Bounded Variation Control of Itô Diffusions with Exogenously Restricted Intervention Times

Abstract

In this paper, bounded variation control of one-dimensional diffusion processes is considered. We assume that the agent is allowed to control the diffusion only at the jump times of an observable, independent Poisson process. The agent's objective is to maximize the expected present value of the cumulative payoff generated by the controlled diffusion over its lifetime. We propose a relatively weak set of assumptions on the underlying diffusion and the instantaneous payoff structure, under which we solve the problem in closed form. Moreover, we illustrate the main results with an explicit example.

Ergodic control of diffusions with random intervention times

“…Proof. The proof of the claim on L is similar to that of Lemma 3.3 in [25]. However, to show that the results are in accordance with the remark 1, we need to adjust the argument on finding a point x 1 < x * such that L πµ (x 1 ) < 0.…”

“…which can be derived using representation (1) and straightforward differentiation (see [25] lemma 3 for details). The claim on I follows similarly as differentiation yields…”

“…Otherwise, when there are no interventions, we assume that the system evolves according to one-dimensional linear diffusion X that is independent of the Poisson process. In literature these types of restriction processes on the controllability of X are often referred as constraints or signals, see [23,25,28,29,30].…”

“…These times can be for example multiples of integers as in [33,22] or given by a signal process. Often, as in our model, the times between the arrivals of the signal process are assumed to be exponentially distributed, see [34,36,25]. In [34] this framework was used as a simple model for liquidity effects in a classical investment optimization problem.…”

“…In [34] this framework was used as a simple model for liquidity effects in a classical investment optimization problem. [36] investigates both discounted cost and ergodic cost criterion while tracking a Brownian motion under quadratic cost and [25] generalizes the discounted problem to a more general payoff and underlying structure. Related studies in optimal stopping are [11,23].…”