arXiv subjects

Mihail Zervos

Publications and source records attributed to Mihail Zervos.

13 recordsLinked to original sources

arXiv2025

A dynamic competitive equilibrium model of irreversible capacity investment with stochastic demand and heterogeneous producers

We formulate a continuous-time competitive equilibrium model of irreversible capacity investment in which a continuum of heterogeneous producers supplies a single non-durable good subject to exogenous stochastic demand. Each producer optimally adjusts both output and capacity over time in response to endogenous price signals, while investment decisions are irreversible. Market clearing holds continuously, with prices evolving endogenously to balance aggregate supply and demand through a constant-elasticity demand function driven by a stochastic base component. The model admits a mean-field interpretation, as each producer's decisions both influence and are influenced by the aggregate behaviour of all others. We show that the equilibrium price process can be expressed as a nonlinear functional of the exogenous base demand, leading to a three-dimensional singular stochastic control problem for each producer. We derive an explicit solution to the associated Hamilton-Jacobi-Bellman equation, including a closed-form characterisation of the free-boundary surface separating investment and waiting regions.

math.PR↗

arXiv2025

A risk-sensitive ergodic singular stochastic control problem

We consider a two-sided singular stochastic control problem with a risk-sensitive ergodic criterion. In particular, we consider a stochastic system whose uncontrolled dynamics are modelled by a linear diffusion. The control that can be applied to the system is modelled by an additive finite variation process. The objective of the control problem is to minimise a risk-sensitive long-term average criterion that penalises deviations of the controlled process from a given interval, as well as the expenditure of control effort. The stochastic control problem has been partly motivated by the problem faced by a central bank who wish to control the exchange rate between its domestic currency and a foreign currency so that this fluctuates within a suitable target zone. We derive the complete solution to the problem under general assumptions by deriving a C2 solution to its HJB equation. To this end, we use the solutions to a suitable family of Sturm-Liouville eigenvalue problems.

math.OC↗

arXiv2025

On the applicability of the maximality principle in optimal stopping: an example involving a diffusion and its running maximum

The maximality principle has proved to be a valuable tool for identifying the optimal stopping boundaries in optimal stopping problems involving one-dimensional time-homogeneous diffusions and their running maximum processes. The principle characterises the optimal stopping boundary as the maximal solution to a first-order nonlinear ODE that remains strictly below the diagonal in $\mathbb{R}^2$ or a "zero-level curve". In this paper, we construct a suitably tailored optimal stopping problem to which the maximality principle is not applicable.

math.PR↗

arXiv2025

Singular stochastic control problems motivated by the optimal sustainable exploitation of an ecosystem

We derive the explicit solutions to singular stochastic control problems of the monotone follower type with (a) an expected discounted criterion, (b) an expected ergodic criterion and (c) a pathwise ergodic criterion. These problems have been motivated by the optimal sustainable exploitation of an ecosystem, such as a natural fishery. Under general assumptions on the diffusion coefficients, the discounting rate function, the running payoff function and the marginal profit of control action, we show that the optimal strategies are of a threshold type. We solve the three problems by first constructing suitable solutions to their associated HJB equations, which take the form of quasi-variational inequalities with gradient constraints. In the cases of the ergodic control problems, we also use a suitable new variational argument. Furthermore, we establish the convergence of the solution of the discounted control problem to the one of the ergodic control problems as the discounting rate function tends to 0 in an Abelian sense.

math.OC↗

arXiv2024

The Solution to an Impulse Control Problem Motivated by Optimal Harvesting

We consider a stochastic impulse control problem that is motivated by applications such as the optimal exploitation of a natural resource. In particular, we consider a stochastic system whose uncontrolled state dynamics are modelled by a non-explosive positive linear diffusion. The control that can be applied to this system takes the form of one-sided impulsive action. The objective of the control problem is to maximise a discounted performance criterion that rewards the effect of control action but involves a fixed cost at each time of a control intervention. We derive the complete solution to this problem under general assumptions. It turns out that the solution can take four qualitatively different forms, several of which have not been observed in the literature. In two of the four cases, there exist only $\varepsilon$-optimal control strategies. We also show that the boundary classification of 0 may play a critical role in the solution of the problem. Furthermore, we develop a way for establishing the strong solution to a stochastic impulse control problem's optimally controlled SDE.

math.OC↗

arXiv2019

Discretionary stopping of stochastic differential equations with generalised drift

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem's value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem's data. Next, we derive the complete explicit solution to the problem that arises when the state process is a skew geometric Brownian motion and the reward function is the one of a financial call option. In this case, we show that the optimal stopping strategy can take several qualitatively different forms, depending on parameter values. Furthermore, the explicit solution to this special case reveals that the so-called "principle of smooth fit" does not hold in general for this type of optimal stopping problems in standard senses that this can be formulated.

