Eugene Feinberg - Academia.edu (original) (raw)

Papers by Eugene Feinberg

arXiv (Cornell University), Mar 7, 2016

As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward... more As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward and forward equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov's equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov's backward and forward equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov's backward equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov's forward equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.

arXiv (Cornell University), Jun 15, 2018

This paper studies Markov Decision Processes (MDPs) with atomless initial state distributions and... more This paper studies Markov Decision Processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total costs, and then it is extended under certain assumptions to MDPs with unbounded rewards. For problems with multiple criteria and constraints, the results of this paper imply that for atomless MDPs studied in this paper it is sufficient to consider only deterministic policies, while without the atomless assumption it is well-known that randomized policies can outperform deterministic ones. We also provide an example of an MDP demonstrating that, if a vector measure is defined on a standard Borel space, then Lyapunov's convexity theorem is a special case of the described results.

arXiv (Cornell University), Feb 6, 2018

Recently Feinberg et al. [6] established results on continuity properties of minimax values and s... more Recently Feinberg et al. [6] established results on continuity properties of minimax values and solution sets for a function of two variables depending on a parameter. Such minimax problems appear in games with perfect information, when the second player knows the move of the first one, in turn-based games, and in robust optimization. Some of the results in [6] are proved under the assumption that the multifunction, defining the domains of the second variable, is A-lower semi-continuous. As shown in [6], the Alower semi-continuity property is stronger than lower semi-continuity, but in several important cases these properties coincide. This note provides an example demonstrating that in general the A-lower semi-continuity assumption cannot be relaxed to lower semi-continuity.

Automatica, 2020

We study discrete-time discounted constrained Markov decision processes (CMDPs) with Borel state ... more We study discrete-time discounted constrained Markov decision processes (CMDPs) with Borel state and action spaces. These CMDPs satisfy either weak (W) continuity conditions, that is, the transition probability is weakly continuous and the reward function is upper semicontinuous in state-action pairs, or setwise (S) continuity conditions, that is, the transition probability is setwise continuous and the reward function is upper semicontinuous in actions. Our main goal is to study models with unbounded reward functions, which are often encountered in applications, e.g., in consumption/investment problems. We provide some general assumptions under which the optimization problems in CMDPs are solvable in the class of randomized stationary policies and in the class of chattering policies introduced in this paper. If the initial distribution and transition probabilities are atomless, then using a general "purification result" of Feinberg and Piunovskiy we show the existence of a deterministic (stationary) optimal policy. Our main results are illustrated by examples.

arXiv (Cornell University), Mar 24, 2021

This paper describes the structure of optimal policies for infinite-state Markov Decision Process... more This paper describes the structure of optimal policies for infinite-state Markov Decision Processes with setwise continuous transition probabilities. The action sets may be noncompact. The objective criteria are either the expected total discounted and undiscounted costs or average costs per unit time. The analysis of optimality equations and inequalities is based on the optimal selection theorem for infcompact functions introduced in this paper.

arXiv (Cornell University), Sep 10, 2021

This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump... more This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "Chebyshev-200", and it describes the results of their joined studies with Manasa Mandava (1984-2019).

arXiv (Cornell University), Apr 15, 2017

This paper provides sufficient conditions for the existence of solutions for two-person zero-sum ... more This paper provides sufficient conditions for the existence of solutions for two-person zero-sum games with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity-type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of the paper imply several classic results, and they are illustrated with the number guessing game. The paper also provides sufficient conditions for the existence of a value and solutions for each player.

arXiv (Cornell University), Jul 20, 2018

This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measur... more This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The note also provides new formulations of uniform Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis theorem, and the fundamental theorem for Young measures based on the equivalence of uniform integrability and the apparently weaker property of asymptotic uniform integrability for sequences of functions and finite measures.

arXiv (Cornell University), Apr 15, 2017

This paper provides sufficient conditions for the existence of solutions for two-person zero-sum ... more This paper provides sufficient conditions for the existence of solutions for two-person zero-sum games with inf/sup-compact payoff functions and with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity-type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of this paper imply several classic facts. The paper also provides sufficient conditions for the existence of a value and solutions for each player. The results of this paper are illustrated with the number guessing game.

