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MARTINGALE SELECTION THEOREM FOR A STOCHASTIC SEQUENCE WITH RELATIVELY OPEN CONVEX VALUES
, 2006
"... Abstract. For a setvalued stochastic sequence (Gn) N n=0 with relatively open convex values Gn(ω) we give a criterion for the existence of an adapted sequence (xn) N n=0 of selectors, admitting an equivalent martingale measure. Mentioned criterion is expressed in terms of supports of the regular co ..."
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Abstract. For a setvalued stochastic sequence (Gn) N n=0 with relatively open convex values Gn(ω) we give a criterion for the existence of an adapted sequence (xn) N n=0 of selectors, admitting an equivalent martingale measure. Mentioned criterion is expressed in terms of supports of the regular conditional upper distributions of the elements Gn. This result is a refinement of the main result of author’s previous paper (Teor. Veroyatnost. i Primen., 2005, 50:3, 480–500), where the sets Gn(ω) were assumed to be open and where were asked if the openness condition can be relaxed.
Descriptive Set Theory
"... en Xn is a Polish space. Suppose dn is a complete metric on Xn , with dn < 1, for n = 0; 1; : : :. De ne b d on Xn by b d(f; g) = n+1 dn (f(n); g(n)): If f 0 ; f 1 ; : : : is a Cauchysequence, then f 1 (i); f 2 (i); : : : is a Cauchysequence in X i for each i. Let g(n) = lim f i ..."
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en Xn is a Polish space. Suppose dn is a complete metric on Xn , with dn < 1, for n = 0; 1; : : :. De ne b d on Xn by b d(f; g) = n+1 dn (f(n); g(n)): If f 0 ; f 1 ; : : : is a Cauchysequence, then f 1 (i); f 2 (i); : : : is a Cauchysequence in X i for each i. Let g(n) = lim f i (n). Then g is the limit of f 0 ; f 1 ; : : :. Suppose x 0 ; x 1 ; : : : is a dense subset of X i . For 2 N f (n) = (n) if i < jj 0 otherwise The ff : 2 N g is dense in X . In particular, the Hilbert cube H = I is Polish. Indeed, it is a universal Polish space. Theorem 1.4 Every Polish space is homeomorphic to a subspace of H . Proof Let X be a Polish space. Let d be a compatible metric on X with d < 1 and let x 0 ; x 1 ; : : : a dense set. Let f : X ! H by f(x) = (d(x; x 1 ); d(x; x 2 ); : : :). If d(x; y) < =2, then jd(x; x i ) d(y; x i )j < and d(f(x); f(y)) < P 1 n+1 < . Thus f is continuous. If d(x; y) = choose x i such that d(x; x i ) <
A Mixed Value and Policy Iteration Method for Stochastic Control with Universally Measurable Policies,” Lab. for Info. and Decision Systems Report LIDSP2905
, 2013
"... We consider the stochastic control model with Borel spaces and universally measurable policies. For this model the standard policy iteration is known to have difficult measurability issues and cannot be carried out in general. We present a mixed value and policy iteration method that circumvents thi ..."
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We consider the stochastic control model with Borel spaces and universally measurable policies. For this model the standard policy iteration is known to have difficult measurability issues and cannot be carried out in general. We present a mixed value and policy iteration method that circumvents this difficulty. The method allows the use of stationary policies in computing the optimal cost function, in a manner that resembles policy iteration. It can also be used to address similar difficulties of policy iteration in the context of upper and lower semicontinuous models. We analyze the convergence of the method in infinite horizon total cost problems, for the discounted case where the onestage costs are bounded, and for the undiscounted case where the onestage costs are nonpositive or nonnegative. For the undiscounted total cost problems with nonnegative onestage costs, we also give a new convergence theorem for value iteration, which shows that value iteration converges whenever it is initialized with a function that is above the optimal cost function and yet bounded by a multiple of the optimal cost function. This condition resembles Whittle’s bridging condition and is partly motivated by it. The theorem is also partly motivated by a result of Maitra
Arsenin  Kunugui Theorem And Weak Forms Of Borel Bimeasurability
, 1999
"... . Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel) space L onto a metric space M such that f(F ) is a Borel subset of M if F is closed in L. We show that then M is also Luzin, that f \Gamma1 (y) is a K oe set for all, except for countably many, y 2 M , and that the Borel class ..."
