37.06.02 · probability / 06-brownian-motion-stochastic-calculus

The Brownian Martingale Representation Theorem

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Anchor (Master): Revuz–Yor, Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999), Ch. V §3 (representation of Brownian martingales, the chaos expansion); Karatzas–Shreve §3.4 (representation, completeness of Black–Scholes §5.8); Nualart, The Malliavin Calculus and Related Topics (2nd ed. 2006), §1.3 (Wiener chaos), §1.3.3 (Clark–Ocone)

Intuition Beginner

Suppose the only source of randomness in your world is one jittery random path — the trajectory of a particle pushed around by countless tiny collisions, the Brownian motion of the peer units. Everything random you could ever measure is built from watching this single path unfold. The question this unit answers is sweeping: can every random quantity in such a world be reconstructed as a running bet placed on that one path?

The answer is yes, and it is sharper than that. Take any fair game whose fairness comes only from watching the Brownian path — a quantity whose best forecast of tomorrow is always its value today. Then that game is nothing but a starting amount plus the total winnings from a trading strategy that holds some changing amount of the path and collects its increments. There is one and only one such strategy. Nothing else is needed, and nothing else is possible.

The everyday picture is a market with a single risky stock driven by random news. The claim is that any payment you might owe in the future, however complicated, can be reproduced exactly by starting with the right amount of cash and continuously rebalancing how much stock you hold. You never need a second stock, an option, or any outside instrument. The single path already carries every contingency, and the trading strategy that reproduces a target payment is unique.

This is why a market with one stock and one source of noise is called complete: every claim can be hedged perfectly.

Visual Beginner

Imagine a single jagged random path running left to right across the page — the Brownian path that is the one source of randomness. To its right sits a target: some number you will only learn at the final time, depending on the whole history of the path. Between them runs a control dial labelled "how much of the path you hold," which you are allowed to turn up and down as the path unfolds, but only using what you have seen so far.

Read it as a promise and a constraint. The promise: by turning the holding dial appropriately as the path moves, the running total of your winnings can be steered to land exactly on the target, no matter how the path wiggles. The constraint: the dial may only react to the past, never to the future, the same no-peeking rule from the integral unit. The surprise of the theorem is that this is always possible, with a starting amount equal to today's best forecast of the target, and that the required dial schedule is unique — two different schedules cannot both hit every target.

Worked example Beginner

Watch Brownian motion from time $0$ to time $T = 1$ , starting at $0$ . Your target, revealed only at time $1$ , is the final position itself: you must end up holding the number $W (1)$ .

Try the simplest strategy: hold exactly one unit of the path at all times. From the integral unit, the running winnings of holding one unit over $[0, 1]$ is just the net change of the path, $W (1) - W (0) = W (1)$ . So starting with $0$ in cash and holding one unit throughout reproduces the target $W (1)$ exactly. The starting amount $0$ is today's best forecast of $W (1)$ , because a fair path has expected future value equal to its current value $0$ .

Now take a curved target: end up holding $W (1)^{2} - 1$ . From the worked example in the integral unit, holding the current path value $W (t)$ and collecting increments produces $\frac{1}{2} W (1)^{2} - \frac{1}{2}$ . Doubling the holding to $2 W (t)$ doubles the winnings to $W (1)^{2} - 1$ . So starting with $0$ cash and holding $2 W (t)$ at time $t$ reproduces $W (1)^{2} - 1$ exactly. The starting amount is again $0$ , the forecast of $W (1)^{2} - 1$ , since the expected square of the path at time $1$ is $1$ .

What this tells us: two different targets needed two different holding schedules — a constant $1$ for the first, the changing $2 W (t)$ for the second — but each target was hit exactly, from a starting cash equal to its forecast. The theorem says this always works, for every target built from the path.

Check your understanding Beginner

Formal definition Intermediate+

Fix a probability space $(Ω, F, P)$ carrying a standard $d$ -dimensional Brownian motion $W = (W^{1}, \dots, W^{d})$ , as constructed in 02.15.01. Let $(F_{t})_{t \geq 0}$ be the Brownian filtration: the $P$ -augmentation of the natural filtration $σ (W_{s} : s \leq t)$ , so that $(F_{t})$ satisfies the usual conditions and $F_{t}$ is generated, up to null sets, by the path of $W$ on $[0, t]$ . Fix a horizon $T \in (0, \infty]$ and write $F_{T} = σ (⋃_{t < T} F_{t})$ when $T = \infty$ . All martingales below are taken with respect to $(F_{t})$ .

