Yamada-Watanabe: Weak Existence and Pathwise Uniqueness

Three Different Questions About an SDE

For an SDE, existence and uniqueness depend on what is held fixed. A weak solution allows the probability space and Brownian motion to be chosen as part of the solution. A strong solution starts with a given Brownian motion and constructs the state process from it. Pathwise uniqueness compares two solutions driven by the same Brownian motion on the same space; uniqueness in law compares probability distributions and does not require the solutions to share a space or noise.

Let \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) be a filtered probability space satisfying the usual conditions, and let \(B_t\) be a Brownian motion relative to this filtration. Consider, on a fixed interval \([0,T]\), the SDE \(dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\), with deterministic initial value \(X_0=x_0\). Here \(\mu\) is the drift coefficient and \(\sigma\) is the diffusion coefficient. The integral form is \(X_t=x_0+\int_0^t\mu(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\). The Ito integral is defined for progressively measurable integrands in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); if only local square integrability is assumed, the integral is understood by localization.

Definition: A weak solution consists of a filtered probability space, a Brownian motion \(B\) on that space, and an adapted continuous process \(X\) satisfying the SDE, with the required integrals well-defined, and \(X_0=x_0\) almost surely. A strong solution is a solution constructed on a specified space carrying \(B\), adapted to the completed filtration generated by \(B\), and expressible as a nonanticipative measurable function of the Brownian path. Thus weak existence permits the space and noise to vary; strong existence does not.

Definition: Pathwise uniqueness holds if, whenever \(X\) and \(\widetilde X\) are two solutions on the same filtered probability space, driven by the same Brownian motion and with the same initial value, they are indistinguishable: \(\mathbb{P}(X_t=\widetilde X_t\text{ for every }t\in[0,T])=1\). Uniqueness in law means that all weak solutions have the same distribution. In the theorem below, uniqueness in law refers to the joint distribution of \((B,X)\), and hence also implies uniqueness of the distribution of \(X\).

These are not interchangeable claims. Pathwise uniqueness is a comparison on one space with one noise path. Uniqueness in law is a comparison of distributions across possibly different spaces. A theorem connecting them is therefore needed; equality of marginal distributions alone does not provide a pathwise comparison.

The Yamada–Watanabe Theorem

Theorem (Yamada–Watanabe): Fix a deterministic initial value \(x_0\) and coefficients \(\mu,\sigma\) for which the SDE is meaningful. Suppose a weak solution exists on \([0,T]\), and suppose pathwise uniqueness holds among weak solutions with this initial value. Then a strong solution exists. Moreover, the joint law of \((B,X)\) is the same for every weak solution, so uniqueness in law holds.

The conclusion is stronger than simply producing one solution: the process can be constructed from the Brownian motion, and allowing a different weak-solution space cannot change the joint law. The theorem does not say that weak existence alone gives uniqueness, nor does uniqueness in law by itself assert that solutions driven by the same noise agree.

This is a deep result. Its proof uses conditional distributions of solutions given the Brownian path and a conditional-independence coupling argument; that argument is not reproduced here. The examples below verify the definitions and conclusions directly in cases where explicit calculations are available.

Worked Examples

Worked Example: The Zero Equation

Consider \(dX_t=0\,dt+0\,dB_t\), \(X_0=3\), on \([0,T]\). Show weak existence, strong existence, pathwise uniqueness and uniqueness in law.

On any filtered probability space carrying a Brownian motion \(B\), take \(X_t=3\) for every \(t\). The drift integral is \(\int_0^t0\,ds=0\), and the Ito integral has integrand \(H_s=0\), which is progressively measurable and satisfies \(\mathbb{E}\int_0^T H_s^2\,ds=0\). Thus \(X_t=3\) satisfies the integral equation \(X_t=3\), giving both a weak solution and a strong solution.

For any solution, the same integral equation forces \(X_t=3\) for every \(t\), almost surely. Therefore two solutions driven by the same Brownian motion agree, which proves pathwise uniqueness. Since every solution has the same constant state path and the same Brownian marginal, the joint law is fixed as well. This example satisfies the Yamada–Watanabe hypotheses and its conclusions directly.

Check: the drift and diffusion coefficients are both zero, so neither the choice of Brownian path nor the choice of probability space can alter \(X\).

Worked Example: Geometric Brownian Motion with Zero Drift

Consider \(dX_t=X_t\,dB_t\), \(X_0=2\). Verify the explicit solution, prove pathwise uniqueness, and check its mean.

Set \(X_t=2\exp(B_t-\tfrac12t)\). To verify it, apply Ito’s formula to \(f(t,x)=2\exp(x-\tfrac12t)\). Its derivatives are \(f_t=-\tfrac12f\), \(f_x=f\), and \(f_{xx}=f\). The Ito correction is \(\tfrac12 f_{xx}\,dt\), so

df(t,B_t)=\left(-\tfrac12f+\tfrac12f_{xx}\right)dt+f_x\,dB_t =0\,dt+f\,dB_t=X_t\,dB_t.

Also \(f(0,B_0)=2\), since \(B_0=0\). The integrand \(X_s\) is progressively measurable, and the normal moment-generating function gives \(\mathbb{E}[X_s^2]=4e^{-s}\mathbb{E}[e^{2B_s}]=4e^{-s}e^{2s}=4e^s\). Consequently \(\mathbb{E}\int_0^T X_s^2\,ds=4(e^T-1)<\infty\), so the stochastic integral is an \(\mathcal{L}^2\) Ito integral.

