Integral option

In the context of diffusion model of the $(B,S)$-market consisting of two assets: riskless bank account $B=(B_t)_{t\ge 0}$ and risky stock $S=(S_t)_{t\ge 0}$ described by (1.1) and (1.2) we consider the option of American type with payment function of “integral type” $f=(f_t)_{t\ge 0}$:
$$ f_t=e^{-\lambda t}\left[\int_0^t S^u\,du+s\psi_0\right]. $$
The paper solves the problem of definition of the fair price of the integral option under consideration. The structure of the expiration time is also described.