Irreversible Capital Accumulation with Economic Impact

Abstract

We consider an irreversible capacity expansion model in which additional investment has a strictly negative effect on the value of an underlying stochastic economic indicator. The associated optimisation problem takes the form of a singular stochastic control problem that admits an explicit solution. A special characteristic of this stochastic control problem is that changes of the state process due to control action depend on the state process itself in a proportional way.

Optimal installation of renewable electricity sources: the case of Italy

“…This is also analogous to other papers modeling price impact: for example, in problems of optimal execution, Becherer et al (2017) and Becherer et al (2018) take into account a multiplicative and transient price impact, whereas (Guo and Zervos 2015) consider an exponential parametrization in a geometric Brownian motion setting allowing for a permanent price impact. Also, a price impact model has been studied by Al Motairi and Zervos (2017), motivated by an irreversible capital accumulation problem with permanent price impact, and by Ferrari and Koch (2019), in which the authors consider an extraction problem with Ornstein-Uhlenbeck dynamics and transient price impact. In all of the aforementioned papers on price impact models dealing with singular stochastic controls (Al Motairi and Zervos 2017;Becherer et al 2017Becherer et al , 2018Ferrari and Koch 2019;Guo and Zervos 2015), the agents' actions can lead to an immediate jump in the underlying price process, whereas in our setting, it cannot.…”

“…Also, a price impact model has been studied by Al Motairi and Zervos (2017), motivated by an irreversible capital accumulation problem with permanent price impact, and by Ferrari and Koch (2019), in which the authors consider an extraction problem with Ornstein-Uhlenbeck dynamics and transient price impact. In all of the aforementioned papers on price impact models dealing with singular stochastic controls (Al Motairi and Zervos 2017;Becherer et al 2017Becherer et al , 2018Ferrari and Koch 2019;Guo and Zervos 2015), the agents' actions can lead to an immediate jump in the underlying price process, whereas in our setting, it cannot. Our model is instead analogous to , , which show how to incorporate a market impact due to cross-border trading in electricity markets, and to Rowińska et al (2018), which models the price impact of wind electricity production on power prices.…”

Optimal installation of renewable electricity sources: the case of Italy

“…Also, the high lifespan of renewable power plants and the underlying infinite time horizon setting allow us to neglect the seasonal patterns. We therefore assume that the electricity's fundamental price has solely a mean-reverting behavior, and evolves according to an Ornstein-Uhlenbeck (O-U) process 1 . We are also neglecting the stochastic and seasonal effects of renewable power production.…”

“…This is also analogous to other papers modelling price impact: for example, in problems of optimal execution, [2] and [3] take into account a multiplicative and transient price impact, whereas [22] considers an exponential parametrization in a geometric Brownian motion setting allowing for a permanent price impact. Also, a price impact model has been studied by [1], motivated by an irreversible capital accumulation problem with permanent price impact, and by [18], in which the authors consider an extraction problem with Ornstein-Uhlenbeck dynamics and transient price impact. In all of the aforementioned papers on price impact models dealing with singular stochastic controls [1,2,3,18,22], the agents' actions can lead to an immediate jump in the underlying price process, whereas in our setting, it cannot.…”

“…Also, a price impact model has been studied by [1], motivated by an irreversible capital accumulation problem with permanent price impact, and by [18], in which the authors consider an extraction problem with Ornstein-Uhlenbeck dynamics and transient price impact. In all of the aforementioned papers on price impact models dealing with singular stochastic controls [1,2,3,18,22], the agents' actions can lead to an immediate jump in the underlying price process, whereas in our setting, it cannot. Our model is instead analogous to [10,11], which show how to incorporate a market impact due to cross-border trading in electricity markets, and to [26], which models the price impact of wind electricity production on power prices.…”