Optimal Investment with High-Watermark Fee in a Multidimensional Jump Diffusion Model

Lifetime Ruin under High-watermark Fees and Drift Uncertainty

“…In practice, there is also a benchmark profit from which the manager's performance is measured, see [27], and the high-watermark fee is deducted when the high-watermark of the fund is higher than the benchmark level. The initial high-watermark fee is denoted by some non-negative constant vector y = [y 1 , y 2 ] ⊤ .…”

“…Meanwhile, the high-watermark process is also mathematically related to wealth drawdown constraints studied in [22], [18], [20] and also discussed in [16] after the transformation into expectation constraint. Recently, the high-watermark fees have been incorporated also into Merton problem for individual investor together with consumption choice in [26] and [27]. In the presence with consumption control, analytical solutions can no longer be promised as in some of the previous work for fund managers.…”

“…The auxiliary controlled state process, the so-called process of distance to pay performance fees defined in (2.7), can no longer be absorbed to simplify the PDE problem. Furthermore, comparing with [26] and [27] or the lifetime ruin problem with ambiguity aversion in [11], we need to handle a genuine multidimensional control problem with reflections as there exist multiple hedge funds in the market. In other words, the distance process itself is already multi-dimensional, which spurs many new mathematical challenges.…”

Lifetime Ruin Under High-Water Mark Fees and Drift Uncertainty

“…Meanwhile, the high-watermark process is also mathematically related to wealth drawdown constraints studied in [22], [17], [19] and also discussed in [15] after the transformation into expectation constraint. Recently, the high-watermark fees have been incorporated also into Merton problem for individual investor together with consumption choice in [26] and [27]. In the presence with consumption control, analytical solutions can no longer be promised as in some of the previous work for fund managers.…”

“…After identifying the state processes, the path-dependent feature from high-watermark fees can be hidden so that the dynamic programming argument can be recalled to derive the HJB equation heuristically. The homogeneity of power utility function in [26] and [27] enables the key dimension reduction of the value function and the associated HJB equations can be reduced into ODE problems. Although the regularity can hardly be expected, classical Perron's method can be applied and the nice upgrade of regularity of the viscosity solution can be exercised afterwards using the convexity property of the transformed one-dimensional value function.…”