Ito's rule and Levy's theorem in vector lattices

作者：Grobler Jacobus J; Labuschagne Coenraad C A^*

来源：Journal of Mathematical Analysis and Applications, 2017, 455(1-2): 979-1004.

DOI：10.1016/j.jmaa.2017.06.011

• Brownian motion
• Ito rule
• Levy's theorem

The change of variable formula, or Ito's rule, is studied in a Dedekind complete vector lattice with weak order unit E. Using the functional calculus we prove that for a Holder continuous semimartingale X-t = X-a + M-t + B-t, t is an element of J, and a twice continuously differentiable function f, the formula
f(X-t) = f(X-a) + (0)integral(t)f'(X-s) dM(s) + (0)integral(t)f'(X-s) dB(s)
+1/2(0)integral(t)f"(X-s)d < M >(s), 0 <= s <= t is an element of J (0.1)
holds. The first integral in the formula is an Ito integral with reference to the local martingale M and the second and third integrals are Dobrakov-type integrals of a vector valued function with reference to a vector valued measure. Using the formula, we prove Levy's characterization of Brownian motion as being a continuous martingale with compensator tE. The proof of this result yields a concrete description of abstract Brownian motion defined in vector lattices.

• 出版日期2017-11-15