What Makes a Solution Weak?
An SDE specifies a relation between a process and its driving noise. A strong formulation fixes the stochastic basis and Brownian motion in advance. A weak formulation allows the basis to be part of the solution: the probability space, filtration, probability measure, and Brownian motion may be chosen, provided together they support a process satisfying the stated SDE. “Weak” therefore refers to the freedom in the probability-space formulation, not to a relaxed equation.
Fix a finite time horizon \(T>0\), deterministic initial value \(x_0\in\mathbb{R}\), and coefficient functions \(\mu,\sigma:[0,T]\times\mathbb{R}\to\mathbb{R}\). The coefficients are assumed Borel measurable, with their compositions below progressively measurable. A process is progressively measurable if, for each \(t\), its restriction to \([0,t]\times\Omega\) is measurable with respect to the product of the Borel sets on \([0,t]\) and \(\mathcal{F}_t\). This condition ensures that the time integrals and Ito integrals in the formulation are defined.
- \(X_0=x_0\) almost surely;
- \(\mathbb{E}\int_0^T|\mu(t,X_t)|\,dt<\infty\) and \(\mathbb{E}\int_0^T\sigma(t,X_t)^2\,dt<\infty\);
- \(\sigma(t,X_t)\) is progressively measurable and belongs to \(\mathcal{L}^2\), the class of progressively measurable \(H_t\) satisfying \(\mathbb{E}\int_0^T H_t^2\,dt<\infty\); and
- for all \(t\in[0,T]\), almost surely, \(X_t=x_0+\int_0^t\mu(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\), where the stochastic integral is the Ito integral under \(\mathbb{P}\).
In a strong formulation, the probability space, filtration, and Brownian motion are prescribed before the solution is sought; the solution is adapted to that basis and is constructed from its specified noise. In the weak formulation, these objects are chosen as part of the solution. In both formulations, \(B\) must be Brownian relative to the filtration: its increments must have the appropriate normal distributions and be independent of the information available at earlier times. Merely having Brownian marginal distributions is not enough.
The displayed integrability assumptions ensure that the drift integral is defined and that the Ito integral is square-integrable on the finite horizon. They are conditions on the solution, not a general existence theorem. In particular, no Lipschitz or linear growth hypothesis is being asserted here. The integrals in this tutorial are Ito integrals under \(\mathbb{P}\); no Stratonovich integral or change to a risk-neutral measure \(\mathbb{Q}\) is involved.
Two Direct Constructions
Worked Example: Constant Drift and Diffusion on Any Brownian Basis
Let \(x_0=1\), and consider \(dX_t=2\,dt+3\,dB_t\) on \([0,1]\). Take any filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})\) carrying a Brownian motion \(B_t\) relative to \((\mathcal{F}_t)\). Define \(X_t=1+2t+3B_t\).
The process is continuous and adapted because \(B_t\) is continuous and adapted. Its drift integrand is the constant \(2\), so \(\mathbb{E}\int_0^1 2\,dt=2\). Its diffusion integrand is the constant \(3\), which is progressively measurable and satisfies \(\mathbb{E}\int_0^1 3^2\,dt=9\). The integral equation is
For a numerical check at \(t=1\), the formula gives \(X_1=3+3B_1\), while substituting \(t=1\) into the integral equation gives \(1+2+3(B_1-B_0)=3+3B_1\), since \(B_0=0\). Thus the same construction works on any eligible Brownian basis; the underlying sample space need not be specified by the equation.
