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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale.
Apr 24, 2022 · The theory of martingales is beautiful, elegant, and mostly accessible in discrete time, when \( T = \N \). But as with the theory of Markov processes, martingale theory is technically much more complicated in continuous time, when \( T = [0, \infty) \).
The above examples illustrate two important kinds of martingales: those obtained as sums of independent random variables (each with mean zero) and those obtained products of independent random variables (each with mean one).
Jun 9, 2021 · Learn the basics of martingale theory, a general form of statistical dependence that subsumes many classical limit theorems. See examples, properties, and criteria for uniform integrability and closure of martingales and submartingales.
The theory of martingales plays a very important ans ueful role in the study of stochastic processes. A formal definition is given below. Definition 5.1. Let (Ω, F, P ) be a probability space. A martingale se-quence of length n is a chain X 1, X 2, , Xn of random variables and corre-. · · ·.
Most real world asset prices are not martingales, even in theory. So why is this theorem relevant to finance? 1. Many asset prices behave approximately like martingales in the short term. 2. According to the fundamental theorem of asset pricing, as
Martingales. Let Fn be increasing sequence of σ-fields (called a filtration). A sequence Xn is adapted to Fn if Xn ∈ Fn for all n. If Xn is an adapted sequence (with E |Xn| < ∞) then it is called a martingale if. E (Xn+1|Fn) = Xn for all n.
