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Estimates for norms of discrete stochastic integrals. (English. Russian original) Zbl 1216.60049

Dokl. Math. 81, No. 3, 414-417 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 432, No. 3, 322-325 (2010).
The author considers a discrete integral of the form
\[ \int_T U\,dV:=\sum_{t\in T,\, t<t_{\max}} U_t(V_{t_+}-V_t) \]
where \(U=\{U_t\}_{t\in[a,b]}\) and \(V=\{V_t\}_{t\in[a,b]}\) are two real stochastic processes, \(T\) is a finite nonempty subset in \([a,b]\) with \(t_{\min}=\min \{t\in T\}\), \(t_{\max}=\max \{t\in T\}\) and for \(t<t_{\max}\), \(t_+\) denotes the first point in \(T\) larger than \(t\). The author presents new estimates for norms of these discrete stochastic integrals and illustrates the sharpness of these estimates. As an interesting byproduct, these estimates allow to define the (Riemann-Stieltjes) stochastic integral \(\int_a^b U\,dV\) as \(\lim_T \int_T U\,dV\) where \(\lim_T\) denotes the limit over all partitions \(T\) of \([a,b]\) (as the mesh seze tends to zero). Applications of this result are given, namely for the fractional Brownian motion with Hurst parameter in \((\frac{1}{2},1)\).

MSC:

60H05 Stochastic integrals
60E15 Inequalities; stochastic orderings
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References:

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