A Convex Stochastic Optimization Problem Arising From Portfolio Selection

Abstract

A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solving a (static) convex stochastic optimization problem in terms of the terminal wealth, with a budget constraint. In literature the latter is solved by assuming a priori that the problem is well-posed (i.e., the supremum value is finite) and a Lagrange multiplier exists (and as a consequence the optimal solution is attainable). In this paper it is first shown that, via various counter-examples, neither of these two assumptions needs to hold, and an optimal solution does not necessarily exist. These anomalies in turn have important interpretations in and impacts on the portfolio selection modeling and solutions. Relations among the non-existence of the Lagrange multiplier, the ill-posedness of the problem, and the non-attainability of an optimal solution are then investigated. Finally, explicit and easily verifiable conditions are derived which lead to finding the unique optimal solution.

Behavioral portfolio selection with loss control

“…Compared with , the positive part problem (8) here is unaffected by the loss constraint L; and a more general version of it has been solved already in Jin and Zhou (2008), Appendix C. The only difference here lies in the negative part problem (9) where there is an additional upper bound constraint on X. This, certainly, is due to the loss control in the original model.…”

“…The positive part problem (8) in this example (where A = {ρ ≤ c}) has been solved explicitly by with the following results:…”

“…We define the optimal value of Problem (8), denoted v + (A, x + ), in the following way. If P (A) > 0, in which case the feasible set of (8) is non-empty (X = x + 1 A /ρP (A) is a feasible solution satisfying all the constraints), then v + (A, x + ) is defined to be the supremum of (8). …”

Portfolio selection of a closed-end mutual fund

“…The unconstrained problem (5.15), which is referred as the static reformulation of Merton's problem in complete markets, has been heavily investigated in the literature with slight differences; see, e.g., Pliska (1986), Cox and Huang (1989), Karatzas et al (1987) and Jin et al (2008). For extensions to incomplete markets, see He and Pearson (1991), Karatzas et al (1991), among others.…”

Weak Insider Trading and Behavioral Finance

“…Moreover, the properties of the minimal probability in [1, Remark 4], especially properties 1. and 3., make it desirable to use Q ν in Problem (CPT-I). 5 Furthermore, dropping the distinction between losses and gains and probability distortions as well, we recover Problem (EU-I). On the other hand, in the extreme case of no additional information ν = Q Y we have V ν ± (X) = V ± (X) thanks to minimal probability's property 3 in [1, Remark 4], so we turn back to Problem (CPT-N).…”