---
tag: 6M852B4RA8
kind: concept
name: Submartingale
msc: [60G42, 60G44]
specializes: [0QH5ENKZM1]
sources:
  - url: https://en.wikipedia.org/wiki/Martingale_(probability_theory)
---

## Summary

A submartingale is an adapted, integrable [[stochastic process|0QH5ENKZM1]] whose [[conditional expectation|0TTDGTEFMN]] of a future value, given the present information, is at least the present value.
It models the fortune in a game that is favourable or fair to the player.
Every [[martingale|0D0GBNDKDH]] is a submartingale, and convex functions of martingales give many more examples.
Doob's inequalities and convergence theorems are proved for submartingales.

## Explanations

### L1 human

A submartingale tends to go up, or at least not down, on average.
Given everything known today, the best forecast of tomorrow's value is at least today's value.
The value can still go down on some days; only the average trend is upward.
If $S_n$ is a fair coin-tossing walk, then $S_n^2$ is a submartingale: squaring adds $1$ on average at each step.