A process $H = (H_{t})_{0 \leq t \leq T}$ with values in $R^{d}$ is a predictable integrand for $W$ if it is progressively measurable and $\int_{0}^{T} ∣ H_{s} ∣^{2} d s < \infty$ almost surely; it lies in $L_{T}^{2}$ if moreover $E \int_{0}^{T} ∣ H_{s} ∣^{2} d s < \infty$ . For such $H$ the vector Itô integral is $\int_{0}^{t} H_{s} \cdot d W_{s} = \sum_{i = 1}^{d} \int_{0}^{t} H_{s}^{i} d W_{s}^{i}$ , with bracket $⟨ \int H \cdot d W ⟩_{t} = \int_{0}^{t} ∣ H_{s} ∣^{2} d s$ by the isometry of 37.06.01.

Predictable representation property. The Brownian filtration is said to have the predictable representation property (PRP) when, for every square-integrable martingale $(M_{t})_{0 \leq t \leq T}$ adapted to $(F_{t})$ with $sup_{t} E [M_{t}^{2}] < \infty$ , there is a unique $H \in L_{T}^{2}$ such that $$ M_t = M_0 + \int_0^t H_s \cdot dW_s, \qquad 0 \le t \le T, \quad \text{a.s.} $$ Equivalently, every $F \in L^{2} (Ω, F_{T}, P)$ admits a representation $$ F = \mathbb{E}[F] + \int_0^T H_s \cdot dW_s $$ for a unique $H \in L_{T}^{2}$ , obtained by setting $M_{t} = E [F ∣ F_{t}]$ . The constant is forced to be $M_{0} = E [F]$ because $F_{0}$ is degenerate (it contains only null sets and their complements) and the Itô integral has mean zero.

The stochastic exponential (Doléans-Dade exponential) of a deterministic $h \in L^{2} ([0, T]; R^{d})$ is $$ \mathcal{E}(h)_t = \exp!\Big(\int_0^t h_s \cdot dW_s - \tfrac12 \int_0^t |h_s|^2,ds\Big), $$ the exponential martingale of 02.15.02; it satisfies $d E (h)_{t} = E (h)_{t} h_{t} \cdot d W_{t}$ , $E (h)_{0} = 1$ . These exponentials are the test functions whose totality in $L^{2}$ drives the proof.

The $n$ -th Wiener chaos $H_{n}$ is the $L^{2}$ -closure of the span of Hermite-type polynomials of degree $n$ in Wiener integrals $\int_{0}^{T} g d W$ ; equivalently it is the range of the $n$ -fold multiple Wiener–Itô integral $I_{n}$ . The chaos decomposition $L^{2} (Ω, F_{T}, P) = ⨁_{n \geq 0} H_{n}$ (orthogonal direct sum) is the Hilbert-space backbone of the representation. The formalization follows Le Gall ^{[Le Gall 2016]} §5.4 and Revuz–Yor ^{[Revuz 1999]} Ch. V §3.

Counterexamples to common slips

The PRP is a property of the Brownian filtration, not of every filtration to which $W$ is adapted. Enlarge the filtration by an independent coin flip $ξ$ : then $W$ is still a martingale, but the martingale $M_{t} = E [ξ ∣ G_{t}]$ cannot be written as $\int H d W$ , since $ξ$ is orthogonal to every Itô integral. Representation needs $(F_{t})$ generated exactly by $W$ .
The integrand $H$ is unique as an element of $L_{T}^{2}$ , i.e. up to $d P \otimes d t$ -null sets, not pathwise everywhere. Two integrands agreeing off a null set give the same integral.
Local martingales adapted to the Brownian filtration also represent, but as $\int H d W$ with $H$ only locally square-integrable; the clean constant-plus-integral form with $H \in L_{T}^{2}$ and an honest $L^{2}$ martingale requires the square-integrability hypothesis. Dropping it, the representing $H$ may fail to be in $L_{T}^{2}$ .

Key theorem with proof Intermediate+

The structural pillar is that the stochastic exponentials are total in $L^{2}$ , and that each one represents; density then carries representation to all of $L^{2}$ .

Theorem (Brownian martingale representation). Let $(F_{t})$ be the Brownian filtration of a $d$ -dimensional Brownian motion $W$ and fix $T \in (0, \infty)$ . For every $F \in L^{2} (Ω, F_{T}, P)$ there is a unique $H \in L_{T}^{2}$ with $F = E [F] + \int_{0}^{T} H_{s} \cdot d W_{s}$ . Equivalently, every square-integrable $(F_{t})$ -martingale $M$ has the form $M_{t} = M_{0} + \int_{0}^{t} H_{s} \cdot d W_{s}$ for a unique $H \in L_{T}^{2}$ .

Proof. Uniqueness. If $\int_{0}^{T} H \cdot d W = \int_{0}^{T} H^{'} \cdot d W$ in $L^{2}$ with $H, H^{'} \in L_{T}^{2}$ , then $G = H - H^{'}$ satisfies $\int_{0}^{T} G \cdot d W = 0$ . By the Itô isometry of 37.06.01, $E \int_{0}^{T} ∣ G_{s} ∣^{2} d s = E [(\int_{0}^{T} G \cdot d W)^{2}] = 0$ , so $G = 0$ as an element of $L_{T}^{2}$ . Hence $H = H^{'}$ .

