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The objective of this paper is to study the optimal consumption and portfolio choice problem of risk-controlled investors who strive to maximize total expected discounted utility of both consumption and terminal wealth. Risk is measured by the variance of terminal wealth, which introduces a nonlinear function of the expected value into the control problem. The control problem presented is no longer a standard stochastic control problem but rather, a mean field-type control problem. The optimal portfolio and consumption rules are obtained explicitly. Numerical results shed light on the importance of controlling variance risk. The optimal investment policy is nonmyopic, and consumption is not sacrificed.
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This content will become publicly available on September 1, 2026
On Merton's optimal portfolio problem with sporadic bankruptcy for isoelastic utility
In this paper, we consider a stock that follows a geometric Brownian motion (GBM) and a riskless asset continuously compounded at a constant rate. We assume that the stock can go bankrupt, i.e. lose all of its value, at some exogenous random time (independent of the stock price) modeled as the first arrival time of a homogeneous Poisson process. For this setup, we study Merton’s optimal portfolio problem consisting in maximizing the expected isoelastic utility of the total wealth at a given finite maturity time. We obtain an analytical solution using coupled Hamilton–Jacobi–Bellman (HJB) equations. The optimal strategy bans borrowing and never allocates more wealth into the stock than the classical Merton ratio recommends. For nonlogarithmic isoelastic utilities, the optimal weights are nonmyopic. This is an example where a realistic problem, being merely a slight modification of the usual GBM model, leads to nonmyopic weights. For logarithmic utility, we additionally present an alternative derivation using a stochastic integral and verify that the weights obtained are identical to our first approach. We also present an example for our strategy applied to a stock with nonzero bankruptcy probability.
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- Award ID(s):
- 2402544
- PAR ID:
- 10657305
- Publisher / Repository:
- World Scientific Publishing
- Date Published:
- Journal Name:
- International Journal of Theoretical and Applied Finance
- Volume:
- 28
- Issue:
- 05n06
- ISSN:
- 0219-0249
- Subject(s) / Keyword(s):
- Markov-switching processes absorbing processes Hamilton–Jacobi–Bellman equation nonmyopic strategies isoelastic utility
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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