Continuous-Time Martingales

Imagine a game with no edge — none for you, none for the house. However the game has gone so far, your best forecast of your fortune at any future moment is exactly what you hold right now. Not "roughly", not "on average over many players" — conditionally, given everything you know, the future expectation is the present. That is a martingale: a fair game written in mathematics.

The word has a disreputable past. La martingale was an eighteenth-century French betting strategy — double your stake after every loss — that "guaranteed" a profit and bankrupted generations of gamblers (we will see exactly why below). The twentieth century rescued the word: Joseph Doob turned the martingale into one of the central objects of probability theory, and modern finance made it the moral centre of the subject. The fundamental insight of derivative pricing is that fair prices are martingales: under the right probability measure, the discounted price of any traded asset must be a fair game — otherwise there is free money on the table. Every pricing formula you will ever meet, Black–Scholes included, is at heart the statement \mathbb{E}[M_t \mid \mathcal{F}_s] = M_s dressed up for work.

The definition

The discrete martingale — a fair game where the best forecast of tomorrow is today's value — lifts straight to continuous time. A process (M_t)_{t\ge 0} is a martingale with respect to a filtration (\mathcal{F}_t) when three things hold:

\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s.

Given everything known at the present time s, the best forecast of any future value M_t is exactly the current value M_s — a fair game running continuously in time. The third condition is the heart; the first two are the fine print that makes it meaningful. Taking expectations of both sides and using the tower property \mathbb{E}\big[\mathbb{E}[M_t \mid \mathcal{F}_s]\big] = \mathbb{E}[M_t], the mean never moves:

\mathbb{E}[M_t] = \mathbb{E}[M_0] \qquad \text{for every } t.

Be careful with the direction of that implication: a constant mean is a consequence of the martingale property, not a substitute for it. The definition is the conditional statement — it must hold given every possible history, not merely on average — and the "Watch out!" box below shows a process with a perfectly constant mean that is nowhere near a martingale.

Two key examples, verified in full

Two facts carry an astonishing amount of the theory. Throughout, fix times s \le t and use two properties of Brownian motion: the increment W_t - W_s is independent of the history \mathcal{F}_s, and it is distributed N(0,\, t-s) — so it has mean 0 and variance t-s. Every martingale verification you will ever do is a variation on the five-step dance below: split the future into present plus increment, use linearity, pull out what is known, drop the conditioning on what is independent, collect.

(a) Brownian motion is a martingale

Step 1 — split the future into present plus increment. Pure algebra:

\mathbb{E}[W_t \mid \mathcal{F}_s] = \mathbb{E}\big[\, W_s + (W_t - W_s) \,\big|\, \mathcal{F}_s \,\big].

Step 2 — split the conditional expectation. Conditional expectation is linear, so the two terms separate:

= \mathbb{E}[W_s \mid \mathcal{F}_s] + \mathbb{E}[\,W_t - W_s \mid \mathcal{F}_s\,].

Step 3 — pull out the known term. The present value W_s is \mathcal{F}_s-measurable (it is part of the history), so conditioning leaves it untouched: \mathbb{E}[W_s \mid \mathcal{F}_s] = W_s.

= W_s + \mathbb{E}[\,W_t - W_s \mid \mathcal{F}_s\,].

Step 4 — drop the conditioning on the increment. Because W_t - W_s is independent of \mathcal{F}_s, conditioning on the history changes nothing — its conditional mean equals its plain mean, which is 0:

\mathbb{E}[\,W_t - W_s \mid \mathcal{F}_s\,] = \mathbb{E}[\,W_t - W_s\,] = 0.

Step 5 — collect the terms.

\mathbb{E}[W_t \mid \mathcal{F}_s] = W_s + 0 = W_s.

So W_t is a martingale: the best forecast of any future value is today's value. Notice which two properties did all the work — independence of the increment let us drop the conditioning, and mean zero made the leftover vanish. Keep score of those two: the next example uses them again, one order higher.

(b) W_t^2 - t is a martingale

On its own W_t^2 drifts upward — it is a submartingale, since \mathbb{E}[W_t^2] = t grows with t. The claim is that subtracting t compensates that drift exactly. We compute \mathbb{E}[W_t^2 \mid \mathcal{F}_s].

