In this paper we give some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation-G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied in continuous time dynamic and coherent risk measures in finance, in particular for path-dependence risky positions under situations of volatility model uncertainty.

The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case. Introduction.Our purpose is to set a framework for dealing with model uncertainty in mathematical finance and to handle the pricing of contingent claims in this context.Let (S t ) t∈[0,T ] be a real-valued process which stands for an asset price. Usually, it is assumed that the set P m of the equivalent probabilities under which S is a martingale is not empty. This is a sufficient condition to preclude pure gambling strategies that never fail and win with a positive probability.In the classical example of the Black-Scholes model, S solves the linear stochastic differential equation dS t = S t dB t , where

Jovian S‐bursts are intense impulsive decameter radio spikes drifting in frequency in tens of milliseconds over several hundreds of kHz up to a few MHz. Their generation scenario has been much debated for 30 years. The automated analysis of an extensive set of digital radio observations at very high temporal and spectral resolutions is presented here. It strongly suggests that S‐bursts are the cyclotron‐maser emission of electron populations with ∼5 keV energy, accelerated near Io and then in quasi‐adiabatic motion along magnetic field lines connecting the Io torus to Jupiter's auroral regions. This scenario is consistent with Voyager observations of Alfvèn waves near Io's wake and with recent infrared observations of the Io flux tube footprint. The total energy, velocity and pitch angle of the radiating electrons, and the extent of the bursts radiosources are deduced from the radio measurements, which appear as a promising remote sensing tool for the Jovian magnetosphere, and possibly that of other “radio” planets.

Using analytical methods, we prove existence uniqueness and estimates for s.p.d.e. of the type