The expectations hypothesis with non-negative rates (Q1849794)

Let \((\Omega, {\mathcal F}_{t}, {\mathcal F},P)\) be a filtered probability space and let \(W(t)\) be an \(d\)-dimensional \({\mathcal F}_{t}\)-Brownian motion. The underlying assets in the market are a saving account and a family of zero-coupon bonds. The value of the saving account at time \(t\) is given by \(B(t)= \exp\left(\int_{0}^{t}r(s)ds\right)\), where \(r(t)\) is the spot rate, measurable non-negative integrable process adapted to the filtration \({\mathcal F}_{t}\). The time \(t\) price \(P(t,T)\) of zero-coupon bond maturing at time \(T\) is described by the equation \[ dP(t,T)=P(t,T)[a(t,T)dt+b(t,T)'dW(t)], \] where \(a(t,T)\in \mathbb{R}\), \(b(t,T)\in \mathbb{R}^{d}\) are measurable and \({\mathcal F}_{t}\) adapted. The \(q\) - expectation hypothesis is the condition \[ a(t,T)-r(t)=(q/2)\|b(t,T)\|^2. \] The author constructs arbitrage free bond markets in which the \(q\) - expectation hypothesis hold and the spot rate is non-negative. He shows that these markets are in equilibrium.