Quickest detection problems for Bessel processes

Johnson, Peter J.; Peškir, Goran

doi:10.1214/16-aap1223

Cited by 40 publications

(57 citation statements)

References 18 publications

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“…To the best of our knowledge, this is a novelty in the literature on singular stochastic control. We also believe that the detailed analysis of the free boundary that we obtain in this paper contributes to the literature on optimal stopping, since examples of solvable two-dimensional optimal stopping problems are quite rare (see [12], Section 3 in [15], and [29] for recent contributions).…”

Section: Introductionmentioning

confidence: 84%

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Ferrari¹

2018

SIAM J. Control Optim.

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Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time-horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such link to characterise the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein-Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation-dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem.Public Debt Control 2 of a crisis. This negative effect on economic growth from high debt levels has been observed in 18 different advanced economies (see [8]).In this paper we propose a continuous-time stochastic model for the control of the debtto-GDP ratio. The problem we have in mind is that of a government aiming to answer the question: How much is too much? 1 Following classical macroeconomic theory (see, e.g., [5]), in any given period the debt ratio stock grows by the existing debt stock multiplied by the difference between real interest rate and GDP growth, less the primary budget balance 2 . We assume that the government can reduce the level of the debt-to-GDP ratio by adjusting the primary budget balance, e.g. through fiscal interventions like raising taxes or reducing expenses. We therefore interpret the cumulative interventions on the debt ratio as the government's control variable, and we model it as a nonnegative and nondecreasing stochastic process. Uncertainty in our model comes through the GDP growth rate of the country, and its inflation and nominal interest rate which, by Fisher law [21], directly affect the growth rate of the debt ratio. We first assume that they are general not necessarily Markovian stochastic processes whose dynamics is not under government control. Indeed, the level of these macroeconomic variables is usually regulated by an autonomous Central Bank, whose action, however, is not modelled in this paper (see, e.g., [10] and [25] for problems related to the optimal control of inflation).Since high debt-to-GDP ratios can constrain economic growth making it more difficult to break the burden of the debt, we assume that debt ratio generates an instantaneous cost/penalty. This is a general nonnegative convex function of the debt ratio level, that the government would like to k...

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Section: Introductionmentioning

confidence: 84%

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Ferrari¹

2018

SIAM J. Control Optim.

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“…This approach has a long and venerable history in optimal stopping theory, with early contributions dating back to work of Shiryaev in the 1960s in the context of quickest detection. See Shiryaev [45] for a survey, and Johnson and Peskir [34] for some recent developments and further references. However, it seems that this model has never been adopted in the context of singular control.…”

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 99%

“…As it is customary in problems involving the process (π t ) (see e.g. Ekström and Lu [24], Klein [38] or Peskir and Johnson [34]), we introduce here the analogue for π t of the so-called likelihood ratio process, namely…”

Section: A Girsanov Transformationmentioning

confidence: 99%

“…Since the process ( X, ) is driven by the same Brownian motion, we can equivalently consider a two-dimensional state dynamics in which only one component has a diffusive part. This is done by a method similar to the one used in several papers addressing partial information, including De Angelis et al [18] and Johnson and Peskir [34].…”

Section: A Parabolic Formulationmentioning

confidence: 99%

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Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Angelis

2019

Finance Stoch

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We study the optimal dividend problem for a firm's manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a two-dimensional degenerate diffusion whose first component is singularly controlled. Moreover, the process is absorbed when its first component hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with 'creation'. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the 'local time' of an auxiliary two-dimensional reflecting diffusion.

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“…Moreover, even the question of continuity becomes difficult to handle for d > 1. In the case of d = 2 and T = ∞, specific examples were addressed in [13] and [19], while a more complete answer was recently provided in [30].…”

Section: Introductionmentioning

confidence: 99%

On Lipschitz Continuous Optimal Stopping Boundaries

Angelis¹,

Stabile²

2019

SIAM J. Control Optim.

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We obtain a probabilistic proof of the local Lipschitz continuity for the optimal stopping boundary of a class of problems with state space [0, T ] × R d , d ≥ 1. To the best of our knowledge this is the only existing proof that relies exclusively upon stochastic calculus, all the other proofs making use of PDE techniques and integral equations. Thanks to our approach we obtain our result for a class of diffusions whose associated second order differential operator is not necessarily uniformly elliptic. The latter condition is normally assumed in the related PDE literature.

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Quickest detection problems for Bessel processes

Cited by 40 publications

References 18 publications

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

On Lipschitz Continuous Optimal Stopping Boundaries

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