Consider the equation below that gives interest rate dynamics in a setting where the time axis [0, T] is subdivided into it equal intervals, each of length ?:
rt+? = rt + αrt + σt(Wt+? - Wt) + σ2(Wt - Wt-?)
where the random error terms
?Wt = (Wt+? - Wt)
are distributed normally as
?Wt ≈ N (0,√(?)).
(a) Explain the structure of the error terms in this equation. In particular, do you find it plausible that ?Wt-? 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?
(e) Now suppose you know, in addition, that long-term interest rates, R, move according to a dynamic given by
Rt+? = Rt + βrt + θ1(Wt+? - Wt) + θ2(Wt - Wt-?),
where we also know the covariance:
E[?W?W] = ρ?.
Can you write a representation for the vector process
such that Xt is a first-order Markov?
(d) Can you write a continuous lime equivalent of this system?
(e) Suppose short or long rates are individually non-Markov. Is it possible that they are jointlyso?