Abstract
This paper deals with an investment–consumption portfolio problem when the current utility depends also on the wealth process. Such problems arise e.g. in portfolio optimization with random horizon or random trading times. To overcome the difficulties of the problem, a dual approach is employed: a dual control problem is defined and treated by means of dynamic programming, showing that the viscosity solutions of the associated Hamilton–Jacobi–Bellman equation belong to a suitable class of smooth functions. This allows defining a smooth solution of the primal Hamilton–Jacobi–Bellman equation, and proving by verification that such a solution is indeed unique in a suitable class of smooth functions and coincides with the value function of the primal problem. Applications to specific financial problems are given.
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Notes
We also should mention the paper [24], where the HJB equation associated to the dual problem is again fully nonlinear, but admits a semi-explicit solution in the form of a power series.
One could try to prove the continuity of V, then show that V is a viscosity solution of the HJB equation, and finally use quite standard analytical techniques to prove a comparison in the viscosity sense for the equation and therefore get uniqueness for it (see e.g. [6, 14, 25]). Otherwise one could try to drop the proof of continuity and deal with discontinuous viscosity solutions, for which the comparison is a bit harder to prove (see [14, Chap. VII]), and then prove continuity a posteriori as a consequence of the characterization as a viscosity solution. We do not do that since our study of the dual HJB equation will be sufficient to prove a characterization of V as a classical solution to the HJB equation within a suitable class of smooth functions. Our uniqueness result is weaker than what can be obtained by the viscosity approach, but will be enough for our purposes.
To this regard, we should mention e.g. [5, 8, 26] for direct results in this direction, when the problem is autonomous and over an infinite horizon, and the equation elliptic. Up to our knowledge, despite a sketch in [26], there are no results of this kind for parabolic HJB equations coming from investment–consumption problems—as the one we deal with in this paper.
Appealing to the dynamic programming principle may seem somehow unfair, as it is usually problematic to prove it if one has not proved before the continuity of the value function (and we are just proving the continuity by invoking it). However, we observe that in this case (where the time t′ is deterministic), the proof of the dynamic programming principle (see e.g. [25, Chap. 4]) only uses the continuity in the space variable y.
This inequality may be also proved using (3.15) and the concavity of V in x which could be proved directly.
The authors are indebted to an anonymous referee who suggested this alternative approach.
See [10] for the rewriting of the term corresponding to G 2 in the general case when τ may be dependent on \(\mathcal {F}_{T}\), in which case one has to consider \(F(t) :=\mathbb{P}\{\tau\leq t\mid \mathcal{F}_{t}\}\).
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This research has been partially supported by the PRIN research project “Metodi deterministici e stocastici nello studio di problemi di evoluzione” of the Italian Minister of University and Research. The authors thank two anonymous referees and the editors for careful scrutiny and useful suggestions that allowed substantially improving the paper.
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Federico, S., Gassiat, P. & Gozzi, F. Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation. Finance Stoch 19, 415–448 (2015). https://doi.org/10.1007/s00780-015-0257-z
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DOI: https://doi.org/10.1007/s00780-015-0257-z
Keywords
- Optimal stochastic control
- Investment–consumption problem
- Duality
- Hamilton–Jacobi–Bellman equation
- Regularity of viscosity solutions