Weak Construction of the Tanaka Equation

The Construction and Its Main Idea

The Tanaka equation is the stochastic differential equation with zero drift and discontinuous diffusion coefficient \(dX_t=\operatorname{sgn}(X_t)\,dW_t\), started at \(X_0=0\). Its weak construction uses one Brownian motion to build both the candidate solution and its driving noise. Start with a Brownian motion \(B\), take \(X=B\), and define the new noise by integrating the sign of \(B\) against \(B\). The central checks are that this new noise is itself Brownian and that integrating against it recovers \(B\).

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\), with the usual augmentation of its natural filtration. Define \(\operatorname{sgn}(x)=1\) for \(x>0\), \(\operatorname{sgn}(x)=-1\) for \(x<0\), and \(\operatorname{sgn}(0)=0\). All stochastic integrals below are Ito integrals. For each finite \(T\), their integrands will be progressively measurable and in \(\mathcal{L}^2\), meaning that \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\).

Here \(W\) will be a Brownian motion with respect to the filtration that also contains \(B\). This is essential: Brownian motion in the definition must have independent future increments relative to the filtration being used. The integral is an Ito integral, not a Stratonovich integral; no Stratonovich correction is being used in the construction. The drift coefficient is zero and the diffusion coefficient is \(\operatorname{sgn}(x)\).

Brownian Motion Spends Zero Time at the Origin

The value assigned to \(\operatorname{sgn}(0)\) matters pointwise: \(\operatorname{sgn}(x)^2=\mathbf{1}_{\{x\ne0\}}\), not \(1\) everywhere. What makes the construction work is that Brownian motion spends zero Lebesgue time at zero.

Proof. The integrand is nonnegative, so Tonelli's theorem gives \(\mathbb{E}\int_0^T\mathbf{1}_{\{B_s=0\}}\,ds=\int_0^T\mathbb{P}(B_s=0)\,ds\). At every \(s>0\), \(B_s\) has the normal distribution \(N(0,s)\), which assigns probability zero to the singleton \(\{0\}\). The value of the integrand at \(s=0\) does not affect a Lebesgue integral. Thus the expectation of the nonnegative random variable is zero, which implies that the random variable itself is zero almost surely. Taking the intersection of these probability-one events over positive integer \(T\) gives the assertion simultaneously on every finite time interval. \(\square\)

The Transformed Noise Is Brownian

Define, for \(t\ge0\), \(W_t=\int_0^t\operatorname{sgn}(B_s)\,dB_s\). The integrand is progressively measurable because \(B\) is adapted and continuous and the sign function is Borel measurable. On every finite interval it belongs to \(\mathcal{L}^2\), since \(\operatorname{sgn}(B_s)^2\le1\). Consequently this Ito integral has a continuous version and is a square-integrable martingale.

Proof. The Ito integral starts at zero and is a continuous martingale. More precisely, for \(0\le s\le t\), its increment is the Ito integral over \([s,t]\); the defining conditional-mean property of the Ito integral gives \(\mathbb{E}[W_t-W_s\mid\mathcal{F}_s]=0\). The Ito isometry and the zero-occupation proposition give \(\mathbb{E}[W_t^2]=\mathbb{E}\int_0^t\operatorname{sgn}(B_u)^2\,du=t\). The standard quadratic-variation identity for Ito integrals gives \([W]_t=\int_0^t\operatorname{sgn}(B_s)^2\,ds=t\) almost surely for every \(t\). Thus \(W\) is a continuous local martingale starting at zero with quadratic variation \(t\). By Lévy's characterization, a continuous local martingale starting at zero with quadratic variation \(t\) is Brownian motion with respect to the same filtration. Lévy's characterization is a standard result; its proof is not repeated here. In this construction \(W\) is in fact a true square-integrable martingale, not merely a local martingale, as the Ito isometry gives \(\mathbb{E}[W_t^2]=t<\infty\) for every finite \(t\). \(\square\)

Recovering the Candidate Solution

Set \(X_t=B_t\). To verify the equation, we need the identity \(\int_0^t\operatorname{sgn}(B_s)\,dW_s=B_t\). The reason is that \(W\) itself was defined using \(\operatorname{sgn}(B)\,dB\); integrating against \(W\) therefore multiplies the two sign factors. The precise tool is associativity of Ito integration.

