P. Williams, Probability with Martingales. New York, NY: Cambridge Press, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the conditional expectation by integration over B. Thus instead of a number we get a # measurable function called the conditional expectation given # and is written E p ( #) It is not at all obvious that such a function should exist. It is a fundamental result of Kolmogorov (see for instance [15], p.84) that such conditional expectations exist. Theorem 1 (Kolmogorov) Let (S, #, p) be a probability triple, X be in and # be a sub # algebra of #, then there exists a Y (S, #, p) such that #B # #. Y dp. 1) Not only does the conditional expectation exist, but it has a lot of ....

David Williams. Probability with Martingales. CUP, Cambridge, 1991.

....will end up in minus infinity almost surely. This means that the possibly existing limit process of the cumulative processes is singular. However, it may be non zero, and the convergence may happen in L 1 . 2. 2 Convergence Recall the following basic properties of discrete martingales (see e.g. [Wil91]) Lemma 1. Let (M n : n 2 N) be a supermartingale bounded in L 1 : sup E(jM n j) 1. Then, almost surely, M = lim n 1 M n exists and is finite. Lemma 2. Let (M n : n 2 N) be a martingale for which M n 2 L 2 for all n. Then it is bounded in L 2 if and only if (M k M k 1 ) 1; 1) ....

D. Williams. Probability with Martingales. Cambridge University Press, 1991.

....it possible for us to apply it for our case. 11 Consider the trivial identity J k IE . By the construction, N 1 is a stopping time with respect to A k , that is, k #(A j : j k) on the other hand, J k k 1 ) is independent of #(A j : j 1) cf. [23]) hence N 1 and J k 1 ) are independent for every k. This yields that = 0. This means that we only need to deal with . Define now the functions Y n and Z n , defined on E as (5.8) Y n (s) IE(J n n 1 = s) and Z n (s) IE(T n n 1 = s) ....

....for every integer m n, F has an absolute continuous component, G#F (m n)# . A.1. Modes of convergence. Let IN be a sequence of random variables, and Y a rn=andom variable, all defined on (#, IP ) Let us summarize the types of convergence used in the following sections (cf. [23]) almost surely: Y n Y a.s. if IP Y = 1 as n . in probability: Y n Y if for every # 0 IP Y # . L convergence: if Y n , Y , n IN and IE( Y n . total variation: Y n tv Y if sup A#E IP Y A Almost sure convergence and ....

[Article contains additional citation context not shown here]

D. Williams, Probability with martingales, 3 rd ed., Cambridge University Press, Cambridge, 1991.

....In this section we give a very brief introduction to the fundamental properties of conditional expectation. The reader wishing to know more is advised to consult any introductory treatise on measure theoretic probability, a beautiful specimen of which is Williams Probability with martingales [29]. However, before we begin the exposition, we would like to eliminate a potential source of confusion concerning the use of the term conditional , as it occurs in the expressions conditional probability and conditional expectation . Conditional probability P (A B) represents a situation ....

D. Williams. Probability with martingales. Cambridge University Press, Cambridge, 1991.

....for player 1 that at all s 2 U plays according to an optimal distribution of the matrix game corresponding to Ppre 1 (w) s) and at all s 2 S n U plays arbitrarily. Fix a state s 0 2 S and an arbitrary strategy 2 2 Pi 2 . The process fH n g n0 defined by H n = w( Theta n ) is a submartingale [Wil91] in fact, from w(s) Ppre 1 (w) s) for s 2 U and from the choice of 1 follows that E s0 fH n 1 j H 0 ; H 1 ; H n g H n for all n 0. Hence, we have E s0 fH n g H 0 = w(s 0 ) Moreover, since w(s) 1 at all s 2 S, by inspection we have E s0 fH n g Pr (2nU ) where 2nU is the ....

D. Williams. Probability With Martingales. Cambridge University Press, 1991. 12

.... ; X t ) note that T is a stopping time for this ltration) Once this is done, we get E[T ] E[Z T ] Z 0 = N P N j=1 1 j fl Gamma (1 Gamma fl)d N(1 log N) fl Gamma (1 Gamma fl)d where the rst inequality is an application of Doob s Optional Stopping Theorem (see e.g. Williams [22]) It remains to prove the supermartingale property of fZ t g 1 t=0 , i.e. that E[Z t 1 Gamma Z t j F t ] 0: 7.3) Suppose that Y t = n. For n = 0, we have T t, so that Z t 1 = Z t = T , and (7.3) becomes trivial. Hence, we can assume that n 2 f1; Ng. Obviously, the 15 increments ....