arXiv (Cornell University), Sep 13, 2016

This paper describes the structure of optimal policies for discounted periodicreview single-commo... more This paper describes the structure of optimal policies for discounted periodicreview single-commodity total-cost inventory control problems with fixed ordering costs for finite and infinite horizons. There are known conditions in the literature for optimality of (s t , S t) policies for finite-horizon problems and the optimality of (s, S) policies for infinite-horizon problems. The results of this paper cover the situation, when such assumption may not hold. This paper describes a parameter, which, together with the value of the discount factor and the horizon length, defines the structure of an optimal policy. For the infinite horizon, depending on the values of this parameter and the discount factor, an optimal policy either is an (s, S) policy or never orders inventory. For a finite horizon, depending on the values of this parameter, the discount factor, and the horizon length, there are three possible structures of an optimal policy: (i) it is an (s t , S t) policy, (ii) it is an (s t , S t) policy at earlier stages and then does not order inventory, or (iii) it never orders inventory. The paper also establishes continuity of optimal value functions and describes alternative optimal actions at states s t and s.

arXiv (Cornell University), Jul 17, 2015

This paper describes results on the existence of optimal policies and convergence properties of o... more This paper describes results on the existence of optimal policies and convergence properties of optimal actions for discounted and average-cost Markov Decision Processes (MDPs) with weakly continuous transition probabilities. It is possible that cost functions are unbounded and action sets are not compact. The following results are established for such MDPs: (i) convergence of value iterations to optimal values for discounted problems with possibly non-zero terminal costs, (ii) convergence of optimal finitehorizon actions to optimal infinite-horizon actions for total discounted costs, as the time horizon tends to infinity, and (iii) convergence of optimal discount-cost actions to optimal average-cost actions for infinite-horizon problems, as the discount factor tends to 1. The general results on MDPs are applied to the classic stochastic periodic-review inventory control problems with backorders, for which they imply the optimality of (s, S) policies and convergence properties of optimal thresholds. In particular we analyze inventory control problems without two assumptions often used in the literature: (a) the demand is either discrete or continuous or (b) backordering is more expensive that the cost of backordered inventory if the backordered amount is large.

Theory of Probability and Its Applications, 2020

This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measur... more This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The note also provides new formulations of uniform Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis theorem, and the fundamental theorem for Young measures based on the equivalence of uniform integrability and the apparently weaker property of asymptotic uniform integrability for sequences of functions and finite measures.

arXiv (Cornell University), Sep 29, 2013

This note generalizes Berge's maximum theorem to noncompact image sets. It is also clarifies the ... more This note generalizes Berge's maximum theorem to noncompact image sets. It is also clarifies the results from E.A. Feinberg, P.O. Kasyanov, N.V. Zadoianchuk, "Berge's theorem for noncompact image sets," J. Math. Anal. Appl. 397(1)(2013), pp. 255-259 on the extension to noncompact image sets of another Berge's theorem, that states semi-continuity of value functions. Here we explain that the notion of a K-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KN-inf-compact function (N stands for "nets" in K-inf-compactness), which coincides with K-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.

arXiv (Cornell University), Sep 13, 2016

This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost... more This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost functions to minimax problems and studies applications of these extensions to two-player zero-sum games with possibly noncompact action sets and unbounded payoffs. For games with perfect information, also known under the name of turn-based games, this paper establishes continuity properties of value functions and solution multifunctions. For games with simultaneous moves, it provides results on the existence of lopsided values (the values in the asymmetric form) and solutions. This paper also establishes continuity properties of the lopsided values and solution multifunctions.

arXiv (Cornell University), Nov 2, 2020

This paper describes the structure of optimal policies for infinite-state Markov Decision Process... more This paper describes the structure of optimal policies for infinite-state Markov Decision Processes with setwise continuous transition probabilities. The action sets may be noncompact. The objective criteria are either the expected total discounted and undiscounted costs or average costs per unit time. The analysis of optimality equations and inequalities is based on the optimal selection theorem for infcompact functions introduced in this paper.

Theory of Probability and Its Applications, Feb 1, 2022

This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump... more This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "Chebyshev-200", and it describes the results of their joined studies with Manasa Mandava (1984-2019).

arXiv (Cornell University), Oct 4, 2013

This note describes examples of all possible equality and strict inequality relations between upp... more This note describes examples of all possible equality and strict inequality relations between upper and lower Abel and Cesàro limits of sequences bounded above or below. It also provides applications to Markov Decision Processes.

