Regularity Results for a Class of Semilinear Parabolic Degenerate Equations and Applications

Abstract

We prove some regularity results for viscosity solutions to strongly degenerate parabolic semilinear problems. These results apply to a specific model used for pricing Mortgage-Backed Securities and allow a complete justification of use of the classical Ito's formula.

On the strong solution of a class of partial differential equations that arise in the pricing of mortgage backed securities

“…They claim that theirs is a non linear relation between the price of an MBS, interest rates, and prepayments. Various regularity results are presented for this equation in [21]. The recent work of Sharp [15] is yet another PDE approach to mortgage pricing.…”

“…In particular, since u(x,t) + h(t) > 0 by financial constraint, that is positive price of an MBS [13,21,22], we have that…”

“…It is shown in [14,21] that (1.1) possesses well defined viscosity solutions [4]. To this end, a high degree of smoothness for the coefficients and the prepayment function h(x,t) is assumed.…”

“…To this end, a high degree of smoothness for the coefficients and the prepayment function h(x,t) is assumed. Also the quadratic non-linearity in (1.1) poses a hindrance to regularity and convergence arguments, and thus requires the authors in [14,21] to employ viscosity techniques.…”

“…In [12] we considered the following non-linear partial differential equation, describing the price evolution u(x,t), of a mortgage backed security and derived in [13,14,21]: ∂u ∂t = 1 2 ∆u + µ(x,t)∇u − ρ |σ T (x,t))∇u| 2 u + h(x,t) + ξ(t) − (r(t) − τ )h(t) − r(t)u, (x,t) ∈ R N × (0,T ),…”