Williams, D. (1991) Probability with Martingales, Cambridge University Press.

....white noise with distribution P 0 ; that is, let X = X x ; x 2 Z d ) be a family of independent identically distributed random elements of E, each X x having distribution P 0 , indexed by the integer lattice. For existence of such an ( F ; P ) and X, see for example section 8. 7 of Williams [33]. Suppose R is a collection of nite subsets ( regions ) of Z d ; for example, R might be the collection of all lattice boxes. Assume R is translation invariant, in the sense that if B 2 R then y B 2 R for all y 2 Z d , where y denotes translation by y, so y B = fx y : x 2 Bg. By a ....

....2 ] E [E [ 0 (B) 0 (1) 2 jF 0 ] E [ 0 (B) 0 (1) 2 ] which is uniformly bounded by the bounded moments condition. Similarly, E [ F 0 (B) F 0 ) 2 ] E [ 0 (B) 0 (1) 2 ] By the stabilization and bounded moments conditions this is uniformly bounded, and moreover (see [33] A 13.2(f) for any sequence ( B n ) n 1 in class C, E [jF 0 ( B n ) 2 F 2 0 j] 0. Returning to the given sequence (B n ) we now use the strong translation invariance of C. Given 0, let B 0 n be a sequence of subregions of B n with relative size at least 1 and with the ....

[Article contains additional citation context not shown here]

Williams, D. (1991) Probability with Martingales. Cambridge University Press, Cambridge.

.... : X (k Gamma1) j = V ( I k ) Gamma V i E( I k j X (1) X (k Gamma1) j = V ( I k ) and so E( V ( I) K X k=1 2 k E( V ( I k ) K X k=1 2 k V ( I k ) V ( I) 2 The values of Q k = P k r=1 k ( I r Gamma I) form a martingale (Williams 1991) with respect to the sigma fields defined by Z (k) X (1) X (k) We make no formal use of martingales, but the ideas underly our methods. 4.3 Square root rule for k This section presents a simple rule for choosing the values k . We propose taking k p k, in the ....

Williams, D. (1991), Probability with Martingales, Cambridge University Press, Cambridge.

....#) # 2 log(n) i.o. 1. The last equation is true because of the following two facts: 1. X i s are independent; 18 2. we have # n#1 P X n (1 #) # 2 log(n) Exponential Inequality # 1 # log(n) n (1 #) 2 , which diverges. and then we apply the second Borel Cantelli Lemma, [Wil91]. So we proved (2.18) To prove (2.17) from Lemma 2.11, and recall 1 2 k 2 1 2 k 2 # 1 2 # (log 2 (n) c 1 , when n is large, we only need to prove that P ## 1 2 # (log 2 (n) c 1 # M I,k (n) 1 #) # 2 log(n) i.o. # = 0. Recall c 1 is close to 1. It is not hard ....

David Williams. Probability with Martingales. Cambridge, 1991.

.... 1 be a strategy for player 1 that at all s 2 U plays according to an optimal distribution of the matrix game corresponding to Ppre 1 (w) s) and at all s 2 S n U plays arbitrarily. Fix a state s0 2 S and an arbitrary strategy 2 2 2 . The process fHng n 0 de ned by Hn = w( n) is a submartingale [30]: in fact, from w(s) Ppre 1 (w) s) for s 2 U and from the choice of 1 follows that E 1 ; 2 s 0 fHn 1 j H0 ; H1 ; Hng Hn for all n 0. Hence, we have E 1 ; 2 s 0 fHng H0 = w(s0 ) Moreover, since w(s) 1 at all s 2 S, by inspection we have E 1 ; 2 s 0 fHng Pr ....

D. Williams. Probability With Martingales. Cambridge University Press, 1991. 9

....approximation algorithms for the value V of Asian options, along with error bounds. For the error analysis we show several large deviation results for random walks that may be of independent interest. To define the value of an American option, we need to use the notion of a stopping time[27]. Let# be the sample space of all possible coin toss paths # defined in Section 1.1. A stopping time is a random variable # ## #f0# 1# 2# # # # # ng # f1g with the property that for each k # 0# 1# # # # # n# 1, the set f# # kg belongs to the # algebra F k . This means that membership in the ....

D. Williams. Probability with Martingales . Cambridge Univ. Press, 1992.

....density function (pdf) p( a k ) is known. For this probability density and the remaining probability densities in the chapter, we implicitly assume there is some underlying probability space, and random variables with densities are constructed using appropriate measurability conditions (see [197] for a treatment of these technical concerns) By using the state transition equation, we can obtain a pdf for x k 1 , which is represented by p(x k 1 jx k ; u k ) 44 In general, we will use the notation, F , to refer to minimal subsets of X that can be inferred from the arguments. The role of ....