Totality of the exponentials. Let $S = {E (h)_{T} : h \in L^{2} ([0, T]; R^{d}) step function}$ . We claim $S$ spans a dense subspace of $L^{2} (Ω, F_{T}, P)$ . Suppose $F \in L^{2}$ is orthogonal to every element of $S$ . For a step function $h = \sum_{j} λ_{j} 1_{(t_{j - 1}, t_{j}]}$ with $λ_{j} \in R^{d}$ , the random variable $E (h)_{T}$ is a fixed multiplicative constant times $exp (\sum_{j} λ_{j} \cdot (W_{t_{j}} - W_{t_{j - 1}}))$ . Orthogonality $E [F E (h)_{T}] = 0$ for all $λ_{j}$ therefore says the analytic function $$ (\lambda_1,\dots,\lambda_m) \longmapsto \mathbb{E}\Big[F\exp\Big(\textstyle\sum_j \lambda_j\cdot(W_{t_j} - W_{t_{j-1}})\Big)\Big] $$ vanishes identically on $R^{d m}$ . This function extends to an entire function on $C^{d m}$ (the Gaussian tails make the integral and all its complex derivatives finite), and vanishing on $R^{d m}$ forces it to vanish on $C^{d m}$ . Restricting to purely imaginary $λ_{j} = i θ_{j}$ shows that the Fourier transform of the signed measure $A \mapsto E [F 1_{A}]$ , viewed on the increments $(W_{t_{j}} - W_{t_{j - 1}})$ , is zero. By Fourier inversion the measure is zero, so $E [F g (W_{t_{1}}, \dots, W_{t_{m}})] = 0$ for every bounded Borel $g$ and every finite set of times. Since such variables generate $F_{T}$ , $F$ is orthogonal to a dense subspace and to itself: $F = 0$ . Thus $S^{⊥} = {0}$ and $span S$ is dense.

Each exponential represents. For a step $h$ , the exponential martingale $E (h)_{t}$ solves $d E (h)_{t} = E (h)_{t} h_{t} \cdot d W_{t}$ with $E (h)_{0} = 1$ , by Itô's formula of 02.15.02. Hence $$ \mathcal{E}(h)_T = 1 + \int_0^T \mathcal{E}(h)_s,h_s \cdot dW_s, $$ which is exactly the representation $F = E [F] + \int_{0}^{T} H \cdot d W$ with $F = E (h)_{T}$ , $E [F] = 1$ , and $H_{s} = E (h)_{s} h_{s} \in L_{T}^{2}$ (the integrand is in $L_{T}^{2}$ because $h$ is bounded and $sup_{s \leq T} E [E (h)_{s}^{2}] < \infty$ for bounded $h$ ).

Density closes the argument. Let $R \subseteq L^{2} (Ω, F_{T}, P)$ be the set of representable $F$ , i.e. those of the form $E [F] + \int_{0}^{T} H \cdot d W$ , $H \in L_{T}^{2}$ . The map $(c, H) \mapsto c + \int_{0}^{T} H \cdot d W$ from $R \oplus L_{T}^{2}$ into $L^{2}$ is, by the isometry, an isometry onto its image once $L_{T}^{2}$ carries the norm $(E \int_{0}^{T} ∣ H ∣^{2} d s)^{1/2}$ and $R$ the absolute value; since $R \oplus L_{T}^{2}$ is complete, $R$ is a closed subspace of $L^{2}$ . It contains the dense set $span S$ , so $R = L^{2}$ . Every $F$ represents; setting $M_{t} = E [F ∣ F_{t}] = E [F] + \int_{0}^{t} H \cdot d W$ gives the martingale form, and uniqueness of $H$ was shown above. $□$

Bridge. This theorem is the foundational reason a one-noise market is complete: it builds toward the hedging identity that an $F_{T}$ -payoff $F = E [F] + \int_{0}^{T} H \cdot d W$ is replicated by holding $H_{s}$ in the driving martingale, and the same representation map appears again in the Clark–Ocone formula, where the integrand is identified explicitly as the predictable projection of the Malliavin derivative. The totality of stochastic exponentials is exactly the Hilbert-space content that $L^{2}$ of Wiener space is generated by the single path, and the orthogonal chaos decomposition $L^{2} = ⨁_{n} H_{n}$ generalises the constant-plus-integral splitting to all polynomial orders at once. Putting these together, the representation property is dual to the fact that the Brownian filtration carries no information beyond $W$ , so that the integral $\int H d W$ — whose bracket is the $\int ∣ H ∣^{2} d s$ clock of 37.06.01 — exhausts every square-integrable functional; that single fact is the central insight tying martingale representation, market completeness, and the Wiener chaos into one statement.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Represent $F = W_{T}^{3}$ (one-dimensional). Identify $E [F]$ and $H$ .

