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Optimizing venture capital investments in a jump diffusion model. (English) Zbl 1151.91049

The authors address the issue of modeling the dynamics of the value process of start-up companies and/or R&D projects via jump diffusions. In order to formulate their model, the authors make use of arithmetic Brownian motion as well as a jump component. The latter is incorporated to take into account the technological breakthroughs or discovery of innovative methods and their impact on the value of the start-up company. Moreover, the associated diffusion model considers the uncertainty in cash flow generation.
The final objective of these venture capital investments is, in all likelihood, to make the company public through initial public offerings (IPOs) or to sell to a third party at a premium. However, in many cases, there are times when the start-up company has to raise new capital. At the same time, the venture capitalist has to make decisions on whether to make additional investments. This problem is referred to as the IPO problem. To address the issue of additional investments in the IPO problem, the authors solve an optimal stopping problem of a reflected jump diffusion. In solving this problem, the venture capitalist does not allow the company’s value to drop below a threshold level with little effort and attempts to find an optimal stopping time to IPO. Next, the authors solve both the problem of choosing the threshold level optimally subject to a budgetary constraint as well as its min-max version. The two scenarios considered here relate to a fixed and varying lower threshold level.
Another problem investigated in the paper is one in which the investor may want to extract values out of the company in the form of dividends until the time when the value becomes zero. The type of problem is referred to as the harvesting problem or dividend payment problem. In this problem, the optimal control policy is a mixture of singular and impulse controls.
In order to distinguish their main findings from each other, the authors provide numerical examples involving the IPO and harvesting problems.
The paper is well-written and of interest to experts in both stochastic analysis and control theory as well as their applications to institutional finance.

MSC:

91G50 Corporate finance (dividends, real options, etc.)
49N25 Impulsive optimal control problems
60G40 Stopping times; optimal stopping problems; gambling theory
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References:

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