A time-varying Markov chain model of term structure.
Term structure is a widespread financial terminology refering to the length of investment, for instance trying to explain the differences between short-term, medium-term and long-term nominal interest rates. In this time-varying Markov chain model, the instantaneous interest rate at time \(s\) is a function \(r(s,X_s)\), where \(X\) is a continuous time Markov chain on a finite state space with time-dependent transition rates: \[ \frac{d}{dt}P(X_t=i)=\sum_j a_{ij} P(X_t=j). \] The paper gives a computation of \(E(\exp(-\int_t^T r(s,X_s)\,ds))\) through differential systems. It amounts to the following. Denote \[ \varphi_i(t,T)= E\left(1_{X_T=i} \exp\Biggl(-\int_t^T r(s,X_s)\,ds \Biggr) \right), \] which is \(E(\langle V(t,T),e_i\rangle)\) in the notations of the paper. Then \[ \frac{d}{dT}\varphi_i(t,T) =\sum_j a_{ij}(T) \varphi_j(t,T) -r(T,i) \varphi_i(t,T). \]
scientific article; zbMATH DE number 1907006