Hint

Apply Itô to $x^{3}$ : $W_{T}^{3} = \int_{0}^{T} 3 W_{s}^{2} d W_{s} + 3 \int_{0}^{T} W_{s} d s$ . Then represent the $d s$ -integral by parts or by conditioning.

Answer

By Itô, $W_{T}^{3} = \int_{0}^{T} 3 W_{s}^{2} d W_{s} + 3 \int_{0}^{T} W_{s} d s$ . The first term is already an Itô integral; the second is a finite-variation term that must be folded in. Write $\int_{0}^{T} W_{s} d s = T W_{T} - \int_{0}^{T} s d W_{s}$ (integration by parts, since $d (s W_{s}) = W_{s} d s + s d W_{s}$ ), so $3 \int_{0}^{T} W_{s} d s = 3 T W_{T} - 3 \int_{0}^{T} s d W_{s} = 3 T \int_{0}^{T} d W_{s} - 3 \int_{0}^{T} s d W_{s} = \int_{0}^{T} 3 (T - s) d W_{s}$ . Hence $W_{T}^{3} = \int_{0}^{T} (3 W_{s}^{2} + 3 (T - s)) d W_{s}$ , giving $E [F] = 0$ and $H_{s} = 3 W_{s}^{2} + 3 (T - s)$ . Rubric: full credit for the Itô expansion, the by-parts rewrite of the $d s$ -term, and the combined integrand.

Exercise 6 (hard, short-answer).

Show that the predictable representation property fails for the filtration $G_{t} = F_{t} \lor σ (ξ)$ , where $ξ$ is a fair coin ( $\pm 1$ ) independent of $W$ . Exhibit a square-integrable $G$ -martingale not of the form constant plus $\int H d W$ .

Hint

Consider $M_{t} = E [ξ ∣ G_{t}]$ . Is $ξ$ orthogonal to every Itô integral?

Answer

Since $ξ$ is $G_{0}$ -measurable, $M_{t} = E [ξ ∣ G_{t}] = ξ$ for all $t$ : a constant-in-time, square-integrable $G$ -martingale. If $ξ = c + \int_{0}^{T} H d W$ held, then taking expectations gives $c = E [ξ] = 0$ , so $ξ = \int_{0}^{T} H d W$ . But $ξ$ is independent of $W$ , hence independent of $\int_{0}^{T} H d W$ and orthogonal to it in $L^{2}$ : $E [ξ \int_{0}^{T} H d W] = E [ξ] E [\int_{0}^{T} H d W] = 0$ . Then $E [ξ^{2}] = E [ξ \int_{0}^{T} H d W] = 0$ , contradicting $E [ξ^{2}] = 1$ . So no representation exists; the enlarged filtration carries information beyond $W$ . Rubric: full credit for $M_{t} = ξ$ , the independence/orthogonality step, and the contradiction with $E [ξ^{2}] = 1$ .

Exercise 7 (hard, short-answer).

Use the representation theorem to prove that every $F_{T}$ -measurable event $A$ with $P (A) \in (0, 1)$ has $1_{A} = P (A) + \int_{0}^{T} H d W$ for some non-zero $H \in L_{T}^{2}$ , and deduce that no non-degenerate $F_{T}$ -event is independent of the whole path.

Hint

Apply representation to $F = 1_{A} \in L^{2}$ . If $H \equiv 0$ , what would $1_{A}$ be?

Answer

Since $1_{A} \in L^{2} (Ω, F_{T}, P)$ , the theorem gives $1_{A} = E [1_{A}] + \int_{0}^{T} H d W = P (A) + \int_{0}^{T} H d W$ with a unique $H \in L_{T}^{2}$ . If $H \equiv 0$ then $1_{A} = P (A)$ almost surely, forcing $1_{A}$ to be the constant $P (A) \in (0, 1)$ , impossible for an indicator; so $H \neq = 0$ . The non-vanishing integral $\int_{0}^{T} H d W$ is a non-constant functional of the path, so $1_{A}$ depends genuinely on $W$ and $A$ cannot be independent of the path. The only $F_{T}$ -events independent of $W$ are the null and full sets. Rubric: full credit for applying representation to $1_{A}$ , ruling out $H \equiv 0$ , and the independence deduction.

Advanced results Master

The representation theorem is the entry point to three deeper structures: the Wiener chaos, the Clark–Ocone formula, and the completeness of the Black–Scholes market.

