A SHARP BOUND FOR ITO PROCESSES

  • Published : 1998.08.01

Abstract

Let X and Y be Ito processes with dX$_{s}$ = $\phi$$_{s}$dB$_{s}$$\psi$$_{s}$ds and dY$_{s}$ = (equation omitted)dB$_{s}$ + ξ$_{s}$ds. Burkholder obtained a sharp bound on the distribution of the maximal function of Y under the assumption that │Y$_{0}$$\leq$│X$_{0}$│,│ζ│$\leq$$\phi$│, │ξ│$\leq$$\psi$│ and that X is a nonnegative local submartingale. In this paper we consider a wider class of Ito processes, replace the assumption │ξ│$\leq$$\psi$│ by a more general one │ξ│$\leq$$\alpha$$\psi$│ , where a $\geq$ 0 is a constant, and get a weak-type inequality between X and the maximal function of Y. This inequality, being sharp for all a $\geq$ 0, extends the work by Burkholder.der.urkholder.der.

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