....given in Section 2.2.2, and also in [26] 27] It is interesting to note that the densities in Figure 2.11 appear to be Gaussian, even though the uncertainty model is specified as a uniform density. The effects of the control uncertainty combine additively; therefore, the Central Limit Theorem [197] implies that the densities will tend toward Gaussian. Even in Figure 2.12 at k = 11, the probability mass appears to be two disjoint Gaussians. We have observed that, as the probability mass divides because of obstacles, the individual components also tend toward being Gaussian. These results ....

P. Williams. Probability with Martingales. Cambridge Press, New York, NY, 1991.

....In this section we give a very brief introduction to the fundamental properties of conditional expectation. The reader wishing to know more is advised to consult any introductory treatise on measure theoretic probability, a beautiful specimen of which is Williams Probability with martingales [23]. We begin with a simple example. Suppose we have a variable X on a sample space # which takes values 0 and 1 both with probability 1 2 . Let A i be the subset of # on which X takes value i, then P (A i ) 1 2 . Suppose furthermore that we cannot measure X directly, but can only measure X Y ....

D. Williams. Probability with martingales. Cambridge University Press, Cambridge, 1991.

....ffi and, fi; respectively. We dene the oe elds G k = oe( p j p k) G = k1 G k : Using the independence of 1 ; n ; and the 0 1 Law, we get that E[i] E 2 4 i fi fi fi fi fi fi k1 G k 3 5 : Since G k is a decreasing sequence of oe elds, L#vy s Downward theorem (cf [43], pp 41 136) tells us that E 2 4 i fi fi fi fi fi fi k1 G k 3 5 = lim k 1 E [i jG k ] Hence we can rst determine E[ijG k ] and then take the limit to obtain the expectation in (3.8) The result is e d Y i=1 1 r Q k1 h fl i t 2 k 2 2 1 i exp X k1 i i k ....

D. Williams, Probability with Martingales, Cambridge University Press, 1991.

....In summary, it is fair to say that the contributions of Chernoff Hoeffding bounds are likely endure, and the value of Chernoff tight bounds is considerable. Moreover, exponential based generating functions have turned out to be fundamental to the theory of martingales (c.f. Do 53] KT 81] [Wi 91]) and have had significant application to specific processes 6 such as Brownian motion. The relationship between Chernoff Hoeffding bounds and martingales is briefly discussed in Section 6. In this paper, we explore the issue of attaining sharper expressive estimates. More importantly, we ....

....the x i are independent and have mean zero. In fact, they need not be independent; the probability distribution for x i can depend on the outcome of the previous x i s, provided the mean is always zero. A rigorous theory rests upon a foundation of measure theory; see [Do 53] KT 81] Lo 77] or [Wi 91]. Among the relevant developments is the following special case of Doob s submartingale inequality. Theorem(Doob) Let Y 1 ; Y 2 ; Yn be a martingale. Then for a 0, P rf max 1kn Y k ag min 0 e Gammaa E[e Yn ] Proof: Actually, the theorem is more general than this statement. ....

[Article contains additional citation context not shown here]

D. Williams. Probability with Martingales. Cambridge University Press, 1991.

....as a paradigm for the construction of numerical schemes for functional di erential equations with unbounded memory, cf. 5] 14] 16] we do not attempt to give a complete list of references here) 1. 2 Additional background For the theoretical prerequisites on probability concepts we refer to [29]. Stochastic calculus and stochastic ordinary di erential equation (SODEs) are treated in [1] and [19] for the theory of stochastic delay di erential equation (SDDEs) see (for example) 21, 24, 25] SDDEs with general in nite memory are treated in [18] the case of fading memory is considered in ....

David Williams, Probability with martingales. Cambridge University Press, Cambridge, 1991.

....E. By (2) the indicator 1E is almost everywhere equal to any version of the conditional expectation P(Ejoe(C; D) The latter equivalence class contains P(EjC) as can be seen by using the independence of oe(E; C) ae E [P] and D, and by applying a standard property of conditional expectation [see Williams (1991), page 88, property (k) Hence there is a C 2 C with E = C [P] 2.3 Comparison of differentiably generated oe algebras. For proving rigorously that certain oe algebras are not equivalent modulo null sets, it appears worthwhile to state the following simple facts. Lemma. Let n; m 1 ; m 2 2 N and ....