Wiener chaos decomposition. The space $L^{2} (Ω, F_{T}, P)$ decomposes as the orthogonal direct sum $⨁_{n \geq 0} H_{n}$ , where $H_{n}$ is the image of the $n$ -fold multiple Wiener–Itô integral $I_{n}$ on symmetric kernels $f_{n} \in L^{2} ([0, T]^{n})$ . Each $F$ has a unique expansion $F = \sum_{n} I_{n} (f_{n})$ with $E [F^{2}] = \sum_{n} n! ∥ f_{n} ∥_{L^{2} ([0, T]^{n})}^{2}$ . Representation is the first-order shadow of this: iterating the constant-plus-integral splitting on the integrand $H$ regenerates the kernels $f_{n}$ , and the chaos expansion is the statement that the iteration terminates in a complete orthogonal system. The integrand of the first-order representation is $H_{t} = \sum_{n \geq 1} n I_{n - 1} (f_{n} (\cdot, t))$ , a multiple-integral series whose leading term is the deterministic kernel $f_{1}$ .

Clark–Ocone formula. When $F$ lies in the Malliavin–Sobolev space $D^{1, 2}$ — the domain of the Malliavin derivative $D$ , the densely defined operator that differentiates a Wiener functional in the direction of a perturbation of the path — the representing integrand is the predictable projection of $D F$ : $$ F = \mathbb{E}[F] + \int_0^T \mathbb{E}!\big[D_s F \mid \mathcal{F}s\big]\cdot dW_s. $$ This makes the abstract $H$ of the representation theorem explicit and computable: differentiate $F$ on Wiener space, then take the optional/predictable projection. The formula recovers Exercises 1–4 — e.g. $D_s W_T = \mathbf{1}{[0,T]}(s) $g i v es$ H \equiv 1 $f or$ F = W_T $, an d$ D_s W_T^2 = 2W_T $p r o j ec t s t o$ \mathbb{E}[2W_T\mid\mathcal{F}_s] = 2W_s$.

Completeness of the Black–Scholes market. In a market with one risky asset $d S_{t} = S_{t} (μ d t + σ d W_{t})$ and a bond, change to the risk-neutral measure $Q$ under which the discounted price $\tilde{S}_{t} = e^{- r t} S_{t}$ is a martingale (Girsanov, using the exponential martingale of 02.15.02). A contingent claim is an $F_{T}$ -measurable payoff $F$ with $E_{Q} [e^{- r T} F]^{2} < \infty$ . Representation under $Q$ gives $e^{- r T} F = E_{Q} [e^{- r T} F] + \int_{0}^{T} H_{s} d \tilde{W}_{s}$ for the $Q$ -Brownian motion $\tilde{W}$ , and reading $H$ against $\tilde{S}$ yields a self-financing portfolio with terminal value $F$ almost surely. Every claim is replicable, so the market is complete and the arbitrage price is the unique $Q$ -expectation $e^{- r (T - t)} E_{Q} [F ∣ F_{t}]$ .

Failure of representation with jumps. Replacing $W$ by a Lévy process with jumps breaks the property: the Itô integrals against the continuous part no longer span $L^{2}$ , and one must adjoin compensated Poisson integrals. The clean single-integrand PRP is special to the Brownian (continuous, Gaussian) filtration, and its breakdown is exactly the incompleteness of markets driven by jumps.

Synthesis. Putting these together, the representation theorem, the Wiener chaos, the Clark–Ocone formula, and Black–Scholes completeness are one statement told at four resolutions. The central insight is that $L^{2}$ of the Brownian path is generated by the single integrator $W$ : the constant-plus-integral form is exactly the first-order term of the chaos expansion $F = \sum_{n} I_{n} (f_{n})$ , the Clark–Ocone integrand $E [D_{s} F ∣ F_{s}]$ is exactly the abstract $H$ made explicit through the Malliavin derivative, and the replicating hedge is exactly that integrand read as a portfolio. The property is dual to the degeneracy of $F_{0}$ and the totality of the stochastic exponentials, and it generalises the elementary fact that $W_{T} = \int_{0}^{T} d W$ to every square-integrable functional at once. This is the foundational reason the Black–Scholes market is complete and appears again in the Girsanov change of measure, the Föllmer–Schweizer decomposition for incomplete markets, and the Malliavin-calculus computation of Greeks; the bridge is that the quadratic-variation clock $\int ∣ H ∣^{2} d s$ of 37.06.01, which measures the integral, is also the metric under which the chaos spaces $H_{n}$ are orthogonal, so a single inner-product geometry organises representation, hedging, and chaos at once.

Full proof set Master

The existence and uniqueness of the representation are proved in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (the representing integrand is the closing martingale's bracket density). Let $F \in L^{2} (Ω, F_{T}, P)$ with representation $F = E [F] + \int_{0}^{T} H \cdot d W$ and closing martingale $M_{t} = E [F ∣ F_{t}]$ . Then for any other square-integrable martingale $N_{t} = N_{0} + \int_{0}^{t} K \cdot d W$ , the covariation is $⟨ M, N ⟩_{t} = \int_{0}^{t} H_{s} \cdot K_{s} d s$ . In particular $⟨ M, W^{i} ⟩_{t} = \int_{0}^{t} H_{s}^{i} d s$ , so $H^{i}$ is the density $d ⟨ M, W^{i} ⟩ / d t$ .