Step 1 — write the future as present plus increment, then square. Using (p+q)^2 = p^2 + 2pq + q^2 with p = W_s and q = W_t - W_s:

\mathbb{E}[W_t^2 \mid \mathcal{F}_s] = \mathbb{E}\big[\, (W_s + (W_t - W_s))^2 \,\big|\, \mathcal{F}_s \,\big] = \mathbb{E}\big[\, W_s^2 + 2 W_s (W_t - W_s) + (W_t - W_s)^2 \,\big|\, \mathcal{F}_s \,\big].

Step 2 — split into three conditional expectations (linearity again):

= \mathbb{E}[W_s^2 \mid \mathcal{F}_s] + 2\,\mathbb{E}[\,W_s (W_t - W_s) \mid \mathcal{F}_s\,] + \mathbb{E}[\,(W_t - W_s)^2 \mid \mathcal{F}_s\,].

Step 3 — the first term. W_s^2 is \mathcal{F}_s-measurable, so it passes through untouched:

\mathbb{E}[W_s^2 \mid \mathcal{F}_s] = W_s^2.

Step 4 — the cross term. The factor W_s is known given \mathcal{F}_s, so it pulls out of the conditional expectation; what remains is the mean-zero increment:

2\,\mathbb{E}[\,W_s (W_t - W_s) \mid \mathcal{F}_s\,] = 2 W_s \, \mathbb{E}[\,W_t - W_s \mid \mathcal{F}_s\,] = 2 W_s \cdot 0 = 0.

Step 5 — the squared term. The increment is independent of \mathcal{F}_s, so its conditional second moment equals its plain second moment — which, since the increment has mean 0, is its variance t-s:

\mathbb{E}[\,(W_t - W_s)^2 \mid \mathcal{F}_s\,] = \mathbb{E}[\,(W_t - W_s)^2\,] = \operatorname{Var}(W_t - W_s) = t - s.

Step 6 — collect the three pieces.

\mathbb{E}[W_t^2 \mid \mathcal{F}_s] = W_s^2 + 0 + (t - s).

Step 7 — subtract t from both sides. Since t is a constant, \mathbb{E}[t \mid \mathcal{F}_s] = t, so

\mathbb{E}[\,W_t^2 - t \mid \mathcal{F}_s\,] = W_s^2 + (t - s) - t = W_s^2 - s.

The right-hand side is the process W_t^2 - t evaluated at the present time s — so W_t^2 - t is a martingale. The compensator that restores the fair-game balance is precisely t, your first glimpse of quadratic variation, \langle W\rangle_t = t.

Let (W_t) be a Brownian motion adapted to (\mathcal{F}_t). Then both W_t \qquad\text{and}\qquad W_t^2 - t are martingales: \mathbb{E}[W_t \mid \mathcal{F}_s] = W_s and \mathbb{E}[\,W_t^2 - t \mid \mathcal{F}_s\,] = W_s^2 - s for all s \le t. The compensator in the second is exactly the quadratic variation, \langle W\rangle_t = t.

The figure below draws both martingales on one set of axes: a fresh Brownian path W_t in the first colour, and W_t^2 - t overlaid in the second. The dashed horizontal line is the fair-game level M_0 = 0. Neither path stays at zero — a martingale is random, often wildly so — but neither has any systematic tendency to leave it: at every moment, up and down are in perfect conditional balance. Hit Refresh a few times and watch fresh paths: individual runs wander all over, some ending well above zero, some well below, yet none of them ever trends. That is what \mathbb{E}[M_t \mid \mathcal{F}_s] = M_s looks like.

Example (b) is more than a curiosity — it is the germ of stochastic calculus. Rearranging Step 6 gives, in increment form,

\mathbb{E}[\,(W_t - W_s)^2 \mid \mathcal{F}_s\,] = t - s,

which says the expected squared increment of Brownian motion equals the length of the time step. Summing such squared increments over a fine partition, the fluctuations cancel and the total converges to the elapsed time — that limit is the quadratic variation

[W]_t = \langle W \rangle_t = t.