Worked Example: A Geometric Brownian Solution on an Arbitrary Basis
On any filtered probability space carrying a Brownian motion \(B_t\) relative to its filtration, consider \(dX_t=X_t\,dt+2X_t\,dB_t\), \(X_0=1\). Define \(X_t=\exp(-t+2B_t)\). To verify the equation, apply Ito’s formula to \(f(t,b)=\exp(-t+2b)\). Its derivatives are \(f_t=-f\), \(f_b=2f\), and \(f_{bb}=4f\). The second-order correction is \(\tfrac12 f_{bb}\,dt=2f\,dt\). Therefore
Also \(X_0=\exp(0)=1\). To check the required integrability, the normal moment-generating function gives \(\mathbb{E}[e^{cB_t}]=e^{c^2t/2}\). Hence \(\mathbb{E}[X_t]=e^{-t}\mathbb{E}[e^{2B_t}]=e^{-t}e^{2t}=e^t\), and \(\mathbb{E}[X_t^2]=e^{-2t}\mathbb{E}[e^{4B_t}]=e^{-2t}e^{8t}=e^{6t}\). Consequently, on \([0,T]\), \(\mathbb{E}\int_0^T X_t\,dt=\int_0^T e^t\,dt=e^T-1<\infty\), and \(\mathbb{E}\int_0^T(2X_t)^2\,dt=4\int_0^T e^{6t}\,dt=\tfrac23(e^{6T}-1)<\infty\). The candidate is therefore a weak solution on each such basis.
Check: its expected value is \(e^t\). The drift term in the SDE predicts this: taking expectations in the integral equation gives \(\mathbb{E}[X_t]=1+\int_0^t\mathbb{E}[X_s]\,ds\), because the square-integrable Ito integral has mean zero. The function \(e^t\) satisfies this identity since \(1+\int_0^t e^s\,ds=e^t\).
What Can Vary—and What Must Not
The probability space itself may differ between weak solutions. The filtration may also differ, but each proposed Brownian motion must remain Brownian relative to the filtration used in its solution. That requirement restricts which extra information may be included. An independent random variable can be added without revealing future Brownian increments; a future Brownian value cannot simply be declared known at time zero.
Proof. Fix \(0\le s<t\le T\). By the Brownian property relative to \((\mathcal{F}_t)\), \(B_t-B_s\) has distribution \(N(0,t-s)\) and is independent of \(\mathcal{F}_s\). It is also independent of \(U\), jointly with \(\mathcal{F}_s\), because \(U\) is independent of \(\mathcal{F}\). Thus \(B_t-B_s\) is independent of \(\mathcal{F}_s\vee\sigma(U)=\mathcal{G}_s\), and retains its \(N(0,t-s)\) distribution. Continuity and \(B_0=0\) are unchanged, so \(B\) is Brownian relative to \((\mathcal{G}_t)\).
Every \(\mathcal{F}_t\)-adapted process is \(\mathcal{G}_t\)-adapted. Its progressive measurability is preserved, and its integrability expectations are unchanged because the probability measure on the original variables has not changed. For the Ito integral, first consider a simple integrand \(H_t=\sum_i \xi_i\mathbf{1}_{(t_i,t_{i+1}]}(t)\), with each \(\xi_i\) measurable with respect to \(\mathcal{F}_{t_i}\). Its integral is the sum \(\sum_i\xi_i(B_{t_{i+1}\wedge t}-B_{t_i\wedge t})\), which is the same whether computed with \((\mathcal{F}_t)\) or \((\mathcal{G}_t)\). General progressively measurable square-integrable integrands are obtained in the Ito integral construction by approximation in \(L^2(\mathbb{P}\times dt)\); the Ito isometry shows that the corresponding integrals agree in the limit. The solution equation and all its other requirements therefore remain valid on the enlarged basis. \(\square\)
Worked Example: Adding an Independent Coin
Start with a Brownian basis and the solution \(X_t=1+2t+3B_t\) of \(dX_t=2\,dt+3\,dB_t\). Enlarge the space by an independent coin \(U\) with \(\mathbb{P}(U=0)=\mathbb{P}(U=1)=\tfrac12\), and set \(\mathcal{G}_t=\mathcal{F}_t\vee\sigma(U)\). Since the coin is independent of the entire original \(\mathcal{F}\), the proposition shows that \(B\) remains Brownian relative to \((\mathcal{G}_t)\). The same \(X\) is adapted to this larger filtration and still satisfies \(X_t=1+2t+3B_t\).
