On a model of investment strategy formation on the bonds market (Q2784977)

Let \(P_k (t;r_{k}(t);T_{k})\), \(k=1,\ldots,l,\) be bounds prices at the time \(t\); let \(r_{k}(t)\) be the rate of return of \(k\)-type bond; let \(T_{k}\) be a maturity date of \(k\)-type bond; let \(r_{k}(t)\) and \(r_{m}(t)\) be correlated; and let \(r_{k}(t), k=1,\ldots,l\) satisfy the stochastic differential equation \(d r_{k}(t)=(a_{k}-b_{k}r_{k}(t)) dt+ 2\sqrt{a_{k}r_{k}(t)} dw_{k}(t)\), where \(a_{k}, b_{k}\) are constants; \(r_{k}(0)=r_{k}^0\); \(w_{k}(t)\) is a Wiener process with \(Ew_{k}(t)=0\), \(Dw_{k}(t)=\sigma^2_{k}t\); \(E(w_{k}(t)w_{m}(t))=\rho_{km}\), \(k,m=1,\ldots,l\), \(t\in(0,\min(T_{k},T_{m}))\). The authors obtain the explicit form of \(E(P(t;r_{k}(t);T_{k}))\) and solve the minimization problem for the portfolio risk under the given mean portfolio return.