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An inequality for the predictable projection of an adapted process

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1613)

Abstract

Let (f n) Nn=1 be a stochastic process adapted to the filtration (F n N n=0 ). Denoting by (g n) Nn=1 the predictable projection of this process, i.e., g n=En−1(fn) we show that the inequality

$$\left[ {E(\sum\limits_{n = 1}^N {|g_n |^q } )^{p/q} } \right]^{1/p} \leqslant \left[ {E(\sum\limits_{n = 1}^N {|f_n |^q } )^{p/q} } \right]^{1/p} $$

or, in more abstract terms

$$\parallel (g_n )_{n = 1}^N \parallel _{Lp(l_N^q )} \leqslant 2\parallel (f_n )_{n = 1}^N \parallel _{Lp(l_N^q )} $$

holds true for 1≤pq≤∞ (with the obvious interpretation in the case of p=∞ or q=∞).

Several similar results, pertaining also to the case p>q, are known in the literature. The present result may have some interest in view of the following reasons: (1) the case p=1 and 2<q≤∞ seems to be new; (2) we obtain 2 as a uniform constant which is sharp in the case p=1, q=∞ and (3) the proof is very easy.

Keywords

  • Sharp Constant
  • Continuous Time Case
  • Uniform Constant
  • Obvious Interpretation
  • Continuous Time Setting

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1995 Springer-Verlag

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Delbaen, F., Schachermayer, W. (1995). An inequality for the predictable projection of an adapted process. In: Azéma, J., Emery, M., Meyer, P.A., Yor, M. (eds) Séminaire de Probabilités XXIX. Lecture Notes in Mathematics, vol 1613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094195

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  • DOI: https://doi.org/10.1007/BFb0094195

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