The coin does not appear in the SDE or in its solution formula. For example, \(\mathbb{P}(U=1)=\tfrac12\) still holds, but conditioning on \(U\) does not alter the Brownian increment distribution: \(\mathbb{P}(B_t-B_s\le z\mid U)=\Phi\bigl(z/\sqrt{t-s}\bigr)\) for \(s<t\), where \(\Phi\) is the standard normal CDF. The added information changes the basis without changing the equation or its Brownian driver.
Check: on \([0,1]\), the drift integral is \(2\) and the stochastic integral is \(3B_1\), so the equation gives \(X_1=1+2+3B_1=3+3B_1\), whether or not \(U\) is included in the filtration.
A Direct Characterization for Constant Coefficients
Proof. For the forward statement, the process \(x_0+at+\sigma B_t\) is continuous and adapted, starts at \(x_0\), and has constant progressively measurable integrands \(a\) and \(\sigma\). The drift integrability is \(\mathbb{E}\int_0^T|a|\,dt=|a|T<\infty\), and the diffusion integrability is \(\mathbb{E}\int_0^T\sigma^2\,dt=\sigma^2T<\infty\). Evaluating the two integrals gives \(x_0+at+\sigma B_t\), as required.
For the converse, the definition of a weak solution requires the integral equation \(X_t=x_0+\int_0^t a\,ds+\int_0^t\sigma\,dB_s\). The first integral equals \(at\), and the second equals \(\sigma(B_t-B_0)=\sigma B_t\), because the Ito integral of a constant is the constant times the Brownian increment and \(B_0=0\). Thus \(X_t=x_0+at+\sigma B_t\) almost surely at each time. Both sides have continuous paths, so equality at all rational times in \([0,T]\), on a single event of probability one, extends by continuity to every time in \([0,T]\). This proves the asserted process equality. \(\square\)
This result illustrates the meaning of flexibility precisely: the stochastic basis may vary, but the equation still ties the solution to the Brownian motion on that basis. The theorem does not assert a general existence result for nonlinear coefficients. For those, hypotheses such as Lipschitz and linear growth bounds are relevant to existence and uniqueness arguments; they are not needed for the constant-coefficient calculation above.
Common Pitfall: Revealing the Future
A filtration cannot be enlarged arbitrarily while retaining the same Brownian motion. For example, begin with Brownian motion \(B_t\) and set \(\mathcal{G}_0=\mathcal{F}_0\vee\sigma(B_T)\). If \(B\) were Brownian relative to this enlarged filtration, the increment \(B_T-B_0=B_T\) would be independent of \(\mathcal{G}_0\). But \(B_T\) is itself \(\mathcal{G}_0\)-measurable. Independence of a random variable from a sigma-field containing that variable would force it to be almost surely constant: for any event \(A\) determined by \(B_T\), independence would give \(\mathbb{P}(A)=\mathbb{P}(A)^2\). This is false, for example for \(A=\{B_T\le0\}\), which has probability \(\tfrac12\).
The wrong reasoning is: “\(B\) has Brownian sample paths and normal increments under \(\mathbb{P}\), so it is still Brownian after adding \(B_T\) to the time-zero information.” The correction is that Brownian increments must also be independent of the filtration at earlier times. An independent coin satisfies that requirement; the future value \(B_T\) does not.
Check Your Understanding
Use the definition and the filtration condition, not just the distributions of individual random variables.
- State the probability space, Brownian motion, adaptation, and integrability requirements in the definition of a weak solution.
- For \(dX_t=4\,dt-dB_t\), \(X_0=2\), write the solution formula and check its two integrability conditions on \([0,T]\).
- Why does adding an independent coin at time zero preserve the Brownian property relative to the enlarged filtration?
- Why does adding \(B_T\) to the time-zero filtration violate the Brownian property?
- In what sense can the stochastic basis vary between weak solutions, and which relation between the solution and its driving Brownian motion must still hold?