Set-Valued and Variational Analysis, 2015

This paper compares and generalizes Berge's maximum theorem for noncompact image sets established... more This paper compares and generalizes Berge's maximum theorem for noncompact image sets established in Feinberg, Kasyanov and Voorneveld [5] and the local maximum theorem established in Bonnans and Shapiro [3, Proposition 4.4].

Theory of Probability & Its Applications, 2009

This paper deals with the minimax quickest detection problem of a drift change for the Brownian m... more This paper deals with the minimax quickest detection problem of a drift change for the Brownian motion. The following minimax risks are studied: C(T) = inf τ ∈M T sup θ E θ (τ − θ | τ θ) and C(T) = inf τ ∈M T sup θ E θ (τ − θ | τ θ), where M T is the set of stopping times τ such that E∞τ = T and M T is the set of randomized stopping times τ such that E∞τ = T. The goal of this paper is to obtain for these risks estimates from above and from below. Using these estimates we prove the existence of stopping times, which are asymptotically optimal of the first and second orders as T → ∞ (for C(T) and C(T), respectively).

Theory of Probability & Its Applications, 2010

Dvoretzky, Wald and Wolfowitz proved in 1951 the existence of equivalent and strongly equivalent ... more Dvoretzky, Wald and Wolfowitz proved in 1951 the existence of equivalent and strongly equivalent mappings for a given transition probability when the number of nonatomic measures is finite and the decision set is finite. This paper introduces a notion of strongly equivalent transition probabilities with respect to a finite collection of functions. This notion contains the notions of equivalent and strongly equivalent transition probabilities as particular cases. This paper shows that a strongly equivalent mapping with respect to a finite collection of functions exists for a finite number of nonatomic distributions and finite decision set. It also provides a condition when this is true for a countable decision set. According to a recent example by Loeb and Sun, a strongly equivalent mapping may not exist under these conditions when the decision set is uncountable. This paper also provides two additional counterexamples and shows that strongly equivalent mappings exist for homogeneous transition probabilities.

arXiv (Cornell University), Mar 7, 2016

As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward... more As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward and forward equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov's equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov's backward and forward equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov's backward equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov's forward equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.

arXiv (Cornell University), Jun 15, 2018

This paper studies Markov Decision Processes (MDPs) with atomless initial state distributions and... more This paper studies Markov Decision Processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total costs, and then it is extended under certain assumptions to MDPs with unbounded rewards. For problems with multiple criteria and constraints, the results of this paper imply that for atomless MDPs studied in this paper it is sufficient to consider only deterministic policies, while without the atomless assumption it is well-known that randomized policies can outperform deterministic ones. We also provide an example of an MDP demonstrating that, if a vector measure is defined on a standard Borel space, then Lyapunov's convexity theorem is a special case of the described results.

arXiv (Cornell University), Feb 6, 2018

Recently Feinberg et al. [6] established results on continuity properties of minimax values and s... more Recently Feinberg et al. [6] established results on continuity properties of minimax values and solution sets for a function of two variables depending on a parameter. Such minimax problems appear in games with perfect information, when the second player knows the move of the first one, in turn-based games, and in robust optimization. Some of the results in [6] are proved under the assumption that the multifunction, defining the domains of the second variable, is A-lower semi-continuous. As shown in [6], the Alower semi-continuity property is stronger than lower semi-continuity, but in several important cases these properties coincide. This note provides an example demonstrating that in general the A-lower semi-continuity assumption cannot be relaxed to lower semi-continuity.