This is exactly the new term that ordinary calculus lacks. In a Taylor expansion of f(W_t) the second-order piece is normally negligible, but here (dW_t)^2 behaves like dt rather than vanishing, leaving a surviving correction. That is the content of Itô's lemma, df(W_t) = f'(W_t)\, dW_t + \tfrac{1}{2} f''(W_t)\, dt, whose \tfrac12 f''\,dt term is the compensator we first met as the -t in W_t^2 - t. (Take f(x) = x^2: then f'' = 2, and the correction is exactly dt.)

When fairness breaks: drift, submartingales, supermartingales

Now break the fair game deliberately. Add a steady drift \mu > 0 to Brownian motion and set X_t = W_t + \mu t — a first crude model of a stock that fluctuates randomly but earns a positive expected return. Run the same five-step dance:

\mathbb{E}[X_t \mid \mathcal{F}_s] = \mathbb{E}[\,W_t \mid \mathcal{F}_s\,] + \mu t = W_s + \mu t = X_s + \mu\,(t - s).

The result is not X_s — it is X_s plus a positive amount \mu(t-s) that grows with the horizon. The drift breaks fairness: conditionally on today, tomorrow is expected to be higher. The game favours the player. Flip the sign of \mu and it favours the house. These half-fair games are important enough to have names:

An adapted, integrable process (X_t) is, for all s \le t:

The names feel backwards to almost everyone at first — sub sounds like "below", yet a submartingale trends up. The terminology is inherited from analysis: submartingales are the probabilistic cousins of subharmonic functions (which sit below the harmonic ones), and the mnemonic that actually works is: submartingale — the current value sits below the conditional forecast. Real markets, with returns to be earned and fees to be paid, are almost never martingales under the probabilities you actually experience — a point sharp enough to earn its own warning box below.

The strategy that named the subject — and why it fails

Here is the original martingale strategy, the doomed betting system the concept is named after. Bet £1 on a fair coin. If you win, stop: you are up £1. If you lose, bet £2. If you lose again, bet £4. Keep doubling until the first win. When it finally comes — and with probability 1 it does — the win repays every loss so far and leaves exactly £1 of profit. Check it after three losses: you are down 1 + 2 + 4 = 7, you stake £8, and a win puts you at 8 - 7 = +1. A guaranteed pound, from a perfectly fair game!

But each individual coin flip is fair, so your wealth process (M_t) is a genuine martingale — and a martingale's expectation never moves: \mathbb{E}[M_t] = M_0 = 0 at every fixed time. Both statements cannot be unconditionally true, and the resolution is where all the insight lives. Let \tau be the (random) time of the first win. The strategy really does achieve M_\tau = +1 with probability 1 — but optional stopping (below) needs a regularity condition, and this \tau flunks it: on the way to the win your losses are unbounded. After n straight losses you are down 2^n - 1 — ten losses is a hole of £1,023, twenty is over a million — and the stopped process (M_{t \wedge \tau}) is not uniformly integrable. The "guarantee" secretly assumes infinite capital, infinite time, and a casino with no table limit.

Give the gambler any finite bankroll — say enough for n doublings — and honesty returns. With probability 1 - 2^{-n} you win £1; with probability 2^{-n} you lose everything, 2^n - 1 pounds. The expectation is

\big(1 - 2^{-n}\big)\cdot 1 \;-\; 2^{-n}\,\big(2^n - 1\big) \;=\; 1 - 2^{-n} - 1 + 2^{-n} \;=\; 0.

Exactly zero, for every n. The strategy doesn't manufacture an edge; it reshapes the fair game into "almost always win small, very occasionally lose catastrophically". The tiny probability of ruin carries precisely the weight needed to cancel the near-certain pound. This is the martingale property flexing: no admissible strategy can turn a fair game into a favourable one — the deep truth that becomes, in finance, the no-arbitrage principle.

The word probably comes from the Provençal phrase jouga a la martegalo — "to play in the manner of Martigues", a village whose inhabitants were (unfairly) mocked as naive. By the 1700s la martingale was the doubling system, famous enough that Casanova describes ruining himself with it at the Venetian Ridotto in 1754. Every casino measure against it is still in place today: table minimums and maximums exist largely so that a doubling sequence slams into the ceiling after eight or nine losses — at which point the bettor can no longer continue the scheme and simply eats the loss. Casinos do not fear martingale bettors; they profit from them, since every real casino game is a supermartingale for the player anyway, and the doubling merely speeds up the volume of play.

