arXiv subjects

Neofytos Rodosthenous

Publications and source records attributed to Neofytos Rodosthenous.

18 recordsLinked to original sources

arXiv2026

Outrunning the Omega Clock: A Singular Control Problem for Dividend Optimisation with Ruin and Time-in-Distress Default

This paper extends the classical dividend problem by incorporating a novel, path-dependent mechanism of firm default. In the traditional framework, ruin occurs when the surplus process first reaches zero. In contrast, default in our model may also arise when the surplus spends an excessive amount of time below a distress threshold, even without ever hitting zero. This occupation-time-based default criterion captures financial distress more realistically, as prolonged periods of low liquidity or capitalisation may trigger regulatory intervention or operational failure. The resulting optimisation problem is formulated as a new singular stochastic control problem with discontinuous state-dependent discounting and killing. We provide a complete analytical solution via a bespoke sequential guess-and-verify method and identify three distinct classes of optimal dividend strategies corresponding to different parameter regimes of the dual-ruin structure. Notably, for certain distress thresholds, the optimal policy features disconnected action and inaction regions. We further show that, unlike in the classical dividend problem, higher effective discounting induced by occupation time below a distress level can lead to delayed, rather than earlier, dividend payments.

math.OC↗

arXiv2025

Time-inconsistent singular control problems: Reflection and Absolutely continuous controls with exploding rates

We study a time-inconsistent singular stochastic control problem for a general one-dimensional diffusion, where time-inconsistency arises from a non-exponential discount function. To address this, we adopt a game-theoretic framework and study the optimality of a novel class of controls that encompasses both traditional singular controls -- responsible for generating multiple jumps and reflective boundaries (strong thresholds) -- and new mild threshold control strategies, which allow for the explosion of the control rate in absolutely continuous controls, thereby creating an inaccessible boundary (mild threshold) for the controlled process. We establish a general verification theorem, formulated in terms of a system of variational inequalities, that provides both necessary and sufficient conditions for equilibrium within the proposed class of control strategies and their combinations. To demonstrate the applicability of our theoretical results, we examine case studies in inventory management. We show that for certain parameter values, the problem admits a strong threshold control equilibrium in the form of Skorokhod reflection. In contrast, for other parameter values, we prove that no such equilibrium exists, necessitating the use of our extended control class. In the latter case, we explicitly construct an equilibrium using a mild threshold control strategy with a discontinuous, increasing, and exploding rate that induces an inaccessible boundary for the optimally controlled process, marking the first example of a singular control problem with such a solution structure.

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

On the Singular Control of a Diffusion and its Running Infimum or Supremum

We study a class of singular stochastic control problems for a one-dimensional diffusion $X$ in which the performance criterion to be optimised depends explicitly on the running infimum $I$ (or supremum $S$) of the controlled process. We introduce two novel integral operators that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional singular control problems. The first operator involves integrals where the integrator is the control process of the two-dimensional process $(X,I)$ or $(X,S)$; the second operator concerns integrals where the integrator is the running infimum or supremum process itself. Using these definitions, we prove a general verification theorem for problems involving two-dimensional state-dependent running costs, costs of controlling the process, costs of increasing the running infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an optimal dividend problem in which the manager's time-preferences depend on the company's historical worst performance.

math.OC↗

arXiv2024

The one-shot problem: Solution to an open question of finite-fuel singular control with discretionary stopping

We resolve a long-standing open question posed by Karatzas, Ocone, Wang, and Zervos (Stochastics, 2000) on finite-fuel singular stochastic control with discretionary stopping. Our approach introduces a novel "one-shot" solution technique, based on reduction to an auxiliary optimal stopping problem. We prove that, despite the convexity of the objective function, the waiting region of the optimal strategy need not be connected. Furthermore, we identify a qualitative transition in the optimal policy: even with arbitrarily small positive fuel, the solution can differ from the zero-fuel (pure optimal stopping) limit. These results reveal a fundamental non-robustness in such control problems, showing that zero-fuel models need not approximate behaviour under small but positive fuel constraints.

