397 research outputs found
Doob's maximal identity, multiplicative decompositions and enlargements of filtrations
In the theory of progressive enlargements of filtrations, the supermartingale
associated with an honest time g,
and its additive (Doob-Meyer) decomposition, play an essential role. In this
paper, we propose an alternative approach, using a multiplicative
representation for the supermartingale Z_{t}, based on Doob's maximal identity.
We thus give new examples of progressive enlargements. Moreover, we give, in
our setting, a proof of the decomposition formula for martingales, using
initial enlargement techniques, and use it to obtain some path decompositions
given the maximum or minimum of some processes.Comment: Typos correcte
On a flow of transformations of a Wiener space
In this paper, we define, via Fourier transform, an ergodic flow of
transformations of a Wiener space which preserves the law of the
Ornstein-Uhlenbeck process and which interpolates the iterations of a
transformation previously defined by Jeulin and Yor. Then, we give a more
explicit expression for this flow, and we construct from it a continuous
gaussian process indexed by R^2, such that all its restriction obtained by
fixing the first coordinate are Ornstein-Uhlenbeck processes
Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions
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.Comment: LaTeX; 40 pages; review pape
Multiplicative decompositions and frequency of vanishing of nonnegative submartingales
In this paper, we establish a multiplicative decomposition formula for
nonnegative local martingales and use it to characterize the set of continuous
local submartingales Y of the form Y=N+A, where the measure dA is carried by
the set of zeros of Y. In particular, we shall see that in the set of all local
submartingales with the same martingale part in the multiplicative
decomposition, these submartingales are the smallest ones. We also study some
integrability questions in the multiplicative decomposition and interpret the
notion of saturated sets in the light of our results.Comment: Typos corrected. Close to the published versio
J. L. Doob (27 February 1910--7 June 2004)
Doob's essential contributions to Probability theory are discussed; this
includes the main early results on martingale theory, Doob's -transform, as
well as a summary of Doob's three books. Finally, Doob's `stochastic triangle'
is viewed in the light of the stochastic analysis of the eighties.Comment: Published in at http://dx.doi.org/10.1214/09-AOP480 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Options on realized variance and convex orders
Realized variance option and options on quadratic variation normalized to unit expectation are analysed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk-neutral densities are said to be increasing in the convex order. For Leacutevy processes, such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability, then the resulting risk-neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally, we consider modeling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Leacutevy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order
The fine structure of asset returns: an empirical investigation
We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks. This leads to the conjecture, confirmed on options data, that the risk-neutral process should be free of a diffusion component. We conclude that the statistical and risk-neutral processes for equity prices are pure jump processes of infinite activity and finite variation
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