Irreversible investment with fixed adjustment costs: a stochastic impulse control approach

Abstract

We consider an optimal stochastic impulse control problem over an infinite time horizon motivated by a model of irreversible investment choices with fixed adjustment costs. By employing techniques of viscosity solutions and relying on semiconvexity arguments, we prove that the value function is a classical solution to the associated quasi-variational inequality. This enables us to characterize the structure of the continuation and action regions and construct an optimal control. Finally, we focus on the linear case, discussing, by a numerical analysis, the sensitivity of the solution with respect to the relevant parameters of the problem.A.M.S. Subject Classification: 93E20 (Optimal stochastic control); 35Q93 (PDEs in connecton woth control and optimization); 35D40 (Viscosity solution); 35B65 (Smoothness and regularity of solutions).Related literature. First of all, it is worth noticing that the stochastic impulse control setting has been widely employed in several applied fields: e.g., exchange and interest rates [21,51,56], portfolio optimization with transaction costs [34,49,57], inventory and cash management [12,20,27,30,31,44,45,58,62,67,68,71], real options [47,54], reliability theory [7]. More recently, games of stochastic impulse control have been investigated with application to pollution [39].From a modeling point of view, the closest works to ours can be considered [3,6,26,35,49]. On the theoretical side, starting from the classical book [17], several works investigated QVIs associated to stochastic impulse optimal control in R n . Among them, we mention the recent [43] in a diffusion setting and [14,29] in a jump-diffusion setting. In particular, [17, Ch. 4] deals with Sobolev type solutions, whereas [43] deals with viscosity solutions. These two works prove a W 2,pregularity, with p < ∞, for the solution of QVI, which, by classical Sobolev embeddings, yields a C 1 -regularity. However, it is typically not easy to obtain by such regularity information on the structure of the so called continuation and action regions, hence on the candidate optimal control. If this structure is established, then one can try to prove a verificiation theorem to prove that the candidate optimal control is actually optimal. In a stylized one dimensional example, [43, Sec. 5] successfully employs this method by exploiting the regularity result proved in [43, Sec. 4] to depict the structure of the continuation and action region for the problem at hand. Concerning verification, we need to mention the recent paper [15], which provides a non-smooth verification theorem in a quite general setting based on the stochastic Perron method to construct a viscosity solution to QVI; also this paper, in the last section, provides and application of the results to a one dimensional problem with an implementable solution. In dimension one other approaches, based on excessive mappings and iterated optimal stopping schemes, have been successfully employed in the context of stochastic impulse control (see [3,6,35,46]). More recently, these ...

Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions

“…The equation (79) gives the formula for the mean E[τ ], and the formula for the variance E[τ 2 ] -(E[τ ]) 2 immediately follows from ( 79) and (80). The standard deviation is then obtained as Std 2 .…”

“…Before the analysis, we check their validity through a comparison with a numerical result by a standard Monte-Carlo combined with the classical Euler-Maruyama discretization of the optimally-controlled SDE. For the nominal parameter values, the formulae ( 79), (80), and (81) lead to E[τ ] = 0.5887, Std[τ ] = 1.0530, and H = 19.63. On the other hand, the numerical counterparts with the Monte-Carlo method (10 9 sample points, the The parameter dependence of the optimally-controlled dynamics on the other parameters are summarized in Table 3, suggesting monotone dependence of the statistical indices of the dynamics on the parameters.…”

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Finite volume computation for the non-stationary probability density function of an impulsively controlled 1-D diffusion process

“…Stochastic impulse control problems arise in a variety of research areas where decision-making plays a central role. Such examples include portfolio management [6,7], central bank operation [1,8,9], consumption and investment [5,10], cash Published: August 31, 2020 management [11], and recently ecological management [12,13].…”