Comparison theorems for stochastic evolution equations

Monotonicity of the collateralized debt obligations term structure model

“…The pairs of conditions (P 1), (P 2) and (M 1), (M 2) correspond to positivity and monotonicity of the solution of the HJMM equation in L 2,γ n . They follow from a generalized version of the result of Milian, see [8], which was concerned with a general SPDE driven by a Wiener process. We show how to pass to a Lévy process in the case of the HJMM equation.…”

On Invariance of Linear Subspaces for Stochastic Evolution Equations

“…We repeat the needed facts from [6] without proofs, thus making our exposition self-contained. The following lemma is a restatement of Lemma 2 from [6].…”

“…The set of positive functions in L 2 is characterized by Tessitore and Zabczyk (1998). Polyhedrons in Hilbert spaces are characterized by Milian (1998).The invariance problem with respect to weak solutions to (1), related to finitedimensional submanifolds in H is studied by Filipovic (1999). Compared to result by Jachimiak, we consider linear subspaces of H however our assumptions on coefficients are more general.…”

“…General invariant closed subsets of H are characterized by Jachimiak (1998), the set of nonnegative functions in L 2 by Kotelenez (1992), Goncharuk and Kotelenez (1996)and by Milian (1998). The set of positive functions in L 2 is characterized by Tessitore and Zabczyk (1998).…”

Forward invariance and Wong–Zakai approximation for stochastic moving boundary problems

“…The main motivation in the literature for forward invariance in the framework of mild solutions seems to come from the question of positivity of solutions for HJM interest rate models; see for example [10,25,39]. Milian [21] used Yosida approximations to extend inward pointing and parallel to the boundary conditions from finite dimensional equations to prove a comparison result for stochastic evolution equations, under Lipschitz conditions of B and C on E. These results have been extended by Filipovic et al [10], to show positivity for HJM equations provided that point-wise versions of inward pointing and parallel to the boundary conditions are satisfied.…”