Wald's Martingale
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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, Wald's martingale is the name sometimes given to a martingale used to study sums of i.i.d. random variables. It is named after the mathematician
Abraham Wald Abraham Wald (; ; , ;  – ) was a Hungarian and American mathematician and statistician who contributed to decision theory, geometry and econometrics, and founded the field of sequential analysis. One of his well-known statistical works was ...
, who used these ideas in a series of influential publications. Wald's martingale can be seen as discrete-time equivalent of the
Doléans-Dade exponential In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale ''X'' is the unique strong solution of the stochastic differential equation dY_t = Y_\,dX_t,\quad\quad Y_0=1,where Y_ denotes the process of left lim ...
.


Formal statement

Let (X_n)_ be a sequence of i.i.d. random variables whose moment generating function M: \theta \mapsto \mathbb(e^) is finite for some \theta > 0, and let S_n = X_1 + \cdots + X_n, with S_0 = 0. Then, the process (W_n)_ defined by :W_n = \frac is a martingale known as ''Wald's martingale''. In particular, \mathbb(W_n) = 1 for all n \geq 0.


See also

* Martingale *
geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It ...
*
Doléans-Dade exponential In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale ''X'' is the unique strong solution of the stochastic differential equation dY_t = Y_\,dX_t,\quad\quad Y_0=1,where Y_ denotes the process of left lim ...
*
Wald's equation In probability theory, Wald's equation, Wald's identity or Wald's lemma is an important identity (mathematics), identity that simplifies the calculation of the expected value of the sum of a random number of random quantities. In its simplest form, ...


Notes

Martingale theory {{probability-stub