## Sunday, December 29, 2019

### An Introduction Of Martingale Theory - 1465 Words

Introduction to Martingale Theory Martingale Theory is a simple mathematical model which models sequence of a fair game . It is a stochastic process on some probability space {ÃŽ ©, F, P}. Originally it was used as a betting strategy during 18th century in France. In 1934, Paul LÃƒ ©vy introduced concept of Martingale in probability theory. It is named after Ã¢â‚¬ËœLa Grande Martingale,Ã¢â‚¬â„¢ which means strategy for even odd bets where bets are doubled every time we lose. Idea behind this betting strategy is tht one cannot expect gains without taking risks. To understand Martingale, we need to understand Filtration first. Filtration Available information is modelled by a sub-ÃÆ'-algebra F. A sequence of ÃÆ'-algebras Fn such that F0 Ã¢Å â€š F1 Ã¢Å â€šÃ¢â‚¬ ¦Ã¢Å â€š F. ThisÃ¢â‚¬ ¦show more contentÃ¢â‚¬ ¦Every time the person lost he had to double his bet so that the first win would help him regain from the previous losses plus win profit equal to the original stake . If X1, X2,Ã¢â‚¬ ¦, is a sequence of independent and identically distributed random variables with P(Xn = 1) = 1/2 , P(Xn = -1)= Ã‚ ½. Filtration (F_n )_n Fn = ÃÆ'(X1,Ã¢â‚¬ ¦,Xn). Then sequence (s_n )_(n=1)^Ã¢Ë†Å¾(simple random variable walk on Z) is martingale w.t.r. (F_n )_n as E(Sn | Fn-1) = E(Sn-1 + Xn | Fn-1) = Sn-1 + E(Xn | Fn-1) = Sn-1 + E(Xn) = Sn-1  PoylaÃ¢â‚¬â„¢s Urn Ã¢â‚¬â€œ A container has balls of two color, say red and blue (r and b). We pick out one ball, observe its color and place it back in the container along with another ball of the same color. Draw another ball from the same container. Probability of drawing red ball in first try is r/(r+b) and that of blue ball is b/(r+b). In second draw, probability of red ball being drawn if a red ball was drawn first time will be (r+1)/(r+b+1). Probability of red ball being drawn if a blue ball was drawn first time will be r/(r+b+x). Probability of a red ball being drawn on second draw P(Red : Draw =2) = ((r+1)/(r+b+1))*(r/(r+b))+((r/r+b+1)*(b/r+b)) = r/(r+b)  Example of Submartingale, Supermartingale : In a coin toss we have three coins. One unbiased and two biased. In the two biased coins, one has P(HEADS) = Ã‚ ¾ and the other has P(HEADS) = Ã‚ ¼. After tossing the coin if heads come, we win, tails, we lose. If we