math.OC↗

arXiv2023

A Stochastic Non-Zero-Sum Game of Controlling the Debt-to-GDP Ratio

We introduce a non-zero-sum game between a government and a legislative body to study the optimal level of debt. Each player, with different time preferences, can intervene on the stochastic dynamics of the debt-to-GDP ratio via singular stochastic controls, in view of minimizing non-continuously differentiable running costs. We completely characterise Nash equilibria in the class of Skorokhod-reflection-type policies. We highlight the importance of different time preferences resulting in qualitatively different type of equilibria. In particular, we show that, while it is always optimal for the government to devise an appropriate debt issuance policy, the legislator should optimally impose a debt ceiling only under relatively low discount rates and a laissez-faire policy can be optimal for high values of the legislator's discount rate.

math.OC↗

arXiv2023

Uncertainty over Uncertainty in Environmental Policy Adoption: Bayesian Learning of Unpredictable Socioeconomic Costs

The socioeconomic impact of pollution naturally comes with uncertainty due to, e.g., current new technological developments in emissions' abatement or demographic changes. On top of that, the trend of the future costs of the environmental damage is unknown: Will global warming dominate or technological advancements prevail? The truth is that we do not know which scenario will be realised and the scientific debate is still open. This paper captures those two layers of uncertainty by developing a real-options-like model in which a decision maker aims at adopting a once-and-for-all costly reduction in the current emissions rate, when the stochastic dynamics of the socioeconomic costs of pollution are subject to Brownian shocks and the drift is an unobservable random variable. By keeping track of the actual evolution of the costs, the decision maker is able to learn the unknown drift and to form a posterior dynamic belief of its true value. The resulting decision maker's timing problem boils down to a truly two-dimensional optimal stopping problem which we address via probabilistic free-boundary methods and a state-space transformation. We completely characterise the solution by showing that the optimal timing for implementing the emissions reduction policy is the first time that the learning process has become ``decisive'' enough; that is, when it exceeds a time-dependent percentage. This is given in terms of an endogenously determined threshold function, which solves uniquely a nonlinear integral equation. We numerically illustrate our results, discuss the implications of the optimal policy and also perform comparative statics to understand the role of the relevant model's parameters in the optimal policy.

math.OC↗

arXiv2022

Two-sided Singular Control of an Inventory with Unknown Demand Trend (Extended Version)

We study the problem of optimally managing an inventory with unknown demand trend. Our formulation leads to a stochastic control problem under partial observation, in which a Brownian motion with non-observable drift can be singularly controlled in both an upward and downward direction. We first derive the equivalent separated problem under full information, with state-space components given by the Brownian motion and the filtering estimate of its unknown drift, and we then completely solve this latter problem. Our approach uses the transition amongst three different but equivalent problem formulations, links between two-dimensional bounded-variation stochastic control problems and games of optimal stopping, and probabilistic methods in combination with refined viscosity theory arguments. We show substantial regularity of (a transformed version of) the value function, we construct an optimal control rule, and we show that the free boundaries delineating (transformed) action and inaction regions are bounded globally Lipschitz continuous functions. To our knowledge this is the first time that such a problem has been solved in the literature.

math.OC↗

arXiv2021

Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities

We introduce a new non-zero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modelling the value of an asset, one player observes and can act on the process continuously, while the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a L\'evy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviour of the players' strategies as well as the quantification of the value of available exercise opportunities.

math.PR↗

arXiv2021

Artificial intelligence applied to bailout decisions in financial systemic risk management

We describe the bailout of banks by governments as a Markov Decision Process (MDP) where the actions are equity investments. The underlying dynamics is derived from the network of financial institutions linked by mutual exposures, and the negative rewards are associated to the banks' default. Each node represents a bank and is associated to a probability of default per unit time (PD) that depends on its capital and is increased by the default of neighbouring nodes. Governments can control the systemic risk of the network by providing additional capital to the banks, lowering their PD at the expense of an increased exposure in case of their failure. Considering the network of European global systemically important institutions, we find the optimal investment policy that solves the MDP, providing direct indications to governments and regulators on the best way of action to limit the effects of financial crises.

