Elements of Martingale Theory

  • Joseph L. Doob
Part of the Classics in Mathematics book series (CLASSICS)


Let (Ω, F, P; F(t), tI be a filtered probability space, and let x(•), F(•) be a process on this space, with state space (ℝ̄, F(ℝ̄)). The process is called a supermartingale if the process random variables are integrable and if the supermartingale inequality
$$ x\left( s \right) \geqslant E\left\{ {x\left( t \right)|F\left( s \right)} \right\}{\text{ a}}{\text{.s}}{\text{. if }}s < t $$
is satisfied. The exceptional null set may depend on s and t. If I is a set of consecutive integers, inequality (1.1) for t=s+1 implies (1.1) for all pairs s, t with s<t. If the inequality is reversed the process is called a submartingale, and if there is equality in (1.1) the process is called a martingale. The martingale definition is sometimes also applied to complex-valued or vector-valued random variables, but in this book the state space will always be (ℝ̄, F(ℝ̄)) unless some other state space is specified. Martingale theory refers to the mathematics of supermartingales and submartingales as well as martingales.


Conditional Expectation Optional Time Uniform Integrability Martingale Theory Countable Dense Subset 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Joseph L. Doob
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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