On Optimal Terminal Wealth Problems with Random Trading Times and Drawdown Constraints

Abstract

We consider an investment problem where observing and trading are only possible at random times. In addition, we introduce drawdown constraints which require that the investor's wealth does not fall under a prior fixed percentage of its running maximum. The financial market consists of a riskless bond and a stock which is driven by a Lévy process. Moreover, a general utility function is assumed. In this setting we solve the investment problem using a related limsup Markov decision process. We show that the value function can be characterized as the unique fixed point of the Bellman equation and verify the existence of an optimal stationary policy. Under some mild assumptions the value function can be approximated by the value function of a contracting Markov decision process. We are able to use Howard's policy improvement algorithm for computing the value function as well as an optimal policy. These results are illustrated in a numerical example.

The maximal drawdown of the Brownian meander

“…where X(s) := sup u∈[0, s] X(u). There has been some recent research interest on the study of drawdowns from probabilistic point of view ( [7], [8]) as well as applications in insurance and finance ( [1], [2], [3], [10], [12]).…”

Path-dependent Risk Measures - Theory and Applications

“…For example, Grossman and Zhou (1993) incorporate a drawdown constraint into a continuous-time investment problem in which a specified drawdown must not be exceeded at any time. This application has attracted significant attention in the literature, including by Cvitanic and Karatzas (1995), Alexander and Baptista (2006), Elie and Touzi (2008), Sekine (2013), Yao et al (2013), Cherny and Obłój (2013), Rieder andWittlinger (2014), Angoshtari et al (2016), Kardaras et al (2017), andRoche (2019), whereas the drawdown constraint has been generalized, other constraints have been added, and results have been extended to a phenomenon continues to exist for a prolonged period, questions of persistence have been addressed in vastly different fields of finance and economics. These include the persistence of inflation (Pivetta and Reis, 2007), the persistence of firm capital structure (Lemmon et al, 2008), the persistence of bank profits (Goddard et al, 2011), the persistence of executive compensation (Cheng et al, 2015), and the persistence of earnings, cash flows, and accruals (Hui et al, 2016).…”

“…These constraints are intended to ensure that an investment's value never falls below a fixed percentage of the running maximum at any time. Subsequent work on investment problems with a drawdown constraint includes Cvitanic and Karatzas (1995), Alexander and Baptista (2006), Elie and Touzi (2008), Sekine (2013), Yao et al (2013), Cherny and Obłój (2013), Rieder and Wittlinger (2014), Angoshtari et al (2016), Kardaras et al (2017), and Roche (2019). A second strand of literature addresses mathematical properties of the drawdown process.…”