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Markov Process Vidya Sagar M. Sc (student of IVth semester) Roll no. :- 5 Department of Statistics CONTENTS Introduction Stochastic process Martingale process classification Martingale process Submartingale Process supermartingale Process Application of Martingale process INTRODUCTION • Martingale process is play vital role in field of management and many others fields. The concept of martingale in probability theory was introduced by Paul Lévy in 1934, though he did not name them: the term "martingale" was introduced later by Ville (1939), who also extended the definition to continuous martingales. Much of the original development of the theory was done by Joseph Leo Doob among others. Part of the motivation for that work was to show the impossibility of successful betting strategies. stochastic process A stochastic process is the collection of random variable {x(t),tϵT} defined on the given probability space indexed by parameter t where t varies from an index of T. Martingale process In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters. The expected position of a martingale some time in the future is exactly where it is now. Martingale process A stochastic process {X1, X2, X3…. } is a martingale if Xt = E[Xt+1 |Ft ] where t≥0 and Ft = {X1, X2, X3…. Xt } =>a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Example of Markov process • • • An unbiased random walk is an example of a martingale. A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair. Suppose Xn is a gambler's fortune after n tosses of a fair coin, where the gambler wins $1 if the coin comes up heads and loses $1 if it's tails. The gambler's conditional expected fortune after the next trial, given the history, is equal to his present fortune. This sequence is thus a martingale. Classification of Martingale process Martingale process has been classified into two type submartingale supermartingale Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that is both a submartingale and a supermartingale is a martingale. Submartingale process A stochastic process {X1, X2, X3…. } is a martingale if E[Xt+1 |Ft ] ≥ Xt where t≥0 and Ft = {X1, X2, X3…. Xt } In this case of subrmartingale, the gain is favourable for player. Supermartingale process A stochastic process {X1, X2, X3…. } is a martingale if E[Xt+1 |Ft ] ≤Xt where t≥0 and Ft = {X1, X2, X3…. Xt } In this case of supermartingale, the gain is unfavourable for player. Example of supermartingale and sub martingale Process •Consider again the gambler who wins $1 when a coin comes up heads and loses $1 when the coin comes up tails. Suppose now that the coin may be biased, so that it comes up heads with probability p. –If p is equal to 1/2, the gambler on average neither wins nor loses money, and the gambler's fortune over time is a martingale. –If p is less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale. –If p is greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale. Application of Martingale process Security price modelling Portfolio insurance Claims forecasting Asset-liability matching Premium rating Asset and Option pricing Investment strategy Control of large funds Application of Martingale process Equitable profit allocation Mortality/Morbidity modelling Macroeconomic growth modelling Market product pricing Risk theory Solvency and Ruin calculations Early warning systems Consumption/Investment problems Dynamic hedging strategies. Thank you