Strong Law of Large Numbers of Pettis-Integrable Multifunctions

Using reversed martingale techniques, we prove the strong law of large numbres for independent Pettis-integrable multifunctions with convex weakly compact values in a Banach space. The Mosco convergence of reversed Pettis-integrable martingale of the form (EBnX)n≥1, where (Bn)n≥1 is a decreasing sequence of the sub σ-algebra of F is provided.

DYNAMIC MEAN–VARIANCE OPTIMIZATION PROBLEMS WITH DETERMINISTIC INFORMATION

We solve the problems of mean–variance hedging (MVH) and mean–variance portfolio selection (MVPS) under restricted information. We work in a setting where the underlying price process [Formula: see text] is a semimartingale, but not adapted to the filtration [Formula: see text] which models the information available for constructing trading strategies. We choose as [Formula: see text] the zero-information filtration and assume that [Formula: see text] is a time-dependent affine transformation of a square-integrable martingale. This class of processes includes in particular arithmetic and exponential Lévy models with suitable integrability. We give explicit solutions to the MVH and MVPS problems in this setting, and we show for the Lévy case how they can be expressed in terms of the Lévy triplet. Explicit formulas are obtained for hedging European call options in the Bachelier and Black–Scholes models.

On certain martingale inequalities for maximal functions and mean oscillations

Let $X$ be a Banach function space over a nonatomic probability space. For a uniformly integrable martingale $f=(f_n)$ with respect to a filtration ${\mathcal F}=({\mathcal F}_n)$, let $Mf =\sup_n |f_n|$ and $\theta_{\mathcal F}f=\sup_n E[|f_{\infty}- f_{n-1}| \mid{\mathcal F}_n]$. We give a necessary and sufficient condition on $X$ for the inequality $\parallel \theta_{\mathcal F}f \parallel_X \leq C\parallel Mf\parallel_X$ to hold.

Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Let for each be an -valued locally square integrable martingale w.r.t. a filtration (probability spaces may be different for different ). It is assumed that the discontinuities of are in a sense asymptotically small as and the relation holds for all , row vectors , and bounded uniformly continuous functions . Under these two principal assumptions and a number of technical ones, it is proved that the 's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes converge in distribution to some , then a sequence () converges in distribution to a continuous local martingale with initial value and quadratic characteristic , whose finite-dimensional distributions are explicitly expressed via those of .

Stochastic Integration in Abstract Spaces

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let , , and be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of into . If is an integrable, -valued predictable process and is an -valued square integrable martingale, then there exists a -valued process called the stochastic integral. The Lebesgue space of these integrable processes is studied and convergence theorems are given. Extensions to general locally convex spaces are presented.

STRONG SOLUTIONS FOR STOCHASTIC POROUS MEDIA EQUATIONS WITH JUMPS

We prove global well-posedness in the strong sense for stochastic generalized porous media equations driven by a square integrable martingale with stationary independent increments.