Elements of Martingale Theory
Let (Ω, F , P; F (t), t∈I be a filtered probability space, and let x(•), F (•) be a process on this space, with state space (ℝ̄, F (ℝ̄)). The process is called a supermartingale if the process random variables are integrable and if the supermartingale ine
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Elements of Martingale Theory
1. Definitions Let (D.,:F, P;:F(t), tel} be a filtered probability space, and let {x(·),:F(·)} be a process on this space, with state space (~, ~(~». The process is called a supermartingale if the process random variables are integrable and if the supermartingale inequality x(s) ~ E{x(t)I:F(s)}
a.s. if s < t
(1.1)
is satisfied. The exceptional null set may depend on sand t. If I is a set of consecutive integers, inequality (1.1) for t = s + I implies (1.1) for all pairs s, t with s < t. If the inequality is reversed the process is called a submartingale, and if there is equality in (1.1) the process is called a martingale. The martingale definition is sometimes also applied to complex-valued or vectorvalued random variables, but in this book the state space will always be (~, ~(~» unless some other state space is specified. Martingale theory refers to the mathematics of supermartingales and submartingales as well as martingales. A supermartingale is a mathematical model for the fortune of a player of a game in which the player has fortune x(s) at time s and given the past up to and including s the player expects his fortune at the later time t to be at most x(s). Thus a supermartingale represents an unfavorable game; in this context a martingale represents a fair game, and a submartingale represents a favorable game. The negative of a supermartingale is a submartingale, and an adapted process is a martingale if and only if it is both a supermartingale and a submartingale. In anyone of the three cases the process is said to be left [right] closed if I has a first [last] element and is said to be left [right] closable if a first [last] element can be adjoined to I, together with an appropriate (J algebra, to get an enlarged process which is left [right] closed and of the same type as the given one. If a process is right closed, say with last parameter element 00, the (J algebra :F(oo) is not involved in the appropriate version of (1.1) and is restricted only by the condition that the process is adapted. A martingale is always left closable by the pair J.L. Doob, Classical Potential Theory and Its Probably Counter © Springer-Verlag Berlin Heidelberg 2001
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2. Examples
[c, (0, Q)], where c is the common expected value of the process random variables. EXAMPLE. Let 1=71. in increasing order, let ~(n) consist, for each n, of the empty set and the whole probability space, and consider the process all of whose sample functions are identically 1. This process is a martingale and becomes a right closed martingale if the constant function 1 is adjoined at the end or a right closed supermartingale if the constant function 0 is adjoined at the end. Choice of Filtration Let {x('), ~(.)} be a supermartingale. The following remarks are made for the supermartingale case but are valid with the obvious changes in the other two cases. Define ~o(t) = ~{x(s),s s; t}. Then ~o(t) c ~(t) and {x('),~o(')} is a supermartingale because if s < t, x(s)
= E{x(s)l~o(s)}
~ E{E{x(t)I~(s)}I~o(s)}
= E{x(t)I~o
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