Markov property
Quantitative Finance glossary · Probability, Statistics & Linear Algebra
Definition
Memorylessness: mathbbP(X_t+h in A,|,mathcalF_t) = mathbbP(X_t+h in A,|,X_t) — the future depends on the past only through the present. Holds for solutions of SDEs driven by Brownian motion (since Brownian increments are independent), which is why one-factor diffusion models admit PDE pricing (Feynman-Kac). Path-dependent payoffs (Asians, barriers) and non-Markovian SDEs (rough volatility, HJM general) lose this and require state augmentation or path-Monte Carlo.
Markov property in plain English
The Markov property is a fancy name for "memorylessness": to predict what happens next, all you need is where you are right now — the whole history of how you got here adds nothing. Picture a board game where your next move depends only on your current square, not the path you took to reach it. Many financial models are built on random motion (Brownian motion) that has exactly this feature, and that's a gift: it lets us price options by solving a clean equation instead of simulating every possible history. The property breaks, though, for payoffs that depend on the entire journey — like an option that pays based on a stock's average price, or one that's void if the price ever touched a barrier — and then you need heavier machinery to keep track of that extra history.
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