TL;DR
A stochastic process where \( E[M_t | \mathcal{F}_s] \geq M_s \) for \( s \leq t \). It has an upward tendency on average. A supermartingale has the reverse inequality.
Submartingale
A stochastic process where \( E[M_t | \mathcal{F}_s] \geq M_s \) for \( s \leq t \). It has an upward tendency on average. A supermartingale has the reverse inequality.
Why it matters for interviews
Stock prices under the physical measure are typically submartingales (positive expected return). The Doob decomposition and maximal inequalities for submartingales are used in probability theory and mathematical finance.
Definition and Mathematical Foundation
A stochastic process where \( E[M_t | \mathcal{F}_s] \geq M_s \) for \( s \leq t \). It has an upward tendency on average. A supermartingale has the reverse inequality.
Application in Quantitative Finance
Stock prices under the physical measure are typically submartingales (positive expected return). The Doob decomposition and maximal inequalities for submartingales are used in probability theory and mathematical finance.
Related Terms
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