Proof. First take \(K\) to be an elementary predictable process, constant with an \(\mathcal{F}_{t_i}\)-measurable value on each interval \((t_i,t_{i+1}]\). On each such interval, the integral against \(\int H\,dB\) is that constant multiplied by the increment \(\int_{t_i}^{t_{i+1}}H_u\,dB_u\). Summing the intervals gives exactly the Ito integral of \(KH\) against \(B\). For general bounded progressively measurable \(K\), choose elementary predictable \(K_n\) converging to \(K\) in \(L^2(\mathbb{P}\times dt)\), the standard approximation used to construct the Ito integral. If \(|H|\le C\), Ito isometry bounds the squared \(L^2(\mathbb{P})\) error on the left by \(C^2\mathbb{E}\int_0^T(K_n-K)^2\,ds\); the squared error on the right has the same bound. Both errors tend to zero. Thus the two integrals agree at each fixed time in \(L^2\); since both have continuous versions, they agree indistinguishably, first on rational times and then on all times by continuity. \(\square\)

Proof. The processes \(H_s=K_s=\operatorname{sgn}(B_s)\) satisfy the lemma's boundedness and measurability assumptions. The integrand \(\operatorname{sgn}(B_s)\) against \(W\) is also in \(\mathcal{L}^2\): its squared integral has expectation \(T\), by the zero-occupation proposition. Associativity yields \(\int_0^t\operatorname{sgn}(B_s)\,dW_s=\int_0^t\operatorname{sgn}(B_s)^2\,dB_s=\int_0^t\mathbf{1}_{\{B_s\ne0\}}\,dB_s\). The difference between this last integral and \(B_t=\int_0^t1\,dB_s\) is \(\int_0^t\mathbf{1}_{\{B_s=0\}}\,dB_s\). Its integrand is in \(\mathcal{L}^2\), and Ito isometry gives \(\mathbb{E}\big[(\int_0^t\mathbf{1}_{\{B_s=0\}}\,dB_s)^2\big]=\mathbb{E}\int_0^t\mathbf{1}_{\{B_s=0\}}\,ds=0\). Thus this integral is zero at each fixed \(t\); its continuous version is indistinguishably zero, by checking rational times and using continuity. The required integral equation holds for all \(t\) almost surely. Since \(X=B\) is adapted and continuous and \(W\) is Brownian in the same filtration, this is a weak solution. \(\square\)

Scope of the Construction and a Common Mistake

This construction proves weak existence: it provides a probability space, a filtration, a Brownian driver \(W\), and an adapted continuous solution \(X\). A strong solution is a different requirement: the solution must be adapted to the filtration generated by the specified driving Brownian motion (with the usual augmentation). Here we have shown that \(X=B\) is adapted to the filtration in which \(W\) is Brownian; we have not shown that \(X\) is a function of \(W\) alone. Nor does one weak construction prove uniqueness in law or pathwise uniqueness. Those are distinct assertions.

Common mistake: using the original Brownian motion as the driver. It is not valid to set \(W=B\) and then claim that \(X=B\) solves the equation because \(\operatorname{sgn}(B_s)^2=1\). For example, whenever \(B_s<0\), the proposed coefficient \(\operatorname{sgn}(B_s)\) is \(-1\), not \(1\), so the proposed integrand against \(B\) is not the integrand \(1\) that produces \(B_t=\int_0^t1\,dB_s\). The actual construction changes the driver to \(W=\int\operatorname{sgn}(B)\,dB\). Integrating \(\operatorname{sgn}(B)\) against this transformed driver supplies the second sign factor and yields \(\operatorname{sgn}(B)^2\), which equals \(1\) for almost every time along almost every path.