Long Time Asymptotics for Optimal Investment

Abstract

This survey reviews portfolio selection problem for long-term horizon. We consider two objectives: (i) maximize the probability for outperforming a target growth rate of wealth process (ii) minimize the probability of falling below a target growth rate. We study the asymptotic behavior of these criteria formulated as large deviations control problems, that we solve by duality method leading to ergodic risk-sensitive portfolio optimization problems. Special emphasis is placed on linear factor models where explicit solutions are obtained. (2000): 60F10, 91G10, 93E20. MSC Classification

Discrete‐time risk sensitive portfolio optimization with proportional transaction costs

“…Second, (1) is directly linked to so-called equivalent safe rate, which reflects the minimal hypothetical safe rate that would encourage the investor to invest in the risky portfolio, see Guasoni et al (2019). Third, for 𝛾 < 0, the risk-sensitive criterion is dual to the downside risk, which is a common investment criterion in the long-run portfolio optimization, see Nagai (2012) or Pham (2015) for details. Fourth, for 𝛾 > 0, the maximization of (1) is related to the studies of power utility asymptotics and can be considered as a dual problem to upside chance probability, see Pham (2015) and Stettner (2011).…”

“…(2019). Third, for $$\gamma &lt;0$$, the risk‐sensitive criterion is dual to the downside risk , which is a common investment criterion in the long‐run portfolio optimization, see Nagai (2012) or Pham (2015) for details. Fourth, for $$\gamma &gt; 0$$, the maximization of (1) is related to the studies of power utility asymptotics and can be considered as a dual problem to upside chance probability , see Pham (2015) and Stettner (2011).…”

“…Third, for $$\gamma &lt;0$$, the risk‐sensitive criterion is dual to the downside risk , which is a common investment criterion in the long‐run portfolio optimization, see Nagai (2012) or Pham (2015) for details. Fourth, for $$\gamma &gt; 0$$, the maximization of (1) is related to the studies of power utility asymptotics and can be considered as a dual problem to upside chance probability , see Pham (2015) and Stettner (2011). Fifth, let us remark that risk sensitive criterion is an acceptability index (also called performance measure ) and has many economically desirable properties, see for example, Cherny and Madan (2009); Bielecki et al.…”

LONG-TERM GROWTH RATE OF EXPECTED UTILITY FOR LEVERAGED ETFs: MARTINGALE EXTRACTION APPROACH

“…Hata and Sekine (2006) studied a long-term optimal investment problem with an objective of maximizing the probability that the portfolio value would exceeed a given level in a market with Cox-Ingersoll-Ross interest rate. Applying the theory of large deviation, Pham (2003) derived the optimal long-term investment strategy under the CARA utility, and Pham (2015) examined the long-term asymptotics for optimal portfolios that involved maximizing the probability for a portfolio to outperform a target growth rate.…”