## Abstract

Let (Ω, F,
is satisfied. The exceptional null set may depend on

*P*; F(*t*),*t*∈*I*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*$$
x\left( s \right) \geqslant E\left\{ {x\left( t \right)|F\left( s \right)} \right\}{\text{ a}}{\text{.s}}{\text{. if }}s < t
$$

(1.1)

*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.## Keywords

Conditional Expectation Optional Time Uniform Integrability Martingale Theory Countable Dense Subset
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer-Verlag Berlin Heidelberg 2001