Optimizing Venture Capital Investments in a Jump Diffusion Model

We study a practical optimization problems for venture capital investments and/or Research and Development (R&D) investments. The first problem is that, given the amount of the initial investment and the reward function at the initial public offering (IPO) market, the venture capitalist wants to maximize overall discounted cash flows after subtracting subsequent (if needed) investments. We describe this problem as a mixture of singular stochastic control and optimal stopping problems and give an explicit solution. The former corresponds to finding an optimal subsequent investment policy for the purpose that the value of the investee company stays away from zero. The latter corresponds to finding an optimal stopping rule in order to maximize the harvest of their investments. The second kind problem is concerned about optimal dividend policy. Rather than selling the holding stock, the investor may extract dividends when it is appropriate. We will find a quasi-explicit optimal solution to this problem and prove the existence and uniqueness of the solution and the optimality of the proposed strategy.