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Unformatted text preview: Continuous Time Martingales Probability space (Ω , F ,P ) and a filtration ( F t , t ≥ 0) satisfying the “usual conditions” F s ⊆ F t ⊆ F for all s < t (increasing) F s = T t>s F t for all s (right continuous) A ⊆ B ∈ F , P ( B ) = 0 ⇒ A ∈ F (complete) (Ω , F ,P, {F t } ) is called a filtered probability space: F t represents history up to time t . We say that a process { X t , t ≥ } is adapted to {F t , t ≥ } if X t is measurable with respect to F t . A filtration {F t , t ≥ } is generated by the process { X t , t ≥ } if F t is the smallest σ field with respect to which all X s , s ≤ t are measurable. Definition . A process M = { M t , t ≥ } adapted to {F t , t ≥ } is called a submartingale (supermartingale) iff (i) E  M t  < ∞ for all t (ii) E ( M t F s ) ( ≤ ) ≥ M s a.s. for all s < t . An adapted process is called a martingale if and only if it is both a submartingale and a supermartingale (i.e. (ii) is of the form E ( M t F s ) = M s ). A martingale is square integrable if EM 2 t < ∞ for each t ≥ 0. 1 • Let us recall that a simple random walk is a martingale with respect to the natural filtration: for a sequence X 1 ,X 2 ,... of i.i.d. rv’s such that X 1 = 1 with probability 1/2 1 with probability 1/2 the stochastic process S n = X 1 + X 2 + ··· + X n satisfies E [ S n +1  σ ( S 1 ,...,S n )] = S n . The process { S n } n ≥ 1 can be used to model gains and losses from a simple game where we repeatedly toss a fair coin and earn $1, when a head occurs, and lose $1, otherwise. • A martingale , in English and French, is a system by which a gambler tries to become rich. The word appears in the dictionary of the French Academy as early as 1762, where it is said to refer to the particular system where the gambler doubles his bet until he finally wins. In older sense the word martingale refers to the strap of a horse’s harness that passes from the noseband through the forelegs. The concept of a martingale has been introduced to prob ability theory by the French mathematician Jean Ville, who initially called any strategy of placing bets a martingale. Later he changed the meaning of the word. In the context of coin tossing he called the capital processes martingales, and he for mulated condition that resembles the modern definition....
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This note was uploaded on 03/05/2012 for the course ACTSC 446 taught by Professor Adam during the Winter '09 term at Waterloo.
 Winter '09
 Adam

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