μ_{u, u} - μ_{0, u} = \int_{0}^{u} c (w, w, u) d w .

Using it on the two parts of the deterministic discounted exponent yields

A (s, T) - A (0, T) = \int_{0}^{s} \int_{0}^{u} c (w, w, u) d w d u + \int_{s}^{T} \int_{0}^{u} c (s \land w, w, u) d w d u = \int_{0}^{T} \int_{0}^{u} c (s \land w, w, u) d w d u = \frac{1}{2} \int_{0}^{T} \int_{0}^{T} c (s \land u \land w, u, w) d u d w = \frac{1}{2} v (s, T) .

r \leq s \leq T

E [Z_{s, T} ∣ F_{r}] = e^{- A (s, T) - Y (r, T) + (v (s, T) - v (r, T)) /2} = e^{- A (r, T) - Y (r, T)} = Z_{r, T} .

All these prices are integrable because their exponents are Gaussian random variables with finite variance. Thus the mean restriction implies the discounted bond martingale condition, completing the equivalence of the first two descriptions.

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.