Nikos E. Frangos

A Wiener Chaos Approach to Hyperbolic SPDEs

We construct generalized weighted Wiener chaos solutions for hyperbolic linear SPDEs driven by a cylindrical Brownian motion. Explicit conditions for the existence, uniqueness, and regularity of generalized (Wiener Chaos) solutions are established in Sobolev spaces. An equivalence relation between the Wiener Chaos solution and the traditional one is established. The Heath–Jarrow–Morton (HJM) forward rate model is used as an example to illustrate the general construction.

On multiparameter ergodic and martingale theorems in infinite measure spaces

SummaryA unified proof is given of several ergodic and martingale theorems in infinite measure spaces.

The continuity of the quadratic variation of two-parameter martingales

It has been known that any L log+L-integrable two-parameter martingale M possesses a quadratic variation [M]. We show that the continuity properties of M are inherited by its quadratic variation. If M has no point jumps, [M] has no point jumps. [M] has at most axial jumps with respect to one of the coordinate axes in parameter space if M possesses this property. Finally, [M] is continuous along with M.