Williams, D. (1991). Probability with Martingales. Cambridge: Cambridge University Press.

....terms in the sum have bounded variation, it is possible to formulate bounds almost as sharp as the ones presented above. The proper technical tool turns out to be a so called martingale. Martingales and conditional expectation have been studied to great extent in the literature see for instance Williams (1991) below we give a restricted definition that su#ces for our purposes. Definition 3.20. Given a function f : Z n # Z the random variable Y =E[f(X 1 , X n ) X 1 , X i ] 3.84) is defined by Y =E[f(X 1 , X n ) X 1 , X i ) x 1 , x i ) 3.85) with probability Pr[ X 1 ....

Williams, D. (1991). Probability with Martingales. Cambridge: Cambridge University Press.

....adequate for this paper. Theorem (Optional Stopping Theorem) Suppose that S 0 , S 1 , is a martingale and # is a bounded stopping time (i.e. #K # # # # K) Then, E(S # ) E(S 0 ) For a proof of this result and proofs of stronger versions of the optional stopping theorem, consult [7]. 3 Harmonic Functions Suppose that f : Z 2 # R. The Laplacian of f is defined as #f(x) # # 1 4 # e =1 f(x e) # # f(x) 1 4 # e =1 f(x e) 1 4 # e =1 f(x) 3 This definition gives a more natural way to think about the Laplacian; namely as the di#erence between the ....

D. Williams. Probability with Martingales. Cambridge University Press, Cambridge, Great Britain, 1991. 10

No context found.

P. Williams, Probability with Martingales. New York, NY: Cambridge Press, 1991.

No context found.

P. Williams. Probability with Martingales. Cambridge Press, New York, NY, 1991. 64

No context found.

David Williams, Probability with Martingales, Cambridge University Press, 1991. 33

No context found.

WILLIAMS, D. (1991) Probability with Martingales. Cambridge University Press.

No context found.

D. Williams, Probability With Martingales. New York: Cambridge Univ. Press, 1991.

No context found.

D. Williams. Probability with Martingales. Cambridge University Press, Cambridge, 1991.

No context found.

Williams, D., 1991. "Probability with Martingales," Cambridge University Press, Cambridge, U.K.

No context found.

D. Williams. Probability with Martingales. Cambridge University Press, 1991.

No context found.

WILLIAMS, D. (1991) Probability with Martingales. Cambridge University Press.

No context found.

D. Williams, (1991), Probability with Martingales. Cambridge University Press.

No context found.

WILLIAMS D. (1992) Probability with Martingales,Cambridge Mathematical textbooks. 29

No context found.

Williams, David., Probability with Martingales. Cambridge University Press (1991), Cambridge. 32

No context found.

Williams, D. (1991): Probability with Martingales. Cambridge U. Press.

No context found.

David Williams. Probability with Martingales. Cambridge University Press, New York, 1991. 17

No context found.

D. Williams. Probability with martingales. Cambridge University Press, Cambridge, 1991.

No context found.

D. Williams, Probability with Martingales, Cambridge University Press, Cambridge, UK, 1991.

No context found.

D. Williams, Probability with martingales, Cambridge University Press, Cambridge (1991).

No context found.

D. Williams, Probability with martingales. Cambridge University Press, Cambridge, 1991. 13

No context found.

D. WILLIAMS, Probability with Martingales, Cambridge University Press, Cambridge, UK, 1991.

No context found.

David Williams. Probability with Martingales. Cambridge University Press, 1991.

No context found.

D. Williams. Probability with Martingales. Cambridge University Press, 1991.

No context found.

D. Williams. Probability with Martingales. CUP, Cambridge, 1991. 10

No context found.

D. Williams, Probability with martingales, Cambridge University Press, Cambridge (1991).

No context found.

D. Williams. Probability with martingales. Cambridge University Press, 1991. 28

No context found.

D. Williams, Probability with Martingales, Cambridge University Press, Cambridge. 1991. 25

No context found.

D. WILLIAMS. Probability with Martingales. University Press, Cambridge (1991).

No context found.

D. Williams, Probability with Martingales (Cambridge University Press, New York, 1991).

No context found.

D. Williams, Probability with Martingales, Cambridge University Press, 1991.

No context found.

D. Williams. Probability With Martingales. Cambridge University Press, 1991.

No context found.

D. Williams. Probability with Martingales, Cambridge Univ. Press, 1991. 23

No context found.

Williams, D., Probability with Martingales, Cambridge University Press, New York, 1991. 18

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC