Lifetime Ruin Under High-Water Mark Fees and Drift Uncertainty
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Lifetime Ruin Under High-Water Mark Fees and Drift Uncertainty Junbeom Lee1
· Xiang Yu2 · Chao Zhou3
Accepted: 20 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper aims to study lifetime ruin minimization problem by considering investment in two hedge funds with high-watermark fees and drift uncertainty. Due to multidimensional performance fees that are charged whenever each fund profit exceeds its historical maximum, the value function is expected to be multi-dimensional. New mathematical challenges arise as the standard dimension reduction cannot be applied, and the convexity of the value function and Isaacs condition may not hold in our probability minimization problem with drift uncertainty. We propose to employ the stochastic Perron’s method to characterize the value function as the unique viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation without resorting to the proof of dynamic programming principle. The required comparison principle is also established in our setting to close the loop of stochastic Perron’s method. Keywords Lifetime ruin · Multiple hedge funds · High-watermark fees · Drift uncertainty · Stochastic Perron’s method · Comparison principle
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Junbeom Lee [email protected] Xiang Yu [email protected] Chao Zhou [email protected]
1
Department of Sales and Trading, Yuanta Securities Korea, Seoul 04538, Korea
2
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
3
Department of Mathematics, Institute of Operations Research and Analytics and Suzhou Research Institute, National University of Singapore, Singapore 119076, Singapore
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Applied Mathematics & Optimization
1 Introduction Hedge funds have existed for many decades in financial markets and have become increasingly popular in recent times. As opposed to the individual investment, hedge funds pool capital and invest in a variety of assets and it is administered by professionals. Hedge fund managers charge performance fees for their service to individual investors as some regular fees proportional to fund’s component assets plus a fraction of the fund’s profits. The most common scheme entails annual fees of 2% of assets and 20% of fund profit whenever the profit exceeds its historical maximum—the so-called high-watermark. In the present paper, we are interested in investment opportunities among several hedge funds and we intend to study a stochastic control problem given the path-dependent trading frictions as multi-dimensional high-watermark fees. The existing research on high-watermark fees mainly has focused on the asset management problem from the point of view of the fund manager, see some examples by [1,21,23,24,29]. Meanwhile, the high-watermark process is also mathematically related to wealth drawdown constraints studied in [17,19,22] and also discussed in [15] after the transformation into expectation constraint. Recently, the high-watermark fees have been incorporated also into Merton proble
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