q-fin.MF↗

arXiv2020

When to sell an asset amid anxiety about drawdowns

We consider risk averse investors with different levels of anxiety about asset price drawdowns. The latter is defined as the distance of the current price away from its best performance since inception. These drawdowns can increase either continuously or by jumps, and will contribute towards the investor's overall impatience when breaching the investor's private tolerance level. We investigate the unusual reactions of investors when aiming to sell an asset under such adverse market conditions. Mathematically, we study the optimal stopping of the utility of an asset sale with a random discounting that captures the investor's overall impatience. The random discounting is given by the cumulative amount of time spent by the drawdowns in an undesirable high region, fine tuned by the investor's personal tolerance and anxiety about drawdowns. We prove that in addition to the traditional take-profit sales, the real-life employed stop-loss orders and trailing stops may become part of the optimal selling strategy, depending on different personal characteristics. This paper thus provides insights on the effect of anxiety and its distinction with traditional risk aversion on decision making.

q-fin.MF↗

arXiv2019

Optimal Control of Debt-to-GDP Ratio in an N-state Regime Switching Economy

We solve an infinite time-horizon bounded-variation stochastic control problem with regime switching between $N$ states. This is motivated by the problem of a government that wants to control the country's debt-to-GDP (gross domestic product) ratio. In our formulation, the debt-to-GDP ratio evolves stochastically in continuous time, and its drift -- given by the interest rate on government debt, net of the growth rate of GDP -- is affected by an exogenous macroeconomic risk process modelled by a continuous-time Markov chain with $N$ states. The government can act on the public debt by increasing or decreasing its level, and it aims at minimising a net expected regime-dependent cost functional. Without relying on a guess-and-verify approach, but performing a direct probabilistic study, we show that it is optimal to keep the debt-to-GDP ratio in an interval, whose boundaries depend on the states of the risk process. These boundaries are given through a zero-sum optimal stopping game with regime switching with $N$ states and are characterised through a system of nonlinear algebraic equations with constraints. To the best of our knowledge, such a result appears here for the first time. Finally, we put in practice our methodology in a case study of a Markov chain with $N=2$ states; we provide a thorough analysis and we complement our theoretical results by a detailed numerical study on the sensitivity of the optimal debt ratio management policy with respect to the problem's parameters.

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↗

arXiv2017

Beating the Omega Clock: An Optimal Stopping Problem with Random Time-horizon under Spectrally Negative Lévy Models

We study the optimal stopping of an American call option in a random time-horizon under exponential spectrally negative Lévy models. The random time-horizon is modeled as the so-called Omega default clock in insurance, which is the first time when the occupation time of the underlying Lévy process below a level $y$, exceeds an independent exponential random variable with mean $1/q>0$. We show that the shape of the value function varies qualitatively with different values of $q$ and $y$. In particular, we show that for certain values of $q$ and $y$, some quantitatively different but traditional up-crossing strategies are still optimal, while for other values we may have two disconnected continuation regions, resulting in the optimality of two-sided exit strategies. By deriving the joint distribution of the discounting factor and the underlying process under a random discount rate, we give a complete characterization of all optimal exercising thresholds. Finally, we present an example with a compound Poisson process plus a drifted Brownian motion.

q-fin.MF↗

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↗

arXiv2016

Perpetual American options in diffusion-type models with running maxima and drawdowns

We study perpetual American option pricing problems in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal exercise times are shown to be the first times at which the underlying asset hits certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. The optimal exercise boundaries for the perpetual American options on the maximum of the market depth with fixed and floating strikes are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations.

math.PR↗

arXiv2015

Robustness of the ${N}$-CUSUM stopping rule in a Wiener disorder problem

We study a Wiener disorder problem of detecting the minimum of $N$ change-points in $N$ observation channels coupled by correlated noises. It is assumed that the observations in each dimension can have different strengths and that the change-points may differ from channel to channel. The objective is the quickest detection of the minimum of the $N$ change-points. We adopt a min-max approach and consider an extended Lorden's criterion, which is minimized subject to a constraint on the mean time to the first false alarm. It is seen that, under partial information of the post-change drifts and a general nonsingular stochastic correlation structure in the noises, the minimum of $N$ cumulative sums (CUSUM) stopping rules is asymptotically optimal as the mean time to the first false alarm increases without bound. We further discuss applications of this result with emphasis on its implications to the efficiency of the decentralized versus the centralized systems of observations which arise in engineering.

math.ST↗

arXiv2014

Optimal stopping problems in diffusion-type models with running maxima and drawdowns

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

math.PR↗