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Optimal consumption and investment under transaction costs *
Author(s) -
Hobson David,
Tse Alex S. L.,
Zhu Yeqi
Publication year - 2019
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12187
Subject(s) - transaction cost , leverage (statistics) , mathematical optimization , database transaction , univariate , quadratic equation , bellman equation , investment (military) , econometrics , asset (computer security) , economics , microeconomics , computer science , mathematical economics , mathematics , multivariate statistics , statistics , computer security , geometry , politics , political science , law , programming language
In this paper, we consider the Merton problem in a market with a single risky asset and proportional transaction costs. We give a complete solution of the problem up to the solution of a first‐crossing problem for a first‐order differential equation. We find that the characteristics of the solution (e.g., well‐posedness) can be related to some simple properties of a univariate quadratic whose coefficients are functions of the parameters of the problem. Our solution to the problem via the value function includes expressions for the boundaries of the no‐transaction wedge. Using these expressions, we prove a precise condition for when leverage occurs. One new and unexpected result is that when the solution to the Merton problem (without transaction costs) involves a leveraged position, and when transaction costs are large, the location of the boundary at which sales of the risky asset occur is independent of the transaction cost on purchases.