### Constrained Dynamic Optimality and Binomial Terminal Wealth

Authors: Jesper Lund Pedersen, Goran Peskir

Journal: SIAM Journal on Control and Optimization

Publication Date: 10 April, 2018

Department of: Mathematics

### In Abstract

**A time-consistent solution to the Markowitz problem**

Imagine an investor who has an initial wealth which he wishes to exchange between a risky stock and a riskless bank account, in a self-financing manner, dynamically in time, so as to minimise his risk (variance) in obtaining a desired return at the given terminal time.

Now, researchers at the University of Copenhagen and the University of Manchester have derived dynamically optimal strategies that are the first known time-consistent trading strategies that are optimal in minimising risk when the wealth is being prevented from going below a desired tolerance level.

The new methodology for solving such nonlinear control problems rests on the concept of dynamic optimality which consists of continuous rebalancing of optimal controls upon overruling all the past controls. The binomial nature of the new trading strategies stands in sharp contrast with other known trading strategies encountered in the literature. A direct comparison shows that the dynamically optimal (time-consistent) strategy outperforms the previously developed statically optimal (time-inconsistent) strategy in solving the problem.

- Markowitz solved the problem in a one-period model in 1952. He received the Nobel prize in economics for this work in 1990. The problem of finding a time-consistent solution in continuous time, now solved by this research, had been open since the 1950s.