Automatica, 2020

We study discrete-time discounted constrained Markov decision processes (CMDPs) with Borel state ... more We study discrete-time discounted constrained Markov decision processes (CMDPs) with Borel state and action spaces. These CMDPs satisfy either weak (W) continuity conditions, that is, the transition probability is weakly continuous and the reward function is upper semicontinuous in state-action pairs, or setwise (S) continuity conditions, that is, the transition probability is setwise continuous and the reward function is upper semicontinuous in actions. Our main goal is to study models with unbounded reward functions, which are often encountered in applications, e.g., in consumption/investment problems. We provide some general assumptions under which the optimization problems in CMDPs are solvable in the class of randomized stationary policies and in the class of chattering policies introduced in this paper. If the initial distribution and transition probabilities are atomless, then using a general "purification result" of Feinberg and Piunovskiy we show the existence of a deterministic (stationary) optimal policy. Our main results are illustrated by examples.

arXiv (Cornell University), Mar 24, 2021

This paper describes the structure of optimal policies for infinite-state Markov Decision Process... more This paper describes the structure of optimal policies for infinite-state Markov Decision Processes with setwise continuous transition probabilities. The action sets may be noncompact. The objective criteria are either the expected total discounted and undiscounted costs or average costs per unit time. The analysis of optimality equations and inequalities is based on the optimal selection theorem for infcompact functions introduced in this paper.

arXiv (Cornell University), Sep 10, 2021

This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump... more This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "Chebyshev-200", and it describes the results of their joined studies with Manasa Mandava (1984-2019).

arXiv (Cornell University), Apr 15, 2017

This paper provides sufficient conditions for the existence of solutions for two-person zero-sum ... more This paper provides sufficient conditions for the existence of solutions for two-person zero-sum games with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity-type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of the paper imply several classic results, and they are illustrated with the number guessing game. The paper also provides sufficient conditions for the existence of a value and solutions for each player.

arXiv (Cornell University), Jul 20, 2018

This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measur... more This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The note also provides new formulations of uniform Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis theorem, and the fundamental theorem for Young measures based on the equivalence of uniform integrability and the apparently weaker property of asymptotic uniform integrability for sequences of functions and finite measures.

arXiv (Cornell University), Apr 15, 2017

This paper provides sufficient conditions for the existence of solutions for two-person zero-sum ... more This paper provides sufficient conditions for the existence of solutions for two-person zero-sum games with inf/sup-compact payoff functions and with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity-type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of this paper imply several classic facts. The paper also provides sufficient conditions for the existence of a value and solutions for each player. The results of this paper are illustrated with the number guessing game.

arXiv (Cornell University), Sep 13, 2016

This paper describes the structure of optimal policies for discounted periodicreview single-commo... more This paper describes the structure of optimal policies for discounted periodicreview single-commodity total-cost inventory control problems with fixed ordering costs for finite and infinite horizons. There are known conditions in the literature for optimality of (s t , S t) policies for finite-horizon problems and the optimality of (s, S) policies for infinite-horizon problems. The results of this paper cover the situation, when such assumption may not hold. This paper describes a parameter, which, together with the value of the discount factor and the horizon length, defines the structure of an optimal policy. For the infinite horizon, depending on the values of this parameter and the discount factor, an optimal policy either is an (s, S) policy or never orders inventory. For a finite horizon, depending on the values of this parameter, the discount factor, and the horizon length, there are three possible structures of an optimal policy: (i) it is an (s t , S t) policy, (ii) it is an (s t , S t) policy at earlier stages and then does not order inventory, or (iii) it never orders inventory. The paper also establishes continuity of optimal value functions and describes alternative optimal actions at states s t and s.

arXiv (Cornell University), Jul 17, 2015

This paper describes results on the existence of optimal policies and convergence properties of o... more This paper describes results on the existence of optimal policies and convergence properties of optimal actions for discounted and average-cost Markov Decision Processes (MDPs) with weakly continuous transition probabilities. It is possible that cost functions are unbounded and action sets are not compact. The following results are established for such MDPs: (i) convergence of value iterations to optimal values for discounted problems with possibly non-zero terminal costs, (ii) convergence of optimal finitehorizon actions to optimal infinite-horizon actions for total discounted costs, as the time horizon tends to infinity, and (iii) convergence of optimal discount-cost actions to optimal average-cost actions for infinite-horizon problems, as the discount factor tends to 1. The general results on MDPs are applied to the classic stochastic periodic-review inventory control problems with backorders, for which they imply the optimality of (s, S) policies and convergence properties of optimal thresholds. In particular we analyze inventory control problems without two assumptions often used in the literature: (a) the demand is either discrete or continuous or (b) backordering is more expensive that the cost of backordered inventory if the backordered amount is large.

