Martingale inequalities are important in the study of stochastic differential equations. For example, martingale inequalities are used to establish bounds of solutions to stochastic differential equations. In this chapter, we present several important martingale inequalities.

**Theorem 24.1** (Doob’s Submartingale Inequality). *Let* {*Y*_{t} : *a* ≤ *t* ≤ *b*} *be a right-continuous submartingale. Then for any* > 0, *we have*

(24.1)

*where Y*_{b}^{+} = max(*Y*_{b}, 0). *In particular, if the Y*_{t} *is a right-continuous martingale, then for any* > 0, *we have*

(24.2)

**24.1.** Let {*X*_{i} : *i* = 0, 1, …, *n*} be a submartingale adapted to a filtration {_{i} : *i* = 0, 1, …, *n*}. Show that for every λ > 0, we have

where *M*_{n} = max{*X*_{i} : *i* = 0, 1, …, *n*}.

**24.2** (Doob’s Maximal Inequality). Let {*X*_{i} : *i* = 1, 2, …, *n*} be a martingale or nonnegative submartingale adapted to the ...

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