Proof. By bilinearity of the bracket 37.06.01 and the rule $⟨ \int H \cdot d W, \int K \cdot d W ⟩_{t} = \sum_{i, j} \int_{0}^{t} H^{i} K^{j} d ⟨ W^{i}, W^{j} ⟩_{s} = \sum_{i} \int_{0}^{t} H_{s}^{i} K_{s}^{i} d s$ , using $⟨ W^{i}, W^{j} ⟩_{s} = δ_{ij} s$ from 02.15.01. Taking $N = W^{i}$ , i.e. $K = e_{i}$ the $i$ -th basis vector, gives $⟨ M, W^{i} ⟩_{t} = \int_{0}^{t} H_{s}^{i} d s$ , whose Lebesgue density in $t$ is $H_{t}^{i}$ for a.e. $t$ . This identifies the integrand intrinsically, independent of the representation procedure, and reproves uniqueness: the bracket density is determined by $M$ . $□$

Proposition (orthogonality of the Wiener chaoses). The subspaces $H_{n} = range I_{n}$ are mutually orthogonal in $L^{2} (Ω, F_{T}, P)$ , and $E [I_{n} (f) I_{m} (g)] = δ_{nm} n! ⟨ \tilde{f}, \tilde{g} ⟩_{L^{2} ([0, T]^{n})}$ for symmetric kernels.

Proof. For $n \neq = m$ , take $f, g$ tensor kernels $f = h^{\otimes n}$ , $g = h^{\otimes m}$ for the same $h$ with $∥ h ∥_{L^{2}} = 1$ ; then $I_{n} (h^{\otimes n}) = H_{n} (\int_{0}^{T} h d W)$ with $H_{n}$ the $n$ -th Hermite polynomial (the multiple integral of a tensor power is a Hermite polynomial of the Wiener integral). Since $\int_{0}^{T} h d W$ is standard Gaussian and the Hermite polynomials are orthogonal under the Gaussian weight with $E [H_{n} (Z) H_{m} (Z)] = δ_{nm} n!$ , orthogonality holds on tensor kernels; these span $H_{n}$ , and polarization extends the identity to all symmetric kernels, giving $E [I_{n} (f) I_{m} (g)] = δ_{nm} n! ⟨ \tilde{f}, \tilde{g} ⟩$ . Density of the iterated stochastic integrals in $L^{2}$ (a consequence of the representation theorem applied recursively to the integrand) gives $L^{2} = ⨁_{n} H_{n}$ . $□$

Proposition (Clark–Ocone for smooth cylinder functionals). Let $F = f (W_{t_{1}}, \dots, W_{t_{n}})$ with $f \in C_{b}^{1} (R^{n})$ and $0 \leq t_{1} < \dots < t_{n} \leq T$ (one-dimensional $W$ for clarity). Then $D_{s} F = \sum_{k} \partial_{k} f (W_{t_{1}}, \dots, W_{t_{n}}) 1_{[0, t_{k}]} (s)$ and $F = E [F] + \int_{0}^{T} E [D_{s} F ∣ F_{s}] d W_{s}$ .

Proof. The Malliavin derivative of a smooth cylinder functional is $D_{s} F = \sum_{k} \partial_{k} f (W_{t_{1}}, \dots, W_{t_{n}}) \partial_{s} W_{t_{k}}$ , and $D_{s} W_{t_{k}} = 1_{[0, t_{k}]} (s)$ since perturbing the path by $\int_{0}^{\cdot} \overset{γ}{˙}$ changes $W_{t_{k}}$ by $\int_{0}^{t_{k}} \overset{γ}{˙}$ . Thus $D_{s} F = \sum_{k : t_{k} \geq s} \partial_{k} f (\dots)$ . For the representation, define $G_{s} = E [D_{s} F ∣ F_{s}]$ , which lies in $L_{T}^{2}$ because $f \in C_{b}^{1}$ bounds $D_{s} F$ . The process $N_{t} = E [F] + \int_{0}^{t} G d W$ is a square-integrable martingale; its bracket with $W$ is $⟨ N, W ⟩_{t} = \int_{0}^{t} G_{s} d s$ . The integration-by-parts (duality) formula on Wiener space, $E [F \int_{0}^{T} u d W] = E \int_{0}^{T} u_{s} D_{s} F d s$ for adapted $u$ , shows $⟨ M, W ⟩_{t} = \int_{0}^{t} E [D_{s} F ∣ F_{s}] d s = ⟨ N, W ⟩_{t}$ where $M_{t} = E [F ∣ F_{t}]$ . By the previous proposition's bracket-density identification, the integrands of $M$ and $N$ coincide, so $M = N$ and $F = M_{T} = E [F] + \int_{0}^{T} E [D_{s} F ∣ F_{s}] d W_{s}$ . $□$

Proposition (completeness of Black–Scholes). In the model $d S_{t} = S_{t} (μ d t + σ d W_{t})$ , $σ > 0$ , with bond $d B_{t} = r B_{t} d t$ , every claim $F \in L^{2} (Ω, F_{T}, Q)$ is replicable by a self-financing portfolio, and its time- $t$ arbitrage price is $V_{t} = e^{- r (T - t)} E_{Q} [F ∣ F_{t}]$ .

