Local martingales form a very important class of processes in the theory of stochastic calculus.This is because the local martingale property is preserved by the stochastic integral, but the martingale property is not.Examples of local martingales which are not proper martingales are given by solutions to the stochastic differential equation We give a necessary and sufficient condition for the process Z to be a true martingale in the case where M t = R t 0 b(Y u)dW u and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a
local martingale M is itself a continuous local martingale. I'll try to give some intuition without too much mathematical details. For example, think about a fair game of roulette (fair as in your expected profit / loss is 0). Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time. Then apply martingale property to the stopped martingale M t_n and since the stopped martingale and the original one coincide up to times less than t_n, you get that M is in fact a martingale. Martingale and semi-martingale have very precise mathematical definitions, so it's definitely not something easy to understand. Now, we prove some theorems. A local martingale is usually de ned in an inclusive manner: it includes all martingales as special cases.
In fact, that's were the name comes from. In teoria della probabilità, una martingala locale è un tipo di processo stocastico che soddisfa una versione locale della proprietà delle martingale.I due concetti non coincidono: ogni martingala è una martingala locale, ma non vale il viceversa, anche se ogni martingala locale limitata è una martingala. If X stoped at time t_n is martingale for every t_n (sequence growing to infinty), then X is a martingale: Pick s,t arbitrary and t_n biger than both. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Theorem 3 (The Optional Stopping Theorem) Let Xbe a continuous local martingale. A martingale process is essentially a betting strategy on a fair game. I think to understand the martingale/local martingale distinction, it helps to bring in a third class of processes, the uniformly integrable martingale. CONDITIONAL EXPECTATION AND MARTINGALES 1. 2 Continuous Local Martingales The first section was supposed to convince you why you should care about continuous local martingales. If S Tare stopping times, and X T^tis a uniformly integrable martingale, then E[X TjF S] = X S. Martingale is a stochastic process that it's expectation is equal to the current value.