Power Utility

Optimal Investment and Consumption in a Stochastic Factor Model ArXiv ID: 2509.09452 “View on arXiv” Authors: Florian Gutekunst, Martin Herdegen, David Hobson Abstract In this article, we study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem, and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an Itô diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including – for the first time – the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme. ...

A greedy algorithm for habit formation under multiplicative utility ArXiv ID: 2305.04748 “View on arXiv” Authors: Unknown Abstract We consider the problem of optimizing lifetime consumption under a habit formation model, both with and without an exogenous pension. Unlike much of the existing literature, we apply a power utility to the ratio of consumption to habit, rather than to their difference. The martingale/duality method becomes intractable in this setting, so we develop a greedy version of this method that is solvable using Monte Carlo simulation. We investigate the behaviour of the greedy solution, and explore what parameter values make the greedy solution a good approximation to the optimal one. ...