Stochastic Perron’s method and verification without smoothness using viscosity comparison: The linear case

Bayraktar, Erhan; Ŝırbu, Mihai

doi:10.1090/s0002-9939-2012-11336-x

Cited by 64 publications

(118 citation statements)

References 7 publications

(16 reference statements)

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“…Bayraktar and Xing [5] characterize one-dimensional time-homogeneous Cauchy problems with a unique solution. For the relevance of super-and sub-solutions in the study of partial differential equations of parabolic type we refer to the recent paper by [4] and the references therein. Let us also note that the characterizations of Propositions 5.4 and 5.5 are impervious to boundary conditions at the endpoints of the state space I = ( , r ) .…”

Section: Proposition 55 (Unique Bounded Solution)mentioning

confidence: 99%

Distribution of the time to explosion for one-dimensional diffusions

Karatzas

Ruf

2015

Probab. Theory Relat. Fields

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We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to the computation of expectations of suitable nonnegative local martingales. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller's criterion for explosions. We discuss in detail several examples for which To the Memory of Marc Yor. for their insightful remarks that improved the manuscript. We thank Alex Mijatović for raising the issues of strict positivity and full support and for prompting us to think about them (Sects. 4.2 and 4.3, respectively). We are grateful to the referees for their meticulous reading of the paper and their many and incisive suggestions. J.R. acknowledges generous support from the Oxford-Man Institute of Quantitative Finance, University of Oxford, where a major part of this work was completed. The problem discussed here was suggested to us by Marc Yor, who also gave us generous expert advice on several of its aspects. We dedicate the paper to his memory with deep sorrow at his passing. I.

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Section: Proposition 55 (Unique Bounded Solution)mentioning

confidence: 99%

Distribution of the time to explosion for one-dimensional diffusions

Karatzas

Ruf

2015

Probab. Theory Relat. Fields

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“…By Proposition 4.1 in Bayraktar and Sîrbu [9] and by Lemma 3.3, we know that v + is the limit of a nonincreasing sequence of stochastic supersolutions {v n } n∈N . Fix δ ′ ∈ (0, δ), and define a sequence of sets…”

Section: Stochastic Supersolutionmentioning

confidence: 89%

Minimizing the Discounted Probability of Exponential Parisian Ruin via Reinsurance

Liang¹,

Young²

2020

SIAM J. Control Optim.

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We study the problem of minimizing the discounted probability of exponential Parisian ruin, that is, the discounted probability that an insurer's surplus exhibits an excursion below zero in excess of an exponentially distributed clock. The insurer controls its surplus via reinsurance priced according to the mean-variance premium principle, as in Liang, Liang, and Young [23]. We consider the classical risk model and apply stochastic Perron's method, as introduced by Bayraktar and Sîrbu [9,10,11], to show that the minimum discounted probability of exponential Parisian ruin is the unique viscosity solution of its Hamilton-Jacobi-Bellman equation with boundary conditions at ±∞. A major difficulty in proving the comparison principle arises from the discontinuity of the Hamiltonian. . V. R. Young thanks the Cecil J. and Ethel M. Nesbitt Professorship of Actuarial Mathematics for financial support.1 problem generalizes the ordinary-ruin problem and retains its one-dimensional property; specifically, because an exponential random variable has a constant hazard rate, we do not need to introduce a time variable. 1 In this paper, we consider the problem of minimizing the discounted probability of exponential Parisian ruin for an insurance company that can purchase per-loss reinsurance, which is priced according to the mean-variance premium principle, as in Han, Liang, and Young [20] and in Liang, Liang, and Young [23]. We consider the classical model, for which we cannot find an explicit expression of the minimum discounted probability of exponential Parisian ruin. We use stochastic Perron's method to prove that the value function is the unique (continuous) viscosity solution of its associated discontinuous HJB equation with boundary conditions (Theorem 3.4).Stochastic Perron's method was introduced in Bayraktar and Sîrbu [9] for linear problems, Bayraktar and Sîrbu [10] for nonlinear problems of HJB equations in stochastic control, and Bayraktar and Sîrbu [11] for problems related to Dynkin games. Stochastic Perron's method provides a way to show that the value function of the stochastic control problem is the unique viscosity solution of the associated HJB equation. Unlike the classical verification approach, it requires neither the dynamic programming principle nor the regularity of the value function. Roughly speaking, stochastic Perron's method consists of the following steps:(1) estimate the value function from below and above by stochastic sub-and supersolutions, (2) prove that the supremum and the infimum of the respective families are viscosity super-and subsolutions, respectively, and (3) prove a comparison principle for viscosity sub-and supersolutions, which has an immediate corollary that the value function is the unique (continuous) viscosity solution of its HJB equation. In contrast to the classical verification method, the comparison principle here plays a role for both verification and uniqueness. More recently, stochastic Perron's method was applied to solve an exit-time problem in Rokhlin [27], a transact...

