Martingales

Motivation: why this matters in quant finance

That distinction matters in finance. A process can have zero formal drift and still fail to be an honest martingale if the admissibility conditions are too loose. The martingale machinery is therefore both a pricing tool and a guardrail against impossible trading strategies.

The informal idea

A martingale is a process whose best current forecast, using current information, is its current value. In discrete time this is

In continuous time it becomes

The filtration $\mathcal{F}_t$ records what is known by time $t$. The martingale property is always relative to that filtration.

Formal definitions

for all $0\le s\le t$.

Lawler's continuous-time stochastic-calculus examples focus on continuous martingales and stochastic integrals of the form

When $A$ is square-integrable enough, $Z_t$ is a martingale. When $A$ can become too large, $Z_t$ may only be a local martingale.

Local martingales

with $\tau_j\uparrow T$ almost surely, such that each stopped process

is a martingale.

For a stochastic integral

Lawler uses the localisation times

Then $Z_{t\wedge\tau_j}$ is square-integrable, so the stopped process is a martingale even if the unstopped integral is not.

Optional sampling

The optional sampling theorem says that stopping a fair game is still fair under the right conditions. Lawler states two useful continuous-martingale versions.

If $Z_t$ is a continuous martingale and $T$ is a stopping time, then

is a continuous martingale, so

If, in addition, there exists $C<\infty$ such that

for all $t$, and $T<\infty$ almost surely, then

The boundedness or integrability condition is not decoration. Without it, stopping rules can smuggle in doubling strategies or other non-admissible behaviour.

Continuous martingales and quadratic variation

Continuous martingales are the processes that behave like time-changed Brownian noise. Lawler states a key characterisation: if $M_t$ is a continuous martingale with $M_0=0$ and quadratic variation

then $M_t$ is Brownian motion.

More generally, stochastic integrals have quadratic variation

This is why quadratic variation is the right clock for martingale volatility.

Martingale representation

In a Brownian filtration, Lawler's martingale representation theorem says that Brownian motion supplies all the randomness. If $V$ is a claim at time $T$ and

then there exists an adapted process $A_t$ such that

This is the bridge to hedging. The martingale $V_t$ can be represented as accumulated exposure to the Brownian source of randomness. Lawler motivates the continuous theorem through the easier random-walk case, where every martingale increment can be written as a predictable coefficient times the next fair coin increment.

Worked examples

Example 1: Brownian motion stopped at barriers

Let $Z_t$ be a continuous martingale with $Z_0=0$, and let

for $a,b>0$. If $T<\infty$ almost surely and the stopped process is bounded, optional sampling gives

Since $Z_T$ is either $-a$ or $b$,

so

This is the continuous-martingale version of gambler's ruin.

Example 2: zero drift is not always enough

Suppose a process is written formally as

The $dt$ drift is zero, but Lawler stresses that this guarantees a local martingale, not automatically a true martingale. If $A_t$ can become too large, expectations may fail to behave well. Localisation fixes the process up to stopping times where the accumulated variance is controlled.

Example 3: pricing as a martingale

If $V$ is a terminal payoff and

then the tower property gives, for $s<t$,

So conditional-value processes are martingales. This is the probabilistic backbone of risk-neutral valuation.

Common confusions and pitfalls

"Zero drift always means martingale." Zero drift in an SDE calculation usually gives a local martingale. Extra integrability or boundedness is needed to conclude it is a true martingale.

"Optional stopping says every stopping strategy is harmless." Optional stopping needs hypotheses such as bounded stopping, bounded stopped values, or uniform integrability. Without them, doubling strategies break the intuition.

"A martingale cannot move." Martingales can be very volatile. The condition is on conditional expectation, not path variation.

"The martingale property is absolute." It depends on the filtration and probability measure. Changing information or changing measure can destroy or create the martingale property.

"Martingale representation gives an explicit hedge for free." It proves an adapted integrand exists in a Brownian filtration. Finding that integrand in a usable form is a separate modelling and calculation problem.

Where this goes next

• Local Martingales and Semimartingales: Develops the local-martingale distinction that continuous-time stochastic calculus needs.
• Stochastic Integrals: Builds martingales by integrating predictable processes against Brownian motion.
• Quadratic Variation: Explains the variance clock behind continuous martingales.
• Girsanov's Theorem: Uses martingales to change probability measures.
• Feynman-Kac Formula: Converts martingale conditional expectations into PDEs.

References

• Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 4 §4.1 (Martingales and local martingales), §4.5 (Continuous martingales), Ch. 5 §5.7 (Martingale representation theorem).