Ricerc@Sapienza

Stochastic delay differential equations (SDDE's) have been used for financial
modeling. In this article, we study a SDDE obtained by the equation of a CIR
process, with an additional fixed delay term in drift; in particular, we prove
that there exists a unique strong solution (positive and integrable) which we
call fixed delay CIR process. Moreover, for the fixed delay CIR process, we
derive a Feynman-Kac type formula, leading to a generalized exponential-affine
formula, which is used to determine a bond pricing formula when the interest
rate follows the delay's equation. It turns out that, for each maturity time T,
the instantaneous forward rate is an affine function (with time dependent
coefficients) of the rate process and of an auxiliary process (also depending
on T). The coefficients satisfy a system of deterministic delay differential
equations.