Proof. Under $Q$ defined by the Girsanov density $E (- \frac{μ - r}{σ})_{T}$ , the process $\tilde{W}_{t} = W_{t} + \frac{μ - r}{σ} t$ is a $Q$ -Brownian motion and $d \tilde{S}_{t} = σ \tilde{S}_{t} d \tilde{W}_{t}$ with $\tilde{S}_{t} = e^{- r t} S_{t}$ , so $\tilde{S}$ is a $Q$ -martingale. The discounted claim $e^{- r T} F$ has, by the representation theorem applied to the $\tilde{W}$ -Brownian filtration (which equals $(F_{t})$ ), the form $e^{- r T} F = E_{Q} [e^{- r T} F] + \int_{0}^{T} H_{s} d \tilde{W}_{s}$ . Set the stock holding $ϕ_{t} = H_{t} / (σ \tilde{S}_{t})$ and the bond holding so that the portfolio is self-financing; then the discounted wealth $\tilde{V}_{t} = E_{Q} [e^{- r T} F] + \int_{0}^{t} ϕ_{s} d \tilde{S}_{s} = E_{Q} [e^{- r T} F ∣ F_{t}]$ satisfies $\tilde{V}_{T} = e^{- r T} F$ , i.e. $V_{T} = F$ almost surely. Replicability gives the arbitrage price $V_{t} = B_{t} \tilde{V}_{t} = e^{- r (T - t)} E_{Q} [F ∣ F_{t}]$ by the no-arbitrage principle. $□$

Connections Master

The bracket and isometry of 37.06.01 are the exact machinery behind both existence and uniqueness here: uniqueness of the representing integrand is the isometry $E [(\int H \cdot d W)^{2}] = E \int ∣ H ∣^{2} d s$ applied to a difference, and the intrinsic identification of $H$ as the density $d ⟨ M, W^{i} ⟩ / d t$ is the covariation pairing built there. The representation theorem is the statement that, over the Brownian filtration, this pairing exhausts $L^{2}$ — the continuous-local-martingale theory specialised to the case where the integrator is Brownian and the filtration carries nothing else.

Brownian motion 02.15.01 supplies the integrator and, crucially, the filtration: the proof's totality argument uses the Gaussian structure of the increments $W_{t_{j}} - W_{t_{j - 1}}$ to pass from real to complex exponentials, and the degeneracy of $F_{0}$ that forces the representation constant to be $E [F]$ is a property of the augmented natural filtration constructed there. Lévy's characterisation from that unit reappears in the Girsanov step of the completeness proof, identifying $\tilde{W}$ as a Brownian motion.

The Itô integral and the exponential martingale of 02.15.02 are the representers: each stochastic exponential $E (h)$ is shown there to be an Itô integral against itself, and the density of these exponentials in $L^{2}$ is what the present theorem converts into representation for all functionals. The Itô formula computing $W_{T}^{2} = T + 2 \int W d W$ supplies the explicit integrands in the exercises, and the Doléans-Dade exponential is the Girsanov density used for market completeness.

Downstream, the representation theorem is the foundation for the Malliavin calculus and the Clark–Ocone formula, where the abstract integrand becomes the predictable projection of the Malliavin derivative; for stochastic-control and backward-SDE theory, where the pair $(Y, Z)$ solving a BSDE is exactly a martingale and its representation integrand; and for the Föllmer–Schweizer and quadratic-hedging decompositions 37.07.03 in incomplete markets, which measure precisely the obstruction to the clean single-integrand representation that holds here.

Historical & philosophical context Master

The representation of Brownian functionals as stochastic integrals originates with Kiyosi Itô's 1951 study of the multiple Wiener integral ^{[Itô 1951]}, which decomposed $L^{2}$ of Wiener space into orthogonal chaoses and so exhibited the first-order term as an Itô integral; Norbert Wiener's earlier homogeneous chaos (1938) was the analytic precursor without the martingale framing. The clean martingale-representation statement — every $L^{2}$ Brownian martingale is a constant plus an Itô integral — was established through the 1950s and 1960s and given its filtration-theoretic form by Kunita and Watanabe in their 1967 work on square-integrable martingales, the same paper that supplied the bracket of 37.06.01. J. M. C. Clark's 1970 paper The representation of functionals of Brownian motion by stochastic integrals (Ann. Math. Statist. 41, 1282–1295) made the integrand explicit for differentiable functionals, the formula completed by Daniel Ocone in 1984 using the Malliavin derivative, now the Clark–Ocone formula.

