This paper derives a closed-form expression for the expected discounted value of aggregate cash flows when arrival times follow a Poisson process but both the time horizon and the number of arrivals are finite. The result provides a tractable analytical formula for the expected discounted sum under simultaneous constraints on time and arrival counts. We show that the expression converges to the well-known infinite-horizon and infinite-arrival results as limiting cases. Numerical illustrations demonstrate the behavior of the formula under different parameter values. The result can be interpreted as the valuation of a discounted compound Poisson process with finite constraints and may be useful in stochastic modeling and risk-analysis applications. The proposed formula provides a simple analytical tool for evaluating discounted losses or revenues in finite risk portfolios.
Kitamura et al. (Thu,) studied this question.