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. Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel) space L onto a metric space M such that f(F ) is a Borel subset of M if F is closed in L. We show that then M is also Luzin, that f \Gamma1 (y) is a K oe set for all, except for countably many, y 2 M , and that the Borel classes of the sets f(F ), F closed in L, are bounded by a countable cardinal. It gives a counterpart to the classical theorem of Arsenin and Kunugui that enables to state it in the form of an equivalence. As a particular case we get a theorem of Taimanov saying that the image of a Luzin space by a closed continuous mapping is a Luzin space. The method is based on a straightforward construction that gives a Hurewicz theorem and on the use of the Jankov  von Neumann selection theorem. A classical theorem of Novikov says that a Borel measurable mapping f : L ! Y of a Borel subset L of a Polish space X to a Polish space Y is Borel bimeasurable (i.e. f(B) is Borel for every Borel subset B of L) if ...
States, Models and Unitary Equivalence I: Representation Theorems and Analogical Reasoning
, 2004
"... I show that virtually any model of decision making under uncertainty is associated to a certain structure. This contains three fundamental ingredients: (1) The domain of the acts; (2) Another set, which is called the set of models for the decision maker; and (3) The decision maker’s information abou ..."
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I show that virtually any model of decision making under uncertainty is associated to a certain structure. This contains three fundamental ingredients: (1) The domain of the acts; (2) Another set, which is called the set of models for the decision maker; and (3) The decision maker’s information about the set of models (an algebra of subsets of the set of models). A consequence of this finding is that that the decision maker’s choices can be viewed as the outcome of a twostage process. First, the set of acts is mapped into a system of hypothetical bets on the set of models. Then, the latter are ranked by the decision maker. I show that this procedure can be thought of as describing a general form of analogical reasoning. I also observe that the appearance of two different sets implies that the decision maker is uncertain about two different objects and that he may receive information about any of them. In particular, information about the set of models affects the decision maker’s ranking of the
THE JENSEN ENVELOPE IS PLURISUBHARMONIC ON ALL MANIFOLDS
, 1999
"... Abstract. The Jensen envelope Jϕ of an upper semicontinuous function ϕ on a complex manifold X is defined at x ∈ X as the infimum of µ(ϕ) over all Jensen measures µ centred at x. The Poisson envelope Pϕ is defined by using only the boundary measures of analytic discs centred at x. One of the main op ..."
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Abstract. The Jensen envelope Jϕ of an upper semicontinuous function ϕ on a complex manifold X is defined at x ∈ X as the infimum of µ(ϕ) over all Jensen measures µ centred at x. The Poisson envelope Pϕ is defined by using only the boundary measures of analytic discs centred at x. One of the main open problems in the theory of disc functionals is whether the Poisson envelope is plurisubharmonic on an arbitrary manifold. This is equivalent to the two envelopes being equal, so plurisubharmonicity of Jϕ is a necessary condition for Pϕ to be plurisubharmonic. We prove that the Jensen envelope is plurisubharmonic, with no assumptions on the manifold X. Hence Jϕ is the largest plurisubharmonic function smaller than ϕ. We also show that the Poisson envelope is plurisubharmonic if and only if boundary measures of analytic discs are dense among Jensen measures.
Constructive
, 2006
"... noarbitrage criterion under transaction costs in the case of finite discrete time ∗ ..."
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noarbitrage criterion under transaction costs in the case of finite discrete time ∗
unknown title
, 2001
"... Filtrations of random processes in the light of classification theory. I. A topological zeroone law. ..."
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Filtrations of random processes in the light of classification theory. I. A topological zeroone law.
unknown title
, 2006
"... Brownian local minima, random dense countable sets and random equivalence classes ..."
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Brownian local minima, random dense countable sets and random equivalence classes
CONTINUOUS FUNCTIONS TAKING EVERY VALUE A GIVEN NUMBER OF TIMES
, 708
"... Abstract. We give necessary and sufficient conditions on a function f: [0, 1] → {1, 2,..., ω, c} under which there exists a continuous function F: [0, 1] → [0, 1] such that for every y ∈ [0, 1], F −1 (y)  = f(y). 1. ..."
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Abstract. We give necessary and sufficient conditions on a function f: [0, 1] → {1, 2,..., ω, c} under which there exists a continuous function F: [0, 1] → [0, 1] such that for every y ∈ [0, 1], F −1 (y)  = f(y). 1.