Mathematical model of investment in bonds (Q2703274)

The authors prove an existence and uniqueness theorem for the solution of the equation NEWLINE\[NEWLINE\int_{a}^{t}u(x)\exp\{a(t-x)\} dx= \int_{a}^{t}S(x)\exp\{a(t-x)\} dx, \quad \forall t\in[a,b],NEWLINE\]NEWLINE where \(a(t)\) is an unknown function connected with the discount coefficient; \(u(\tau)\) is the investment density which satisfies the relation NEWLINE\[NEWLINEu(\tau)=\sum\limits_{v\in V^{n}(t)}\sum\limits_{k=1}^{k_{v}} P_{v}(t_{vk})x_{v}(t_{vk})\delta(\tau-t_{vk}),\quad \tau\leq t,NEWLINE\]NEWLINE \(V^{n}(t)\) is the set of bonds in the portfolio; \(t_{vk}, k= 1,\dots, k_{v}\) are the buying times for the bonds of the \(v\)-th type; \(P_{v}(t_{vk})\) is the buying price of the \(v\)-th bond at the time \(t_{vk}\); \(x_{v}(t_{vk})\) is the number of bonds of \(v\)-th type bought at the time \(t_{vk}\); \(\delta(\cdot)\) is the Dirac function; \(S(\tau)\) is the payment density for the purchased bonds.