Local Martingales and Semimartingales

Motivation: why this matters in quant finance

Derivative pricing arguments often say that a discounted gain process is a martingale. Lawler's continuous-time construction shows why that sentence is slightly too clean: an Itô integral with uncontrolled bet size can fail to be a true martingale, even though it becomes one after stopping before the quadratic variation gets too large.

That distinction matters when a trading strategy is allowed to scale positions aggressively. A model can have the formal differential

dX_t=R_t\,dt+A_t\,dW_t,

but the

A_t\,dW_t

term is honestly only a local martingale unless the integrability conditions make it a true martingale. In finance language, the predictable

R_t\,dt

term is drift or carrying cost, while the stochastic integral is the fair-game risk term after localisation.

This lesson sits after stochastic integrals and stopping times, and before Girsanov's theorem, jump-diffusion processes, and martingale pricing. The key habit is precise: "martingale part" usually means "local martingale part" unless a theorem proves more.

The informal idea

Lawler motivates local martingales with a continuous-time version of a doubling strategy. A stochastic integral

Z_t=\int_0^t A_s\,dW_s

is a square-integrable martingale if $\mathbb{E}\int_0^t A_s^2\,ds<\infty$ . If that condition fails, the integral can still be defined path by path up to the time its quadratic variation explodes, but the expectation need not stay fixed. The process behaves like a martingale only while the accumulated variance is kept under control.

Localisation does exactly that. Instead of asking whether $Z_t$ is a martingale for all paths and all times, stop it at the first time its quadratic variation reaches level $j$ . Each stopped process is a proper martingale; the original process is recovered as $j\to\infty$ .

For Itô processes, the semimartingale idea is the decomposition already present in Lawler's SDE notation:

X_t=X_0+\int_0^t R_s\,ds+\int_0^t A_s\,dW_s.

The first integral is finite variation. The second is a local martingale. Full semimartingale theory is broader than Lawler's introductory treatment, but this decomposition is the version that students need for continuous diffusion models and the one supported by the source material.

Formal definitions

Local martingale. A continuous adapted process $M_t$ on $[0,T)$ is a local martingale if there exists an increasing sequence of stopping times $\tau_1\le \tau_2\le\cdots$ such that $\tau_j\to T$ almost surely and, for each $j$ , the stopped process
$M_t^{(j)}=M_{t\wedge \tau_j}$
is a martingale.

For an Itô integral $Z_t=\int_0^t A_s\,dW_s$ , Lawler uses the stopping times

\tau_j=\inf\left\{t:\langle Z\rangle_t=\int_0^t A_s^2\,ds=j\right\}.

Then $Z_{t\wedge\tau_j}$ is square-integrable for each $j$ , so $Z$ is a local martingale up to

T=\inf\left\{t:\int_0^t A_s^2\,ds=\infty\right\}.

Itô semimartingale form used in this lesson. A continuous process is in Itô semimartingale form if it can be written as
$X_t=X_0+\int_0^t R_s\,ds+\int_0^t A_s\,dW_s,$
where the first integral is the finite-variation part and the second is the local-martingale part.

This is the natural continuous-path process class generated by Lawler's Chapter 3 SDE decomposition. General semimartingales can include more integrators and jump terms; those are outside this specific Lawler-grounded note.

Key properties

Localisation turns variance control into martingales

For

Z_t=\int_0^t A_s\,dW_s

, the process may not be a true martingale when

\mathbb{E}\int_0^t A_s^2\,ds

is infinite. After stopping at

\tau_j

, the quadratic variation is bounded by

j

, and the stopped integral is square-integrable. Finance consequence: a self-financing gain process may be locally fair while still failing the expectation identity needed for valuation.

The drift term decides martingality in an Itô decomposition

dX_t=R_t\,dt+A_t\,dW_t,

then a nonzero

R_t

prevents

X

from being a martingale. If

R_t\equiv0

, the stochastic integral is at least a local martingale, and becomes a true martingale under stronger integrability. Finance consequence: risk-neutral pricing sets the drift of discounted prices to zero, but still needs integrability or localisation to justify expectations.

Optional sampling survives under the right hypotheses

Lawler's optional sampling theorem states that if $Z_t$ is a continuous martingale and $T$ is a stopping time, then $Z_{t\wedge T}$ is a martingale and $\mathbb{E}[Z_{t\wedge T}]=\mathbb{E}[Z_0]$ . If additionally $\mathbb{E}[Z_{t\wedge T}^2]\le C$ for all $t$ and $T<\infty$ almost surely, then $\mathbb{E}[Z_T]=\mathbb{E}[Z_0]$ .