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“…From Proposition 4.1 in [BS12], there existw n ∈ L such that w − = sup nwn . We define the increasing sequence w n =w 1 ∨ · · · ∨w n ∈ L w − .…”

Section: Proposition 34 (Asymptotic Perron) the Function W − Is An mentioning

confidence: 99%

Asymptotic Perron's Method and Simple Markov Strategies in Stochastic Games and Control

Ŝırbu¹

2015

SIAM J. Control Optim.

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We introduce a modification of Perron's method, where semisolutions are considered in a carefully defined asymptotic sense. With this definition, we can show, in a rather elementary way, that in a zero-sum game or a control problem (with or without model uncertainty), the value function over all strategies coincides with the value function over Markov strategies discretized in time. Therefore, there are always discretized Markov ε-optimal strategies (uniform with respect to the bounded initial condition). With a minor modification, the method produces a value and approximate saddle points for an asymmetric game of feedback strategies versus counterstrategies. Introduction.The aim of the paper is to introduce the asymptotic Perron's method, i.e., constructing a solution of the Hamilton-Jacobi-Belman-Isaacs (HJBI) equation as the supremum/infimum of carefully defined asymptotic semisolutions. Using this method we show, in a rather elementary way, that the value functions of zero-sum games/control problems can be (uniformly) approximated by some simple Markov strategies for the weaker player (the player in the exterior of the sup/inf or inf/sup). From this point of view, we can think of the method as an alternative to the shaken coefficients method of Krylov [Kry00] (in the case of only one player, under slightly different technical assumptions) or to the related method of regularization of solutions of HJBIs byŚwiȩch in [Świ96a] and [Świ96b] (for control problems or games in Elliott-Kalton formulation). The method of shaken coefficients has been recently used to study games in Elliott-Kalton formulation in [BN] under a convexity assumption (not needed here).While the result on zero-sum games (under our standing assumptions) is rather new, but certainly expected, the goal of the paper is to present the method. To the best of our knowledge, this modification of Perron's method does not appear in the literature. In addition, we believe that it applies to more general situations than we consider here and using either a stochastic formulation (as in the present work) or an analytic one (see Remark 3.1). Compared to the method of shaken coefficients of Krylov, or to the regularization of solutions byŚwiȩch, the analytic approximation of the value function/solution of HJB by smooth approximate solutions is replaced by the Perron construction. The careful definition of asymptotic semisolutions allows us to prove that such semisolutions work well with Markov strategies. The idea of restricting actions to a deterministic time grid is certainly not new; we just provide a method that works well for such strategies/controls. The arguments display once again the *

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Stochastic Perron’s method and verification without smoothness using viscosity comparison: The linear case

Cited by 64 publications

References 7 publications

Distribution of the time to explosion for one-dimensional diffusions

Distribution of the time to explosion for one-dimensional diffusions

Minimizing the Discounted Probability of Exponential Parisian Ruin via Reinsurance

Asymptotic Perron's Method and Simple Markov Strategies in Stochastic Games and Control

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