (Bt; t ≥ 0) be a one dimensional Brownian motion, with local time process (Lx t; t ≥ 0, x ∈ R). We determine the rate of decay of Z V [ { t (x): = Ex exp − 1 L
The Most Visited Sites of Symmetric Stable Processes
, 2000
"... this paper remain unchanged if we replace ..."
The First Exit Time of Planar Brownian Motion from the Interior of a Parabola
 Ann. Probab
, 2001
"... Introduction For x 2 IR n nf0g, we let (x) be the angle between x and the point (1; 0; : : : ; 0). The right circular cone of angle 0 < < is the domain = fx 2 IR n : (x) < g. Let fB t : t 0g be the n{dimensional Brownian motion and denote by E x and P x the expectation and probab ..."
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Cited by 8 (4 self)
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Introduction For x 2 IR n nf0g, we let (x) be the angle between x and the point (1; 0; : : : ; 0). The right circular cone of angle 0 < < is the domain = fx 2 IR n : (x) < g. Let fB t : t 0g be the n{dimensional Brownian motion and denote by E x and P x the expectation and probability associated with this motion starting at x and denote by = infft > 0 : B t = 2 g its rst exit time from . The following result was
On the fundamental solution of the KolmogorovShiryaev equation. The Shiryaev Festschrift (Metabief 2005
, 2006
"... We derive an integral representation for the fundamental solution of the Kolmogorov forward equation: ft = −((1+µx)f)x + (ν x 2 f)xx associated with the Shiryaev process X solving: dXt = (1+µXt) dt + σXt dBt where µ ∈ IR, ν = σ 2 /2> 0 and B is a standard Brownian motion. The method of proof is b ..."
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Cited by 3 (2 self)
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We derive an integral representation for the fundamental solution of the Kolmogorov forward equation: ft = −((1+µx)f)x + (ν x 2 f)xx associated with the Shiryaev process X solving: dXt = (1+µXt) dt + σXt dBt where µ ∈ IR, ν = σ 2 /2> 0 and B is a standard Brownian motion. The method of proof is based upon deriving and inverting a Laplace transform. Basic properties of X needed in the proof are reviewed. 1.
INFINITELY DIVISIBILITY OF SOLUTIONS OF SOME SEMISTABLE INTEGRODIFFERENTIAL EQUATIONS AND EXPONENTIAL FUNCTIONALS OF LÉVY PROCESSES
, 2006
"... We provide the increasing qharmonic functions associated to the following family of integrodifferential operators, for any α> 0, γ ≥ 0 and f ∈ D(L (α,ψ,γ)), (0.1) L (α,ψ,γ) f(x) = x −α ..."
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Cited by 1 (1 self)
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We provide the increasing qharmonic functions associated to the following family of integrodifferential operators, for any α> 0, γ ≥ 0 and f ∈ D(L (α,ψ,γ)), (0.1) L (α,ψ,γ) f(x) = x −α
Credit derivatives · Implied volatility skew
, 2006
"... Abstract We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link betw ..."
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Abstract We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and equity volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a constant elasticity of variance (CEV) specification for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our stock price process to a standard Bessel process with killing. This reduction permits the development of completely explicit closed form solutions for riskneutral survival probabilities, CDS spreads, corporate bond values, and Europeanstyle equity options. Furthermore, our valuation model is sufficiently flexible so that it can be calibrated to exactly match arbitrarily given term structures of CDS spreads, interest rates, dividend yields, and atthemoney implied volatilities.