A general principle and some questions about penalisations

Abstract

In the preceding chapters, we have shown that, in a large class of frameworks, certain families $\{ P^{(t)} ,t \ge 0\}$ of probabilities defined on a filtered space $\{ \Omega ,(F_s )_{S \ge 0} \}$ converge, as t → ∞, to a probability Q, at least in the sense that :

$$\forall \;s\; > 0,\;\forall \Lambda _s \in \;F_s ,\;P^{(t)} (\Lambda _S )\mathop \to \limits_{t \to \infty } Q(\Lambda _S )$$

The typical question which we consider in this chapter is, roughly :

to which extent can we replace $\Lambda _s \in \;F_s ,\;by\;\Lambda \in \;F_\infty \;?$

The answer is, to say the least, subtle, and not every Λ ∈ F_∞ is suitable. This led us to detect some non-uniform integrability results, as well as the convergence of certain (Hellinger, total variation⋯) distances between P ^(t) and $Q_{|F_t }$, as t → ∞.

We also prove, in the Brownian framework, the convergence in law of $X^{(t)\;} : = \left( {\frac{1}{{\sqrt t }}X_{tu} ;u \le 1} \right)$, under P ^(t) or Q, as t → ∞. The limits in law are different, and may be expressed in terms of the BES(3) process or the Brownian meander.