Consider the equation below that gives interest rate dynamics in a setting where the time axis [0, T] is subdivided into n equal intervals, each of length ˆ†:

where the random error terms
ˆ†Wt = (Wt+ˆ† ˆ’Wt)
are distributed normally as
ˆ†Wt ˆ¼ N(0, ˆšˆ†)
(a) Explain the structure of the error terms in this equation. In particular, do you find it plausible that ˆ†Wtˆ’ˆ† may enter the dynamics of observed interest rates?
(b) Can you write a stochastic differential equation that will be the analog of this in continuous time? What is the difficulty?
(c) Now suppose you know, in addition, that long-term interest rates, Rt, move according to a dynamic given by

where we also know the covariance:

Can you write a representation for the vector Process

such that Xt is a first-order Markov?
(d) Can you write a continuous-time equivalent of this system?
(e) Suppose short or long rates are individually non-Markov. Is it possible that they are jointly so?

Transcribed Image Text:

+02 (W-W-A) Tt