An inequality for the predictable projection of an adapted process

Abstract

Let (f _n) ^N_n=1 be a stochastic process adapted to the filtration (F _n ^N _n=0 ). Denoting by (g _n) ^N_n=1 the predictable projection of this process, i.e., g _n=E_n−1(f_n) 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≤p≤q≤∞ (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.