31 research outputs found
On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes
No full textWe obtain closed-form expressions for the value of the joint Laplace transform of therunning maximum and minimum of a diffusion-type process stopped at the first time at which theassociated drawdown or drawup process hits a constant level before an independent exponentialrandom time. It is assumed that the coefficients of the diffusion-type process are regular functionsof the current values of its running maximum and minimum. The proof is based on the solution tothe equivalent inhomogeneous ordinary differential boundary-value problem and the applicationof the normal-reflection conditions for the value function at the edges of the state space of theresulting three-dimensional Markov process. The result is related to the computation of probabilitycharacteristics of the take-profit and stop-loss values of a market trader during a given time period.</jats:p
On the Singular Control of a Diffusion and Its Running Infimum or Supremum
No full textFerrari G, Rodosthenous N. On the Singular Control of a Diffusion and Its Running Infimum or Supremum. Center for Mathematical Economics Working Papers. Vol 745. Bielefeld: Center for Mathematical Economics; 2025.We study a class of singular stochastic control problems for a one-dimensional
diffusion in which the performance criterion to be optimised depends explicitly on the running
infimum (or supremum ) 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 or ; 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.MSC2020 subject classification: 93E20; 60J60; 49L12; 91B7
A Stochastic Non-zero-Sum Game of Controlling the Debt-to-GDP Ratio
No full textDammann F, Rodosthenous N, Villeneuve S. A Stochastic Non-zero-Sum Game of Controlling the Debt-to-GDP Ratio. Applied Mathematics and Optimization . 2024;90(3): 52.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
A Stochastic Non-Zero-Sum Game of Controlling the Debt-to-GDP Ratio
No full textDammann F, Rodosthenous N, Villeneuve S. A Stochastic Non-Zero-Sum Game of Controlling the Debt-to-GDP Ratio. Center for Mathematical Economics Working Papers. Vol 730. Bielefeld: Center for Mathematical Economics; 2024.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 minimiz-
ing 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 opti-
mally 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.MSC2010 subject classification: 91A05, 93E20, 90B0
Uncertainty over Uncertainty in Environmental Policy Adoption: Bayesian Learning of Unpredictable Socioeconomic Costs
No full textBasei M, Ferrari G, Rodosthenous N. Uncertainty over Uncertainty in Environmental Policy Adoption: Bayesian Learning of Unpredictable Socioeconomic Costs. Center for Mathematical Economics Working Papers. Vol 677. Bielefeld: Center for Mathematical Economics; 2023.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
show 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 uniquely solving a
nonlinear integral equation, which we can solve numerically. We 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.OR/MS subject classification: Environment: Pollution; Probability: Stochastic model ap-
plications; Dynamic programming/optimal control: Applications, Markov, Model
Optimal Control of Debt-To-GDP Ratio in an N-State Regime Switching Economy
No full textFerrari G, Rodosthenous N. Optimal Control of Debt-To-GDP Ratio in an N-State Regime Switching Economy. Center for Mathematical Economics Working Papers. Vol 589 Aktual. Version Februar 2019. Bielefeld: Center for Mathematical Economics; 2019.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 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
we completely characterise them as solutions to 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
Uncertainty over uncertainty in environmental policy adoption: Bayesian learning of unpredictable socioeconomic costs
No full textBasei M, Ferrari G, Rodosthenous N. Uncertainty over uncertainty in environmental policy adoption: Bayesian learning of unpredictable socioeconomic costs. Journal of Economic Dynamics and ControlL. 2024;161: 104841.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 onceand -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
Regulation in a Mean-Field Investment Game with Climate Damage
No full textAid R, Federico S, Ferrari G, Rodosthenous N. Regulation in a Mean-Field Investment Game with Climate Damage. Center for Mathematical Economics Working Papers. Vol 705. Bielefeld: Center for Mathematical Economics; 2025.We study the problem of optimal investment in brown (carbon-intensive) production amid climate change and the impact of rising global temperatures. Our approach is based
on a mean-field model of firms that produce goods whose productivity is adversely affected by
temperature-related damages, which are in turn linked to the global stock of greenhouse gas
(GHG) emissions. Each firm controls its investment rate in view of increasing its capital stock,
which evolves stochastically due to idiosyncratic Gaussian shocks and is subject to exponential
depreciation in the absence of investment. Firms aim to maximize their expected discounted
profits, net of investment costs, by choosing investment strategies that respond to the level of
aggregate GHG emissions and their adverse impact. We constructively establish the existence
and uniqueness of a mean-field equilibrium, by characterising it as the unique solution to a
bespoke three-dimensional system of forward-backward ordinary differential equations. This
characterisation enables the implementation of the model to support numerical analyses for
exploring the implications of climate damage on equilibrium outcomes and policy design in