The rescue of the word is one of mathematics' better ironies. Jean Ville, in his 1939 thesis, formalised gambling strategies and used martingale for the fair-game processes themselves; Joseph Doob seized on the idea and built the modern theory — convergence theorems, optional stopping, the decomposition that bears his name. A term coined for a strategy that cannot beat a fair game now names the mathematical proof of why nothing can.

Two traps catch nearly every newcomer, and both matter enormously in finance.

Trap 1 — "martingale" is measure-relative. A martingale has no drift under the probability measure you are conditioning with. A stock with positive expected return — which is to say, essentially every stock, since investors demand compensation for risk — is not a martingale under the real-world measure \mathbb{P}: it is a submartingale, like W_t + \mu t above. The astonishing move at the heart of derivative pricing is to change the measure: there is a different, artificial probability measure \mathbb{Q} — the risk-neutral measure — under which the discounted stock price is a martingale, and prices computed as \mathbb{Q}-expectations are exactly the no-arbitrage ones. The machine that converts \mathbb{P}-drift into \mathbb{Q}-fairness is Girsanov's theorem, waiting a few concepts downstream. Until then, whenever you say "martingale", silently append "…under this measure, with this filtration".

Trap 2 — a constant mean is not enough. \mathbb{E}[M_t] = M_0 for all t is a consequence of the martingale property, necessary but nowhere near sufficient. The definition is the conditional statement. For a counterexample, let Z be a mean-zero random variable revealed at time 0 (so Z is \mathcal{F}_0-measurable) and set X_t = t\,Z. Then \mathbb{E}[X_t] = t\,\mathbb{E}[Z] = 0 for every t — the mean is perfectly constant. But conditionally,

\mathbb{E}[X_t \mid \mathcal{F}_s] = t\,Z \;\ne\; s\,Z = X_s \quad \text{whenever } Z \ne 0:

once you have seen Z, you know precisely which way X is heading, and the game is not remotely fair. Averaging over the two halves of the world (positive Z, negative Z) hides a drift that every individual observer can see. A martingale must be driftless along every history, not just on the grand average.

A martingale stays fair even if you quit at a random, non-clairvoyant time. For a continuous martingale (M_t) and a stopping time \tau satisfying a regularity condition (e.g. \tau bounded, or (M_{t \wedge \tau}) uniformly integrable),

\mathbb{E}[M_\tau] = \mathbb{E}[M_0].

Combined with the two martingales above this is a workhorse. Stopping W_t at the first exit time \tau from an interval (-b, a) gives the exit probabilities, and stopping W_t^2 - t at the same \tau yields \mathbb{E}[\tau] = \mathbb{E}[W_\tau^2] — the expected time to escape — straight from "a fair game stays fair when you quit on a rule". The regularity condition is not decoration: the doubling strategy above is exactly what goes wrong without it. There, M_\tau = 1 almost surely while M_0 = 0, and the theorem is not contradicted — its hypothesis fails, because the losses en route are unbounded. Whenever an optional-stopping argument gives you something for nothing, go back and check the hypothesis.

The integrators of stochastic calculus

Continuous martingales are not just examples — they are the integrators on which the stochastic integral is built. The Itô integral \int_0^t H_s\, dM_s is defined precisely so that, when the integrand H is adapted, the result is itself a martingale: a fair bet against a fair game stays fair. That single structural fact — martingale in, martingale out — is what makes the Itô calculus well posed, and it is the engine room of quantitative finance. The trading gains of any strategy H against a martingale price M are the integral \int H\, dM; that this is again a martingale is the continuous-time statement that no strategy beats a fair game — the doubling gambler's lesson, industrialised.

From here the road to pricing is short. The fundamental theorem of asset pricing says a market is free of arbitrage exactly when there exists a measure under which discounted asset prices are martingales; a derivative's fair price is then the martingale's defining property read as a formula, V_s = \mathbb{E}^{\mathbb{Q}}[\,V_t \mid \mathcal{F}_s\,] (discounting aside). Everything in this course that follows — quadratic variation, the Itô integral, Girsanov, Black–Scholes — is machinery for finding that measure and computing that expectation.