Lecture 3: Martingales: deﬁnition, examples 3 EX 3.11 (Product of iid RVs with mean 1) Same setup with X 0 = 1, X i 0 and E[X 1] = 1.Deﬁne M n = Y i n X i: Note that EjM submartingale, supermartingale). Elementary Proof of Bounded Convergence Theorem 60 6.23. A martingale is a process where ExpectedValue(Mn+1) = Mn. (b) … In fact, one may replace IR by any ... For martingale theory, we will generally use IN for the index set, and we ... i=1 µi is a martingale. This is a trivial consequence of the deﬁnition of a martingale. Martingale Proof of Kolmogorov 0-1 Law 63 7.27. Proof. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. (a) Show that {Zn} is a martingale. References [Dur10]Rick Durrett. Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time. So I think I understand the idea of the martingale and the objective of the proof. Proposition 1. Then the processes 1. Example 2.2 Another construction which is often used is what might be called For a proof, see either Ash or Billingsley. We will return to many of these examples in subsequent sections. submartingale, supermartingale), and τ is an arbitrary F-stopping time. But E[X 0] = 1 6= 0 . The proof is not di cult, but the details are not particularly enlightening from our current perspective. More general symetric random walks. The martingale difference sequence {»n} has the following properties: (a) the random variable »n is a function of Fn; and (b) for every n ‚0, (5) E(»n¯1 jFn) ˘0. We give an example that shows that the conditions of the Martingale Convergence Theorem do not guarantee convergence of expec-tations. Example 172 (Examples of continuous martingales) Let Wt be a standard Brownian motion process. Proposition 1. Martingale Theory Problem set 3, with solutions Martingales The solutions of problems 1,2,3,4,5,6, and 11 are written down. The Dazzber brand specializes in producing high quality,safe and comfortable pet products,we focus on every single detail,devoting ourselves to provide the best service for you and your pet. The martingale difference sequence {»n} has the following properties: (a) the random variable »n is a function of Fn; and (b) for every n ‚0, (5) E(»n¯1 jFn) ˘0. Then for every n ‚0, The rest will come soon. 6.22. ngis a martingale, then the stopped process XT = fX T^ngis also a martingale. Then for every n ‚0, Necessary and Su cient Conditions for L1 convergence 60 UI Martingales 62 7.24. Wt 2. Levy's 'Upward' Theorem 62 7.26. E(X n+1jF n) = R2n + E(2 n+1) + 2R nE(n+1) (n+ 1)˙2 = R2 n+ ˙2 (n+ 1)˙2 = R2 n n˙2 = X n: 3. Martingale Proof of the Strong Law 64 3 Let fS ngbe SRW started at 1 and T= inffn>0 : S n= 0g: Then fS T^ngis a nonnegative MG. This is a trivial consequence of the deﬁnition of a martingale. CONDITIONAL EXPECTATION AND MARTINGALES 1. The goal for the remainder of this section is to give some classical examples of martingales, and by doing so, to show the wide variety of applications in which martingales occur. Let {Xn} be a martingale relative to {Yn}, with martingale difference sequence {»n}. At each stage a ball is drawn, and is then replaced in the urn along with another ball of the same color. Let Zn be the fraction of white balls in the urn after the nth iteration. Optional sampling theorem: Suppose X =(Xn,Fn) is a martingale (resp. Corollary 1. In particular, for all nwe have E(X T^n) = E(X 0): This is part (ii) of [4, Theorem 10.9], and an outline of the proof can be found there. So to prove something is a martingale we can prove that or equivalently that ExpectedValue(Mn+1) - ExpectedValue (Mn) = 0. Martingale collars are designed to bo escape proof and correcting behavior the dogs perfect for trainging, give you gentle control over your pet. UI Martingales 62 7.25. Examples. Example: An urn initially contains one white and one black ball. Corollary 1. Proof. Let {Xn} be a martingale relative to {Yn}, with martingale difference sequence {»n}. Then the stopped process Xτ =(Xτ∧n,Fn; n ≥ 0) is also a martingale (resp. It can only converge to 0 . Levy's 'Downward' Theorem 63 7.28.