P. Dupuis and R. Ellis. A weak convergence approach to the theory of large deviations. John Wiley & Sons Inc., New York, 1997. A Wiley-Interscience Publication.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....One advantage of such an uncertainty description is that it allows for stochastic uncertainty inputs to depend dynamically on the uncertainty outputs. In addition, by making use of the duality relationship between a stochastic game and the risk sensitive stochastic control (RSSC) problem [3] [4], 12] it was possible to synthesize the robust LQG controller from the associated algebraic or differential Riccati equations (from a certain specially parameterized RSSC problem) In the infinite horizon case of [16] the uncertainty was described by an approximating sequence of martingales, ....

....the optimal value denoted by # ## . Analogous to the definitions given earlier, one can also define local optimality and global inverse optimality with respect to the standard RSSC problem, as in [2] Utilizing the duality relationship between free energy and relative entropy established in [3] [4], we arrive at the following conclusions. Theorem 1. Consider the linear stochastic system (10) 6) with cost function (11) as well as the system (18) with associated risk sensitive cost function (29) Suppose that the set L is nonempty. Then: J # ( # ; x) inf # ## where ....

P. Dupuis and R. Ellis, A Weak Convergence Approach to Theory of Large Deviations. Wiley, 1997.

....information I(P Q) in P given Q. Now we need to look at I(Q P) called sometimes the relative entropy in P given Q (the term used in the theory of large deviations to characterize this quantity as the average relative entropy in the experiment given a dominating measure Q ; cf e.g. [3], section 1.4; for a di#erent, statistical context, see e.g. 13] Contrary to (2.3) we then will need the condition 0 I(Q P) #. Actually, we only apply this to the particular dominating measure P# , so it su#ces to require 0 I( P) 4.4) where P# is the product measure # ....

Dupuis, P. and Ellis, R.S. (1997), A Weak Convergence Approach to the Theory of Large Deviations, Wiley.

....Kullback Leibler information I(P Q)inP given Q. Now we need to look at I(Q P) called sometimes the relative entropy in P given Q (the term used in the theory of large deviations to characterize this quantity as the average relative entropy in the experiment given a dominating measure Q ;cfe.g. [4], section 1.4; for a di#erent, statistical context, see e.g. 16] Contrary to (2.3) we then will need the condition 0 I(Q P) #. Actually, we only apply this to the particular dominating measure P # , so it su#ces to require 0 I( P) 4.4) where P # is the product measure # ....

P. Dupuis and R.S. Ellis (1997), A Weak Convergence Approach to the Theory of Large Deviations, Wiley.

....may de ne D# # P #,# P # # # # = lim n inf 1 n D # P n #,# #P n ,# # # = sup n inf 1 n D # P n #,# #P n ,# # # . where the in mum is taken over the initial distribution for P n ,# # . Since D # # # is lower semi continuous, see for instance Lemma 1.4. 3 in [9], D# # P #,# P # # # is also a lower semi continuous function of #, #, # # . 2.4. The Dirichlet prior. The Dirichlet prior on # k has already proved to be an important tool in universal data compression of Hidden Markov Sources, in the OPTIMAL ERROR EXPONENTS IN HIDDEN MARKOV MODELS ....

....(2) Lower bound : For any measurable open subset G of E, lim inf n## 1 n log P n (G) # inf x#G I(x) Note that in this text, we will consider many di erent spaces E. The following Lemma is the cornerstone of our argument. Relying on classical theorems from Large Deviations theory [6, 9], it asserts that the empirical measure n # m=1 1 n # xm ,y m ,q 1 m , q j m de ned by the extended chain induced by a tuple of parameters # 1 , # j satis es an ldp. The topological space E is here the set of probability measures on X Y M 1 (X ) j provided with the topology of ....

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P. Dupuis and R.S. Ellis, A weak convergence approach to the theory of large deviations, Wiley and Sons, New York, 1997

....Wiener process W in equation (1.1) with a smaller noise process # #W and simultaneously scale the payoff by raising it to the power of 1 #. Through a generalization of the classical Laplace method for integrals known as Varadhan s Integral Lemma (Dembo and Zeitouni 1993) or a Laplace Principle (Dupuis and Ellis 1997), we will see in Section 2 that we frequently have E[e F( # #Z) # 1 D ] # e #(z # ) # ,## 0, in a sense to be made precise, where z # maximizes the function #(z) F(z) 1 2 z # z over D. Maximizing # is equivalent to maximizing the product of the payoff and the probability ....

....become equivalent and, in fact, optimal in an asymptotic sense. 2.2. Large Deviations and Laplace Principles We begin with brief generalities on large deviations, then specialize to our context. For additional background see Dembo and Zeitouni (1993) Deuschel and Stroock (1989) and Dupuis and Ellis (1997). A sequence of probability measures # # on a topological space (X, B) B the completion of the Borel sets on X) satisfies a large deviations principle with good rate function I if the function I : X # [0, #] is lower semicontinuous with compact level sets and if the following conditions ....

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DUPUIS, P., and R. ELLIS (1997): A Weak Convergence Approach to the Theory of Large Deviations.

....are compacts; 1) for every closed (open) in the metric # 2 set F 2 (G 2 ) from S 2 lim sup n 1 n log P (# 2 n # F 2 ) # inf ##F 2 J 2 (#) lim inf n 1 n log P (# 2 n # G 2 ) # inf ##G 2 J 2 (#) Theorem 1.2 (comp. Ellis [5] and Dupui and Ellis [6] [19]) Under (F) I ) H # ) RM ) and (H 0 ) the family (# 2 n , n # 1) obeys the LDP in (S 2 , # 2 ) with the rate function J 2 (#) # #, marginals of # are di#erent H(# # # # ) marginals of # are the same ( #) 5. Despite Theorem 1.2 is closed to corresponding results from [5] ....

P. Dupuis and R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations. John Willey and Sons, to be published. 26

.... Delta d B Gamma Z t 0 jh j 2 2 d dt = ffi Z 1 0 EQ exp( Gammaffit) Z t 0 jh j 2 2 d dt = ffi Z 1 0 EQ jh u j 2 2 Z 1 u exp( Gammaffit)dtdu = Z 1 0 exp( Gammaffiu)E Q jh u j 2 2 du: 8) Relative entropy is convex in the measure Q (e.g. see Dupuis and Ellis (1997)) 9 This follows from the concavity of the function log y for positive values of y. Relative entropy is nonnegative and zero only when the probability distributions P and Q agree, which for the representation under Claim 3.2 is true only when the process fh t : t 0g is zero. 9 Other ....

....and zero only when the probability distributions P and Q agree, which for the representation under Claim 3.2 is true only when the process fh t : t 0g is zero. 9 Other measures of relative entropy for stochastic processes occur in the literature on large deviations. For example, see Dupuis and Ellis (1997) page 299. 7 Notice that if R(Q) is finite, then Q aeZ t 0 jh u j 2 du 1 oe = 1 for each positive t. It follows from Claim 3.2 that if R(Q) is finite, Q is locally absolutely continuous with respect to P . If in addition, Q aeZ 1 0 jh u j 2 du 1 oe = 1 (9) then Q is absolutely ....

Dupuis, P. and R. S. Ellis (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics. New York: John Wiley and Sons.

.... : p ny dd dd y ny 11 1=2 11 : y ny dd 1=2 dd # : 6) That is, we will show that the decay rate of q n (t; ff; fi) is equal to the decay rate of the largest term of q n (t; ff; fi) also known as Laplace s principle (see for an extensive treatment of this principle Dupuis and Ellis [12]) 7 Let us first find an upper bound on the number of y ij such that ny ij 2 X n . As we have that ny ij must attain an integer value between 0 and n, such an upper bound is (n 1) d 2 : This means that an upper bound for the decay rate of q n (t; ff; fi) is expression (6) increased by ....

....state. The link rate C is denoted by nfi (corresponding with the notation introduced in Section 2) Without loss of generality, we assume the flows peak rates to equal 1. Our reasoning will be of a heuristic nature. As justified in [27] we will extensively use the so called Laplace s principle [12], stating that the decay rate of an integral equals the decay rate of the maximum of the integrand. We denote by A n (t) the number of flows (out of n) in the on state at time t, assuming that they do not leave the system. Notice that this definition slightly differs from the one in Section 2, ....

P. Dupuis and R. Ellis. A weak convergence approach to the theory of large deviations. Wiley, New York, USA, 1997.

....converges weakly to the probability measure Q T . Indeed, consider the Polish space of probability measures on the measurable space (#, F T ) endowed with the topology of weak convergence of probability measures. Note that # is a metric space. Hence, such a topology can be defined; e.g. see [6]. For the sequence Q T i to converge weakly to Q T , it is required that equation (4) holds for all bounded continuous random variables #. Obviously, this requirement is satisfied if Q T i # Q T . As in the finite horizon case [10, 18] we describe the class of admissible ....

....continuous random variables #. Obviously, this requirement is satisfied if Q T i # Q T . As in the finite horizon case [10, 18] we describe the class of admissible uncertainties in terms of the relative entropy functional h(#) for the definition and properties of the functional h(#) see [6]. Definition 1 Let d be a given positive constant. A martingale #( #Mis said to define an admissible uncertainty if there exists a sequence of continuous positive martingales # i (t) F t ,t # 0 # i=1 #Mwhich satisfies the following conditions: i) For each i, h(Q T i #P T ) ....

P. Dupuis and R. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. Wiley, 1997.

....function Ip M 2716 ; 3 5 by Ip rr p rM 27 16 16 = K K log , # if otherwise . 13) where , r r pr pr 1 6 16 16 16 = K K if otherwise 0 1 (14) It is straightforward to prove the following lemma using the results in [21]. Lemma 1: Assume that xr, 16 is bounded as a function of r for each fixed x . Then inf log , # inf sup , ## , # , # # u rM Arx Brur u M rM rre Arx Brur r Ir 16 16 16 27 16 16 49 L 16 16 27 16 = K K 3 . 15) Furthermore, the maximum of the ....

P. Dupuis and R.S. Ellis, A weak convergence approach to the theory of large deviations, Wiley (1997).

.... the mean excursion operator, T # g(y) E y [g(x r (# 1 ) Fo r smoo th f(y)on#D, define the boundary generator G # f(y) N # u # f (y) and the boundary local time operator B # f(y) E y [ Z #1 0 f(x r (t) d# # ] Mathematical Approaches to the Problem of Noise Induced Exit 9 both for y # #D. Applying Ito s formula (1.20) to u # f , integrating over [0,# 1 ] and taking the expectation produces the relationship among these operators: T # f f = B # [G # f]on#D. 1.23) The asymptotic behavior of the exit location distribution is given by the behavior of T ....

Paul Dupuis and Richard S. Ellis (1997). A Weak Convergence Approach to the Theory of Large Deviations,J.Wiley,New York.

.... Z t 0 h t Delta dB t Gamma jh t j) 2 2 dt dt = d Z 0 EQ exp( Gammadt) Z t 0 jh t j 2 2 dt dt = d Z 0 EQ jh t j 2 2 Z t exp( Gammadt)dtdt = Z 0 exp( Gammadt)E Q jh t j 2 2 dt: 6) Relative entropy is convex in the measure Q (e.g. see Dupuis and Ellis (1997)) 6 This follows from the convexity of the function ylogy for positive values of y. Relative entropy is nonnegative and zero only when the probability distributions P and Q agree. In our case this is true only when the process fh t g is zero. 4 Two Robust Control Problems In this section ....

....are closely related. As is typical in penalty formulations of decision problems, we can interpret the robustness parameter q in the first problem as an implied Lagrange 6 Other measures of relative entropy for a stochastic process occur in the literature on large deviations. For example, see Dupuis and Ellis (1997) page 299. 5 multiplier on the specification error constraint R (Q) h. This connection has been explored in informally in Hansen, Sargent, and Tallarini (1999) and formally in Hansen and Sargent (2001) both in the context of linear quadratic control problem. This connection is also ....

Dupuis, P. and R. S. Ellis (1997). A Weak Convergence Approach to the Theory of Large Deviations.

....#(b b 0 ct) d X i=1 f i,0 log M i (#; t) 13) let # # be the supremizing argument. Expression (11) provides information on the state of the sources at time 0, as well as the amount of fluid that entered the system in the interval [0, t] However, by invoking Laplace s principle [11], implicitly the entire most likely trajectory of the distribution of the sources in the interval [0, t] is given. This can be explained in the following 3 steps. 1) First we introduce the matrix G n (s) its (i, j)th entry is the fraction of the sources that were in state i at time 0 that is ....

.... A # (t) # n(b b 0 ct) F n (0) f 0 , G n (s) # g(s) P(G n (s) g(s) F n (0) f 0 )dg(s) 14) 2) Now consider both probabilities in the integrand of (14) Both of them can be evaluated asymptotically by means of the standard Large Deviations theorems of Cramer and Sanov [11, 29]. Denote by # and # the exponential decay rates of both probabilities; then # = sup # 0 #(b b 0 ct) d X i,j=1 f 0,i g ij (s) log B ij (#; s)M j (#; t s) p ij (s) 1 A , 15) # = d X i,j=1 f 0,i g ij (s) log f 0,i g ij (s) p ij (s) 16) 3) Now we apply ....

P. Dupuis and R. Ellis. A weak convergence approach to the theory of large deviations. Wiley, New York, USA, 1997.

....large deviations limit theorem justifying the approximation (7) thus, it is only a heuristic. To have such a theorem, we need to prove a large deviations principle for the scaled processes (Xn ( Delta) Zn ( Delta) which have discontinuous jump rates across a curved boundary. Dupuis and Ellis [9] have proved a large deviations principle for discrete time processes with jump rates across a straight boundary, and Alanyali and Hajek [10] has proved one for continuos time processes. The local rate function we use is a natural generalization of theirs. Our main result about the overload ....

P. Depuis and R. Ellis, A Weak Convergence Approach to the Theory of Large Deviations. to be published.

....[9] proceed to establish what they call a Chernof bound for the occupancy problem. It shows that the Poisson tail estimate (1) is correct even if we do not resort to a conditioning argument, i.e. that the p n factor is spurious. 2. The Large Deviation Approach The large deviation approach (see [2, 5, 7] for recent presentations) aims at identifying the right exponents for tail probability. It provides the right touchstone for the occupancy problem. Rather than using the martingale structure of the occupancy problem, the large deviation approach relies on the Markovian structure of the occupancy ....

Dupuis (Paul) and Ellis (Richard S.). { A weak convergence approach to the theory of large deviations. { John Wiley & Sons, New York, 1997, xviii+479p. A Wiley-Interscience Publication.

....theory of large deviations is one of the classical topics in probability and statistics. For historical background and fundamental results in this area we refer the reader to Varadhan [24] Deuschel and Stroock [7] Dembo and Zeitouni [6] Freidlin and Wentzell [13] Ellis [12] In a recent book [10] a new methodology was introduced for the analysis of large deviation problems. A crucial ingredient of this approach is the representation of the expectations whose asymptotic behavior is to be analyzed by value functions (minimal cost functions) of associated optimal stochastic control problems. ....

....one then obtains the desired large deviation result. In the approach to large deviations just described, the variational representation for the pre limit expectations is the starting point of the analysis. The canonical example of such a representation is the following (Proposition 1.4. 2 [10]) Let (V, A) be a measurable space, k a bounded measurable function mapping V into IR and # a probability measure on V. Then log # V e k d# = inf ##P(V) # R(###) # V kd# # , 1.1) where P(V) is the space of all probability measures on (V, A) and R(#) Representations for ....

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P. Dupuis and R. S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. John Wiley and Sons, Inc., 1997.

....a Laplace principle with rate function I if for every bounded continuous function h mapping X into IR lim n 1 1 n log E fexp[ nh(X n ) g = inf x2X fh(x) I(x)g : A Laplace principle is equivalent to a large deviations principle with the same rate function (see Theorems 2.2.1 and 2.2. 3 in [4] for a proof) For our model of interest, we will prove the former in the theorem that follows, obtaining the latter as a direct consequence of it. Theorem 2.2 For n 2 IN , let X n be the piecewise linear interpolation of fX n j ; j = 0; 1; ng as de ned in (1.4) Under Hypothesis H.1 ....

....: 1=k n 2; n (1 1 kn ) B 1 B 2 ) if t 2 [1 1=k n ; 1] Finally we de ne the admissible control measure n to be the random probability measure de ned for Borel subsets B 1 , B 2 of S and C of [0; 1] through n (B 1 B 2 C) Z C n (B 1 B 2 jt)dt: Theorem A.5. 6 in [4] provides us with a convenient decomposition for the measures n (d dy dt) in terms of the rst marginal of n (d dyjt) More precisely, if for B 1 2 B(S) we de ne the rst marginal n 1 (d jt) of n (d dyjt) through n 1 (B 1 jt) n (B 1 Sjt) then for ....

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P. Dupuis, R. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. John Wiley & Sons, 1996.

.... This research was supported in part by the National Science Foundation (NSF DMS9704426) and the Army Research O#ce (DAAD19 99 1 0223) 1 1 Introduction The theory of large deviations is one of the classical areas in probability and statistics (see for example [23, 7, 6, 13, 11] The book [10] develops an approach to this topic that is based on proving the convergence of solutions to variational problems. The starting point for this approach is the fact that the large deviation principle (LDP) is equivalent to what is called a Laplace principle (see Definition 4.2 below) if the ....

....this approach is the fact that the large deviation principle (LDP) is equivalent to what is called a Laplace principle (see Definition 4.2 below) if the underlying space is Polish. This is a consequence of Varadhan s Lemma [24] and Bryc s converse to Varadhan s Lemma [2] We refer the reader to [10] for the elementary proof. A key step in the approach is the representation of the pre limit normalized expectations in the statement of the Laplace principle by value functions (minimal cost functions) of certain stochastic optimal control problems. The large deviation problem then reduces to ....

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P. Dupuis and R. S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. John Wiley and Sons, Inc., 1997.

.... 3 = f(x 1 ; x 2 ; x 3 ) x 1 0; x 3 = 0g; B 4 = f(x 1 ; x 2 ; x 3 ) x 1 0; x 3 0g: To analyze the probability that a local model follows a linear trajectory, one can consider a representation for the pre limit objects q n (y; in terms of a stochastic control problem (see also 7 [4] for a full exposition of this approach) We recall this representation below in Theorem 1. To state it we need a few de nitions. Let n 2 IN be xed. A control u n (y; v; t) is a measurable function mapping S n ZZ N [0; 1] into [0; 1) We will impose the following condition on a control: ....

....1 (t)dt = t 2 t 1 )L (t 1 ) t 2 ) t 1 ) t 2 t 1 = Z t 2 t 1 L( t) t) dt; where is the linear interpolation of between the times t 1 and t 2 . With this estimate established the construction of the trajectory is straightforward (e.g. as in Lemma 7.5. 4 of [4]) and is omitted. In Section 5 we will verify all the assumptions of Theorem 4 for some interesting models. The following monotonicity result of the regularity of the SM shows that if part 4 of Condition 4 holds for the full model, then it holds for all corresponding local models as well (for a ....

P. Dupuis and R. S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. John Wiley & Sons, New York, 1997.

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P. Dupuis and R. Ellis. A weak convergence approach to the theory of large deviations. John Wiley & Sons Inc., New York, 1997. A Wiley-Interscience Publication.

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Dupuis, P. and Ellis, R. (1997) A Weak Convergence Approach to the Theory of Large Deviations. Join Wiley & Sons, New York.

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P. Dupuis and R. S. Ellis, A weak convergence approach to the theory of large deviations, John Wiley & Sons, 1997. 17

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P. Dupuis and R.S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. LCDS Report #93-6, Brown University, RI 1993.

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, 60 (2), 353-394. Dupuis, P. and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations Wiley Series in Probability and Statistics, New York: John Wiley and Sons,

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P. Dupuis and R.S. Ellis, Weak Convergence Approach to the Theory of Large Deviations. New York: John Wiley and Sons, 1997.

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