The square-integrability condition is not cosmetic. It is the barrier between legitimate stopping arguments and gambling-system paradoxes.

Quadratic variation measures the clock of the martingale part

For a stochastic integral,

\langle Z\rangle_t=\int_0^t A_s^2\,ds.

This is why stopping by quadratic-variation level is natural: it stops the process by its accumulated stochastic risk, not by calendar time. Later continuous-martingale theory sharpens this into the result that a continuous martingale is a time-changed Brownian motion.

Worked examples

Example 1: a stopped Itô integral is a martingale

Let

Z_t=\int_0^t A_s\,dW_s

for an adapted, piecewise-continuous process $A$ . Suppose $\int_0^t A_s^2\,ds$ can be large enough that $\mathbb{E}\int_0^t A_s^2\,ds$ is not finite. Define

\tau_j=\inf\left\{t:\int_0^t A_s^2\,ds=j\right\}.

Then

Z_{t\wedge\tau_j}=\int_0^t A_s\mathbf{1}_{s<\tau_j}\,dW_s,

and its quadratic variation is at most $j$ . The variance rule for Itô integrals gives square integrability, so $Z_{t\wedge\tau_j}$ is a martingale. The unstopped $Z_t$ is therefore a local martingale even when it is not a true martingale.

Example 2: separating drift and local martingale risk in GBM

For geometric Brownian motion,

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t,

the integral form is

S_t=S_0+\int_0^t \mu S_s\,ds+\int_0^t \sigma S_s\,dW_s.

The first integral is finite variation; it accumulates predictable drift. The second integral is the local-martingale part; it accumulates Brownian innovation. Under a risk-neutral measure, the discounted process has zero drift:

d(e^{-rt}S_t)=e^{-rt}\sigma S_t\,dW_t^{\mathbb{Q}},

so the discounted stock is at least locally a martingale. To use it inside an expectation-pricing formula, one still checks conditions that make it a true martingale.

Example 3: continuous gambler's ruin from optional sampling

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

T=\inf\{t:Z_t=-a \text{ or } Z_t=b\}, \qquad a,b>0.

If $T<\infty$ almost surely, then the stopped process is bounded. Optional sampling gives

0=\mathbb{E}[Z_T]=-a\,\mathbb{P}(Z_T=-a)+b\,\mathbb{P}(Z_T=b).

Solving,

\mathbb{P}(Z_T=b)=\frac{a}{a+b}.

This is Lawler's continuous martingale version of gambler's ruin: stopping a fair game at two barriers gives a probability determined by the starting point and boundaries, not by a drift term.

Common confusions and pitfalls

"Local martingale means martingale over a short calendar interval." Localisation is by stopping times, not by taking a small deterministic interval. The stop can depend on the path, such as stopping when quadratic variation reaches $j$ .

"The stochastic integral is always a martingale." Lawler's variance rule gives a square-integrable martingale when the integrability condition holds. Without it, an Itô integral may only be a local martingale.

"Zero drift is enough for pricing by expectation." Zero drift gives a local-martingale candidate. Pricing formulas need enough integrability to justify optional sampling or conditional expectation identities.

"Semimartingale means any process used in finance." In this lesson, the term is used narrowly for the Itô decomposition supported by Lawler: finite variation plus local martingale. General semimartingale theory is wider and includes jump structure not developed here.

"The martingale part is always a true martingale part." The phrase is standard shorthand, but Lawler explicitly warns that one should often say "local martingale part."

Where this goes next

Infinitesimal Generators and Kolmogorov Equations: uses the drift term produced by Itô's formula to build PDEs.
Feynman-Kac Formula: turns a discounted expectation into a PDE by forcing a discounted process to be a martingale.
Girsanov's Theorem: changes drift while preserving the stochastic-integral structure.
Jump-Diffusion Processes: adds jumps to the same drift-plus-martingale intuition.
Black-Scholes PDE: uses the vanishing-drift condition on discounted values in the finance setting.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 3 §3.2 (Stochastic integral), Ch. 3 §3.4 (Itô process decomposition), Ch. 4 §4.1 (Martingales and local martingales).