The conceptual content is that a single source of continuous Gaussian noise generates, through integration alone, every square-integrable observable of its history. This is the probabilistic counterpart of a spanning theorem: the increments of one Brownian path form a complete system, and the stochastic integral is the synthesis operation. Fischer Black, Myron Scholes, and Robert Merton's 1973 option-pricing theory rests on exactly this completeness — the reason a European option has a unique arbitrage-free price is that its payoff is a representable functional, hence perfectly replicable by trading the underlying. The Malliavin calculus, developed by Paul Malliavin from 1976 for a probabilistic proof of Hörmander's hypoellipticity theorem, gave the representation its differential face and turned the abstract integrand into a computable derivative.

Bibliography Master

@article{Ito1951chaos,
  author  = {It\^o, Kiyosi},
  title   = {Multiple {W}iener Integral},
  journal = {Journal of the Mathematical Society of Japan},
  volume  = {3},
  pages   = {157--169},
  year    = {1951}
}

@article{Clark1970,
  author  = {Clark, J. M. C.},
  title   = {The Representation of Functionals of {B}rownian Motion by Stochastic Integrals},
  journal = {Annals of Mathematical Statistics},
  volume  = {41},
  number  = {4},
  pages   = {1282--1295},
  year    = {1970}
}

@article{Ocone1984,
  author  = {Ocone, Daniel},
  title   = {Malliavin's Calculus and Stochastic Integral Representations of Functionals of Diffusion Processes},
  journal = {Stochastics},
  volume  = {12},
  number  = {3--4},
  pages   = {161--185},
  year    = {1984}
}

@article{KunitaWatanabe1967rep,
  author  = {Kunita, Hiroshi and Watanabe, Shinzo},
  title   = {On Square Integrable Martingales},
  journal = {Nagoya Mathematical Journal},
  volume  = {30},
  pages   = {209--245},
  year    = {1967}
}

@book{Nualart2006,
  author    = {Nualart, David},
  title     = {The Malliavin Calculus and Related Topics},
  edition   = {2nd},
  series    = {Probability and Its Applications},
  publisher = {Springer-Verlag, Berlin},
  year      = {2006}
}

@article{BlackScholes1973,
  author  = {Black, Fischer and Scholes, Myron},
  title   = {The Pricing of Options and Corporate Liabilities},
  journal = {Journal of Political Economy},
  volume  = {81},
  number  = {3},
  pages   = {637--654},
  year    = {1973}
}

Prerequisites

37.06.01
02.15.01
02.15.02

Tier anchors

beginner: Le Gall, Brownian Motion, Martingales, and Stochastic Calculus (Springer GTM 274, 2016), Ch. 5 (informal statement of representation); Øksendal, Stochastic Differential Equations (6th ed.) Ch. 4 (the representation theorem and market completeness, informal)
intermediate: Le Gall, Brownian Motion, Martingales, and Stochastic Calculus (Springer GTM 274, 2016), §5.4 (the predictable representation property, proof via stochastic exponentials); Karatzas–Shreve, Brownian Motion and Stochastic Calculus, §3.4 (martingale representation Thm 3.4.15)
master: Revuz–Yor, Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999), Ch. V §3 (representation of Brownian martingales, the chaos expansion); Karatzas–Shreve §3.4 (representation, completeness of Black–Scholes §5.8); Nualart, The Malliavin Calculus and Related Topics (2nd ed. 2006), §1.3 (Wiener chaos), §1.3.3 (Clark–Ocone)

References

Le Gall — Brownian Motion, Martingales, and Stochastic Calculus (Springer GTM 274, 2016) · Ch. 5 §5.4 (predictable representation property of Brownian motion, Thm 5.18; proof via total set of stochastic exponentials in L^2)
Karatzas, Shreve — Brownian Motion and Stochastic Calculus (Springer GTM 113, 2nd ed. 1991) · §3.4 (Thm 3.4.15 martingale representation, Problem 3.4.16 multidimensional), §5.8 (completeness of the Black–Scholes market via representation)
Revuz, Yor — Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999) · Ch. V §3 (Thm V.3.4 representation of Brownian local martingales, Wiener chaos decomposition, multiple Wiener–Itô integrals)
Nualart — The Malliavin Calculus and Related Topics (Springer, 2nd ed. 2006) · §1.1.2 (Wiener chaos), §1.3 (the Clark–Ocone formula, Prop 1.3.5–1.3.14)
Itô — Multiple Wiener integral (J. Math. Soc. Japan 3, 1951) · pp. 157–169; the orthogonal decomposition of L^2 of Wiener space into multiple integrals (the Wiener–Itô chaos)

Estimated time

beginner: 17m
intermediate: 48m
master: 90m