Theory of Probability and Its Applications, 2020

This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measur... more This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The note also provides new formulations of uniform Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis theorem, and the fundamental theorem for Young measures based on the equivalence of uniform integrability and the apparently weaker property of asymptotic uniform integrability for sequences of functions and finite measures.

arXiv (Cornell University), Sep 29, 2013

This note generalizes Berge's maximum theorem to noncompact image sets. It is also clarifies the ... more This note generalizes Berge's maximum theorem to noncompact image sets. It is also clarifies the results from E.A. Feinberg, P.O. Kasyanov, N.V. Zadoianchuk, "Berge's theorem for noncompact image sets," J. Math. Anal. Appl. 397(1)(2013), pp. 255-259 on the extension to noncompact image sets of another Berge's theorem, that states semi-continuity of value functions. Here we explain that the notion of a K-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KN-inf-compact function (N stands for "nets" in K-inf-compactness), which coincides with K-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.

arXiv (Cornell University), Sep 13, 2016

This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost... more This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost functions to minimax problems and studies applications of these extensions to two-player zero-sum games with possibly noncompact action sets and unbounded payoffs. For games with perfect information, also known under the name of turn-based games, this paper establishes continuity properties of value functions and solution multifunctions. For games with simultaneous moves, it provides results on the existence of lopsided values (the values in the asymmetric form) and solutions. This paper also establishes continuity properties of the lopsided values and solution multifunctions.

arXiv (Cornell University), Nov 2, 2020

This paper describes the structure of optimal policies for infinite-state Markov Decision Process... more This paper describes the structure of optimal policies for infinite-state Markov Decision Processes with setwise continuous transition probabilities. The action sets may be noncompact. The objective criteria are either the expected total discounted and undiscounted costs or average costs per unit time. The analysis of optimality equations and inequalities is based on the optimal selection theorem for infcompact functions introduced in this paper.

Theory of Probability and Its Applications, Feb 1, 2022

This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump... more This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "Chebyshev-200", and it describes the results of their joined studies with Manasa Mandava (1984-2019).

arXiv (Cornell University), Oct 4, 2013

This note describes examples of all possible equality and strict inequality relations between upp... more This note describes examples of all possible equality and strict inequality relations between upper and lower Abel and Cesàro limits of sequences bounded above or below. It also provides applications to Markov Decision Processes.

Set-Valued and Variational Analysis, 2015

This paper compares and generalizes Berge's maximum theorem for noncompact image sets established... more This paper compares and generalizes Berge's maximum theorem for noncompact image sets established in Feinberg, Kasyanov and Voorneveld [5] and the local maximum theorem established in Bonnans and Shapiro [3, Proposition 4.4].

Theory of Probability & Its Applications, 2009

This paper deals with the minimax quickest detection problem of a drift change for the Brownian m... more This paper deals with the minimax quickest detection problem of a drift change for the Brownian motion. The following minimax risks are studied: C(T) = inf τ ∈M T sup θ E θ (τ − θ | τ θ) and C(T) = inf τ ∈M T sup θ E θ (τ − θ | τ θ), where M T is the set of stopping times τ such that E∞τ = T and M T is the set of randomized stopping times τ such that E∞τ = T. The goal of this paper is to obtain for these risks estimates from above and from below. Using these estimates we prove the existence of stopping times, which are asymptotically optimal of the first and second orders as T → ∞ (for C(T) and C(T), respectively).

Theory of Probability & Its Applications, 2010

Dvoretzky, Wald and Wolfowitz proved in 1951 the existence of equivalent and strongly equivalent ... more Dvoretzky, Wald and Wolfowitz proved in 1951 the existence of equivalent and strongly equivalent mappings for a given transition probability when the number of nonatomic measures is finite and the decision set is finite. This paper introduces a notion of strongly equivalent transition probabilities with respect to a finite collection of functions. This notion contains the notions of equivalent and strongly equivalent transition probabilities as particular cases. This paper shows that a strongly equivalent mapping with respect to a finite collection of functions exists for a finite number of nonatomic distributions and finite decision set. It also provides a condition when this is true for a countable decision set. According to a recent example by Loeb and Sun, a strongly equivalent mapping may not exist under these conditions when the decision set is uncountable. This paper also provides two additional counterexamples and shows that strongly equivalent mappings exist for homogeneous transition probabilities.