For pathwise uniqueness, let \(\widetilde X\) be any other solution driven by the same \(B\), with the same initial value. Define \(F_t=\exp(-B_t+\tfrac12t)\). Ito’s formula, with \(g(t,x)=\exp(-x+\tfrac12t)\), gives \(g_t=\tfrac12g\), \(g_x=-g\), \(g_{xx}=g\), and hence \(dF_t=F_t\,dt-F_t\,dB_t\). The product rule, including the covariation term, yields

d(F_t\widetilde X_t) =F_t\widetilde X_t\,dB_t+\widetilde X_t(F_t\,dt-F_t\,dB_t) -F_t\widetilde X_t\,dt=0.

The last term is \(d[F,\widetilde X]_t=-F_t\widetilde X_t\,dt\), because their diffusion coefficients are \(-F_t\) and \(\widetilde X_t\). Thus \(F_t\widetilde X_t=F_0\widetilde X_0=2\), and \(\widetilde X_t=2\exp(B_t-\tfrac12t)\). The equality holds for all \(t\) outside one null set, proving pathwise uniqueness. The localization implicit in the product rule is valid because the processes are continuous and can first be stopped when their magnitudes exceed a fixed bound.

For the mean, the verified solution satisfies \(X_t=2+\int_0^tX_s\,dB_s\). Its integrand is in \(\mathcal{L}^2\), so the Ito integral is a square-integrable martingale and has expectation zero. Therefore \(\mathbb{E}[X_t]=2\). Independently, using \(B_t\sim N(0,t)\),

\mathbb{E}[2e^{B_t-t/2}] =2e^{-t/2}e^{t/2}=2.

Thus the candidate’s mean agrees with the SDE calculation. In particular, there is no drift term in this equation: taking expectations does not produce an integral involving \(\mathbb{E}[X_s]\).

Worked Example: Weak Existence Without Pathwise Uniqueness

Consider the deterministic-coefficient SDE \(dX_t=2\sqrt{|X_t|}\,dt+0\,dB_t\), \(X_0=0\), on \([0,2]\). Exhibit two solutions and identify which Yamada–Watanabe hypothesis fails.

For any Brownian motion \(B\), let \(X_t=0\). Its drift integrand is \(2\sqrt{|0|}=0\), and its diffusion integrand is zero, so the integral equation holds. A second adapted continuous process is \(Y_t=t^2\). For \(t\ge0\), its drift integral is

\int_0^t2\sqrt{|Y_s|}\,ds =\int_0^t2\sqrt{s^2}\,ds =\int_0^t2s\,ds=t^2=Y_t.

Its diffusion integral is again zero. Both processes start at zero and can be placed on the same filtered space with the same Brownian motion, but \(X_t\ne Y_t\) for every \(t>0\). Hence pathwise uniqueness fails. The equation has weak solutions, but the theorem’s pathwise-uniqueness hypothesis is absent; this example does not contradict the theorem. The coefficient is not Lipschitz at zero, which is consistent with the failure, but no Lipschitz criterion is needed for this direct verification.

What the Theorem Does—and Does Not—Conclude

The implication in the theorem has a useful logical order:

Weak existence.
At least one probability space, Brownian motion and adapted process satisfy the SDE.

Pathwise uniqueness.
Any two solutions with the same Brownian motion and initial value agree almost surely on their common space.

Yamada–Watanabe conclusion.
The solution can be constructed from the driving Brownian motion, and all weak solutions have the same joint law of noise and state.

Strong existence is a construction statement, while pathwise uniqueness is a comparison statement. Uniqueness in law is distributional. A strong solution need not be introduced by choosing a new Brownian motion: it is adapted to the given Brownian motion’s completed natural filtration. Conversely, a weak solution may use a larger filtration and need not initially be presented as a function of its Brownian path.

A common mistake is to apply the ordinary chain rule to the explicit solution of \(dX_t=X_t\,dB_t\) and write \(X_t=2e^{B_t}\). This calculation omits Brownian quadratic variation. For \(f(t,x)=2e^x\), one has \(f_t=0\), \(f_x=f\), and \(f_{xx}=f\); Ito’s formula therefore gives \(d(2e^{B_t})=2e^{B_t}\,dB_t+\tfrac12(2e^{B_t})\,dt\), which has a nonzero drift and does not solve the stated equation. The correct exponent includes \(-t/2\), whose time derivative cancels the \(\tfrac12 f_{xx}\,dt\) correction. In Ito notation, this correction reflects \(dB_t\,dB_t=dt\), while \(dB_t\,dt=0\) and \(dt\,dt=0\).

The distinction among the four notions can be summarized as follows.

Notion	What is fixed or compared?	Question answered
Weak existence	The equation and initial value; the space may be chosen	Does at least one solution exist?
Strong existence	A specified Brownian motion	Can the solution be constructed from that noise?
Pathwise uniqueness	Same space and same Brownian motion	Must two solutions agree almost surely?
Uniqueness in law	Probability distributions of weak solutions	Do all weak solutions have the same distribution?

Check Your Understanding

Use the definitions and theorem to decide which hypotheses and conclusions are involved.

In the zero equation example, which part of the integral equation forces every solution to equal the initial value?
For \(dX_t=X_t\,dB_t\), why is the Ito integral square-integrable on every finite interval?
State the difference between pathwise uniqueness and uniqueness in law without referring to a particular example.
In the square-root drift example, verify directly that \(Y_t=t^2\) satisfies the integral equation for every \(t\in[0,2]\).
Does weak existence alone meet the hypotheses of the Yamada–Watanabe theorem? Explain which additional condition is required.

Yamada-Watanabe: Weak Existence and Pathwise Uniqueness

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