---
tag: 2E8MQXDYNV
kind: statement
role: theorem
name: Martingale representation theorem
msc: [60H05, 60G44]
uses: [22735FGJV6, HNK1Y62G7V, JA95NH206E, 0D0GBNDKDH, 5MDK7HSYRM, 3D88622PBQ]
about: [0D0GBNDKDH, JA95NH206E]
sources:
  - url: https://en.wikipedia.org/wiki/Martingale_representation_theorem
same_as:
  wikidata: Q6777133
  wikipedia: Martingale representation theorem
---

## Statement

Let $T > 0$ and let $(B_t)_{0 \le t \le T}$ be a standard [[Brownian motion|22735FGJV6]] on a probability space $(\Omega, \mathcal{F}, P)$.
Let $(\mathcal{F}_t)$ be the [[filtration|HNK1Y62G7V]] generated by $B$ and completed with the $P$-null sets.
Let $X$ be an $\mathcal{F}_T$-measurable random variable with $E[X^2] < \infty$.
Then there is a progressively measurable process $H$ with $E\left[\int_0^T H_t^2 \, dt\right] < \infty$ such that
$$X = E[X] + \int_0^T H_t \, dB_t,$$
where the integral is the [[Itô integral|JA95NH206E]].
The process $H$ is unique up to a set of measure zero for $dt \otimes dP$.

## Remarks

As a consequence, every square-integrable [[martingale|0D0GBNDKDH]] $M$ for $(\mathcal{F}_t)$ has the form $M_t = M_0 + \int_0^t H_s \, dB_s$, and so has a continuous version.
For the proof, the random variables $\mathcal{E}\left(\int_0^{\cdot} f \, dB\right)_T$ with deterministic $f$ are [[stochastic exponentials|3D88622PBQ]] with explicit representations.
Their linear span is dense in $L^2(\mathcal{F}_T)$, and the [[Itô isometry|5MDK7HSYRM]] passes the representation to limits.
Example: Itô's lemma gives $B_T^2 = T + \int_0^T 2 B_t \, dB_t$, so $H_t = 2 B_t$.
The filtration must be generated by $B$: if $\mathcal{F}_0$ contains a nonconstant variable $\xi$ independent of $B$ with $E[\xi] = 0$, then $\xi$ has no such representation, because $E\left[\int_0^T H \, dB \mid \mathcal{F}_0\right] = 0 \neq \xi$.

## Explanations

### L1 human

In a world where all randomness comes from one Brownian motion, every random payoff is a constant plus a stochastic integral against that motion.
In betting terms, any such payoff can be produced from its mean by a suitable trading strategy in the Brownian motion.

### L2 human

The theorem is the mathematical basis of hedging in complete markets, such as the Black–Scholes model.
Combined with the Girsanov theorem, it shows that each claim can be replicated by trading in the stock.
The Clark–Ocone formula of Malliavin calculus identifies the integrand $H$ for differentiable $X$.
The result also holds for a $d$-dimensional Brownian motion, with a vector-valued integrand.