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A general principle and some questions about penalisations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1969)

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.

Keywords

  • Brownian motion and Bessel processes
  • penalisations
  • local times
  • unilateral maximum process
  • uniform integrability
  • convergence in law of processes

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Correspondence to Bernard Roynette .

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Roynette, B., Yor, M. (2009). A general principle and some questions about penalisations. In: Penalising Brownian Paths. Lecture Notes in Mathematics(), vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89699-9_5

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