math.PR↗

arXiv2018

An Investment Model with Switching Costs and the Option to Abandon

We develop a complete analysis of a general entry-exit-scrapping model. In particular, we consider an investment project that operates within a random environment and yields a payoff rate that is a function of a stochastic economic indicator such as the price of or the demand for the project's output commodity. We assume that the investment project can operate in two modes, an "open" one and a "closed" one. The transitions from one operating mode to the other one are costly and immediate, and form a sequence of decisions made by the project's management. We also assume that the project can be permanently abandoned at a discretionary time and at a constant sunk cost. The objective of the project's management is to maximise the expected discounted payoff resulting from the project's management over all switching and abandonment strategies. We derive the explicit solution to this stochastic control problem that involves impulse control as well as discretionary stopping. It turns out that this has a rather rich structure and the optimal strategy can take eight qualitatively different forms, depending on the problems data.

math.OC↗

arXiv2017

Valuation of Employee Stock Options (ESOs) by means of Mean-Variance Hedging

We consider the problem of ESO valuation in continuous time. In particular, we consider models that assume that an appropriate random time serves as a proxy for anything that causes the ESO's holder to exercise the option early, namely, reflects the ESO holder's job termination risk as well as early exercise behaviour. In this context, we study the problem of ESO valuation by means of mean-variance hedging. Our analysis is based on dynamic programming and uses PDE techniques. We also express the ESO's value that we derive as the expected discounted payoff that the ESO yields with respect to an equivalent martingale measure, which does not coincide with the minimal martingale measure or the variance-optimal measure. Furthermore, we present a numerical study that illustrates aspects or our theoretical results.

q-fin.PR↗

arXiv2016

Necessary and sufficient conditions for the $r$-excessive local martingales to be martingales

We consider the decreasing and the increasing $r$-excessive functions $φ_r$ and $ψ_r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $α< β$. We prove that the $r$-excessive local martingale $\bigl( e^{-r (t \wedge T_α)} φ_r (X_{t \wedge T_α}) \bigr)$ $\bigl($resp., $\bigl( e^{-r (t \wedge T_β)} ψ_r (X_{t \wedge T_β}) \bigr) \bigr)$ is a strict local martingale if the boundary point $α$ (resp., $β$) is inaccessible and entrance, and a martingale otherwise.

math.PR↗

arXiv2016

Watermark Options

We consider a new family of derivatives whose payoffs become strictly positive when the price of their underlying asset falls relative to its historical maximum. We derive the solution to the discretionary stopping problems arising in the context of pricing their perpetual American versions by means of an explicit construction of their value functions. In particular, we fully characterise the free-boundary functions that provide the optimal stopping times of these genuinely two-dimensional problems as the unique solutions to highly non-linear first order ODEs that have the characteristics of a separatrix. The asymptotic growth of these free-boundary functions can take qualitatively different forms depending on parameter values, which is an interesting new feature.

math.PR↗

arXiv2015

A zero-sum game between a singular stochastic controller and a discretionary stopper

We consider a stochastic differential equation that is controlled by means of an additive finite-variation process. A singular stochastic controller, who is a minimizer, determines this finite-variation process, while a discretionary stopper, who is a maximizer, chooses a stopping time at which the game terminates. We consider two closely related games that are differentiated by whether the controller or the stopper has a first-move advantage. The games' performance indices involve a running payoff as well as a terminal payoff and penalize control effort expenditure. We derive a set of variational inequalities that can fully characterize the games' value functions as well as yield Markovian optimal strategies. In particular, we derive the explicit solutions to two special cases and we show that, in general, the games' value functions fail to be $C^1$. The nonuniqueness of the optimal strategy is an interesting feature of the game in which the controller has the first-move advantage.

math.PR↗

arXiv2012

On the Optimal Stopping of a One-dimensional Diffusion

We consider a one-dimensional diffusion which solves a stochastic differential equation with Borel-measurable coefficients in an open interval. We allow for the endpoints to be inaccessible or absorbing. Given a Borel-measurable function $r$ that is uniformly bounded away from 0, we establish a new analytic representation of the $r$-potential of a continuous additive functional of the diffusion. We also characterize the value function of an optimal stopping problem with general reward function as the unique solution of a variational inequality (in the sense of distributions) with appropriate growth or boundary conditions. Furthermore, we establish several other characterisations of the solution to the optimal stopping problem, including a generalisation of the so-called "principle of smooth fit".

math.PR↗

arXiv2007

A Singular Control Model with Application to the Goodwill Problem

We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a performance criterion that rewards high values of the utility derived from the system's controlled state but penalises any expenditure of control effort. This problem has been motivated by applications such as the so-called goodwill problem in which the system's state is used to represent the image that a product has in a market, while control expenditure is associated with raising the product's image, e.g., through advertising. We obtain the solution to the optimisation problem that we consider in a closed analytic form under rather general assumptions. Also, our analysis establishes a number of results that are concerned with analytic as well as probabilistic expressions for the first derivative of the solution to a second order linear non-homogeneous ordinary differential equation. These results have independent interest and can potentially be of use to the solution of other one-dimensional stochastic control problems.

math.PR↗