Impact of Time Illiquidity in a Mixed Market Without Full Observation

Abstract

We study a problem of optimal investment/consumption over an infinite horizon in a market with two possibly correlated assets: one liquid and one illiquid. The liquid asset is observed and can be traded continuously, while the illiquid one can be traded only at discrete random times, corresponding to the jumps of a Poisson process with intensity λ, is observed at the trading dates, and is partially observed between two different trading dates. The problem is a nonstandard mixed discrete/continuous optimal control problem, which we solve by a dynamic programming approach. When the utility has a general form, we prove that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and characterize the optimal allocation in the illiquid asset. In the case of power utility, we establish the regularity of the value function needed to prove the verification theorem, providing the complete theoretical solution of the problem. This enables us to perform numerical simulations, so as to analyze the impact of time illiquidity and how this impact is affected by the degree of observation.

Viscosity Characterization of the Value Function of an Investment-Consumption Problem in Presence of an Illiquid Asset

“…We observe, as expected, a monotone convergence to the constrained Merton problem (see also [4] for comments). The difference between the different values of λ can be taken as an absolute measure of the cost of illiquidity.…”

Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation

“…A related application of our results is the mixed liquid/illiquid investment model studied in [12,13]. We refer to the latter references for details on the model.…”

Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

“…by the boundary condition (28). The continuity of p N at x = x gives c 1 = S −1 c. We then get p W (x) = Fc(e S −1 (x−x) − e S −1 (x−x) ).…”

“…26 This partial observation framework leads to a control problem subject to discrete and random observations that has been used in some financial models under restricted observation chances. [27][28][29] Another difference between the ideal and real problems is the execution delay. Interventions that have much shorter time-scales than those of the target dynamics are reasonably considered to be impulsive.…”