terms of taxes and phase-out dates for brown production.MSC2020 subject classifications: 49N80, 91A07, 49N10, 91B38, 91B70, 91B7
Optimal stopping problems in mathematical finance
No full textThis thesis is concerned with the pricing of American-type contingent claims. First, the explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model for financial markets are presented. Compound options are financial contracts which give their holders the right (but not the obligation) to buy or sell some other options at certain times in the future by the strike prices given. The method of proof is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification for ordinary differential operators. The closed form solution to the perpetual American chooser option pricing problem is also obtained, by means of the analysis of the equivalent two-sided free-boundary problem. Second, an extension of the Black-Merton-Scholes model with piecewise-constant dividend and volatility rates is considered. The optimal stopping problems related to the pricing of the perpetual American standard put and call options are solved in closed form. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. As a result, the explicit algorithms determining the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options, are presented. Third, the optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under different information flows are studied. In this type of contracts, the writers have a right to withdraw the bonds before the holders can exercise them, by converting the bonds into assets. The value functions and the stopping boundaries' expressions are derived in closed-form in the case of observable dividend rate policy, which is modelled by a continuous-time Markov chain. The analysis of the associated parabolic-type free-boundary problem, in the case of unobservable dividend rate policy, is also presented and the optimal exercise times are proved to be the first times at which the asset price process hits boundaries depending on the running state of the filtering dividend rate estimate. Moreover, the explicit estimates for the value function and the optimal exercise boundaries, in the case in which the dividend rate is observable by the writers but unobservable by the holders of the bonds, are presented. Finally, the 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 its maximum drawdown, are studied. The latter process represents the difference between the running maximum and the current asset value. The optimal stopping times for exercising 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. The closed-form solutions to the equivalent free-boundary problems for the value functions are obtained 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 of the perpetual American call, put and strangle options are obtained as solutions of arithmetic equations and first-order nonlinear ordinary differential equations
Optimal stopping problems in mathematical finance
No full textThis thesis is concerned with the pricing of American-type contingent claims. First, the explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model for financial markets are presented. Compound options are financial contracts which give their holders the right (but not the obligation) to buy or sell some other options at certain times in the future by the strike prices given. The method of proof
is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification for ordinary differential operators. The closed form solution to the perpetual
American chooser option pricing problem is also obtained, by means of the analysis of the equivalent two-sided free-boundary problem. Second, an extension of the Black-Merton-Scholes model with piecewise-constant dividend
and volatility rates is considered. The optimal stopping problems related to the pricing of the perpetual American standard put and call options are solved in closed form. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. As a result, the explicit algorithms determining the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options,
are presented. Third, the optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under different information flows are studied. In this type of contracts, the writers have a right to withdraw the bonds
before the holders can exercise them, by converting the bonds into assets. The value functions and the stopping boundaries' expressions are derived in closed-form in the case of observable dividend rate policy, which is modelled by a continuous-time Markov chain. The analysis of the associated parabolic-type free-boundary problem, in the case of unobservable dividend rate policy, is also presented and the optimal exercise times are proved to be the first times at which the asset price process hits boundaries depending on the running state of the filtering dividend rate estimate. Moreover, the explicit estimates for the value function and the optimal exercise boundaries, in the case in which the dividend rate is observable by the writers but unobservable by the holders of the bonds, are presented. Finally, the 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 its maximum
drawdown, are studied. The latter process represents the difference between the running maximum and the current asset value. The optimal stopping times for exercising 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. The closed-form solutions to the equivalent free-boundary problems for the value functions are obtained 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 of the perpetual American call, put and strangle options are obtained as solutions of arithmetic equations and first-order nonlinear ordinary differential equations
