Continuous Martingales and Brownian Motion

Schramm-Loewner Evolutions (SLE) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of SLE and the free field with appropriate boundary conditions; this involves ζ \zeta -regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of SLE with the free field, showing that, in a precise sense, chordal SLE is the solution of a stochastic “differential” equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general κ > 0 \kappa >0 . This identifies SLE curves as local observables of the free field.

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“…The proof shows that ℰ is, up to a multiplicative constant, the exponential martingale of (see e.g. [30]). …”

Section: Variations Of Harmonic Quantitiesmentioning

confidence: 98%

SLE and the free field: Partition functions and couplings

Dubédat

2009

J. Amer. Math. Soc.

173

276

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“…Thus, the results in this example will only be postulated, since the computations and proofs are almost the same as in Example 1. Let (X t ) be the square of a Bessel process of dimension α > 0 (see [7]) satisfying the stochastic differential equation…”

Section: Example 2 Bessel Processmentioning

confidence: 99%

Discounted optimal stopping problems for the maximum process

Pedersen

2000

J. Appl. Probab.

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The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

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“…It was also known [18] that the three-dimensional hyperbolic Bessel process can be realized via a Doob transform as a Brownian motion with negative unit drift conditioned to stay positive. This work was inspired by the search and discovery of an inversion I(X t ) = 1 2 ln coth X t of the 3-dimensional hyperbolic Bessel process (X t ).…”

Section: Motivations and Main Resultsmentioning

confidence: 99%

On Inversions and Doob h-Transforms of Linear Diffusions

Alili

Graczyk

Żak

2015

Lecture Notes in Mathematics

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Let X be a regular linear diffusion whose state space is an open interval E ⊆ R. We consider the dual diffusion X * whose probability law is obtained as a Doob h-transform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. We provide a construction of X * as a deterministic inversion I(X) of X, time changed with some random clock. Such inversions generalize the Euclidean inversions that intervene when X is a Brownian motion. The important case where X * is X conditioned to stay above some fixed level is included. The families of deterministic inversions are given explicitly for the Brownian motion with drift, Bessel processes and the 3-dimensional hyperbolic Bessel process.

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Continuous Martingales and Brownian Motion

Abstract: 3rd ed. p. cm. -(Grundlehren der mathematischen Wissenschaften; 293) Includes bibliographical references and index. ISBN 978-3-642-08400-3 ISBN 978-3-662-06400-9 (eBook)

Cited by 4,020 publications

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SLE and the free field: Partition functions and couplings

SLE and the free field: Partition functions and couplings

Discounted optimal stopping problems for the maximum process

On Inversions and Doob h-Transforms of Linear Diffusions

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