Martingales: The Non-Arbitrage Principle in Discounted Asset Prices

Students often confuse the physical measure

\mathbb{P}

with the risk-neutral measure

\mathbb{Q}

. Crucially,

S_t

is typically a submartingale under

\mathbb{P}

(reflecting risk premium), but must be a martingale under

\mathbb{Q}

to enforce market equilibrium.

Institutional Reference: Advanced Stochastic Processes