Why Extend the Generator?
For a diffusion, the differential expression \(\mathcal{L}f=\mu f'+\tfrac12\sigma^2f''\) describes the instantaneous change in a smooth test function \(f\). The generator defined through the diffusion semigroup, however, may have a restricted domain: its definition involves a limit of expectations, and the limit need not exist for every function of interest. The extended generator uses a different test. It asks whether subtracting the time integral of a candidate drift from \(f(X_t)\) leaves a local martingale.
This formulation is useful when \(f\), its derivatives, or the diffusion coefficients are unbounded. Ito's formula still applies after stopping the process inside bounded regions, where the relevant quantities are controlled. The resulting local martingale need not be a true martingale; that stronger conclusion requires additional integrability.
Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\). Consider a nonexplosive continuous solution of the time-homogeneous SDE \(dX_t=\mu(X_t)\,dt+\sigma(X_t)\,dB_t\), with \(X_0=x\). Here \(\mu\) is the drift and \(\sigma\) is the diffusion coefficient. A standard sufficient condition for a unique nonexplosive strong solution is that \(\mu\) and \(\sigma\) are globally Lipschitz and satisfy linear growth: for some \(C<\infty\), \(|\mu(y)-\mu(z)|+|\sigma(y)-\sigma(z)|\le C|y-z|\) and \(|\mu(y)|+|\sigma(y)|\le C(1+|y|)\). The examples below satisfy these conditions.
All stochastic integrals in this tutorial are Ito integrals. On a finite interval \([0,T]\), an integrand \(H_s\) is in \(\mathcal{L}^2\) when it is progressively measurable and \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). Integrals with integrands that are only locally square-integrable are defined by stopping on intervals where this condition holds.
The phrase “along \(X\)” matters: this definition tests \(f\) only at the states visited by the process. It does not, by itself, determine values of \(g\) at states that \(X\) never visits. The local martingale requirement also does not assert that \(M_t\) is integrable or that \(\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s\); those are requirements for a true martingale.
Localization and the Smooth-Function Formula
For \(n>|x|\), let \(\tau_n=\inf\{t\ge0:|X_t|\ge n\}\), with the infimum of the empty set defined to be infinity. Nonexplosion and continuity imply \(\tau_n\to\infty\) almost surely. If \(\mu,\sigma\) are continuous and \(f\in C^2(\mathbb{R})\), then \(\mu,\sigma,f'\), and \(f''\) are bounded on every compact interval. Thus stopping at \(\tau_n\) controls the coefficients and derivatives appearing in Ito's formula.
Proof. Fix \(n>|x|\). Ito's formula, with \(f\) as the function and \(X\) as the process, uses \(f'(X_s)\) for the first derivative and \(f''(X_s)\) for the second derivative. Its quadratic-variation correction is \(\tfrac12 f''(X_s)\sigma^2(X_s)\,ds\), since \(dB_s\,dB_s=ds\), \(dB_s\,ds=0\), and \(ds\,ds=0\). Stopping the resulting identity at \(\tau_n\) gives $$ f(X_{t\wedge\tau_n})-f(X_0) =\int_0^{t\wedge\tau_n}\left(\mu(X_s)f'(X_s)+\tfrac12\sigma^2(X_s)f''(X_s)\right)ds +\int_0^t \mathbf{1}_{\{s\le\tau_n\}}\sigma(X_s)f'(X_s)\,dB_s. $$ The integrand in the last term is progressively measurable. Before exit, \(X_s\) lies in \([-n,n]\), and the continuous functions \(\sigma\) and \(f'\) are bounded there. Hence, for every finite \(T\), \(\mathbb{E}\int_0^T\mathbf{1}_{\{s\le\tau_n\}}\sigma^2(X_s)(f'(X_s))^2\,ds\le T\sup_{|y|\le n}\sigma^2(y)(f'(y))^2<\infty\). This stopped integrand is in \(\mathcal{L}^2\); its Ito integral is a square-integrable martingale. As \(\tau_n\to\infty\) almost surely, these stopping times localize the unstopped difference in the definition, proving the claim. \(\square\)
The proof explains why an unbounded expression \(\sigma(X_s)f'(X_s)\) is not an obstacle to membership in the extended domain: it is bounded after each localization. It also identifies the limit of this argument. It proves a local martingale, not a true martingale, unless a separate integrability argument is available.
Proof. Set \(H_s=\sigma(X_s)f'(X_s)\). It is progressively measurable, and the additional assumption puts \(H\) in \(\mathcal{L}^2\) on \([0,T]\). The Ito integral \(N_t=\int_0^t H_s\,dB_s\) is square-integrable by the Ito isometry. To check the martingale property, first take a progressively measurable step integrand. Independence and zero conditional mean of each Brownian increment give \(\mathbb{E}[N_t\mid\mathcal{F}_s]=N_s\). General \(\mathcal{L}^2\) integrands are limits of such step integrands in the norm \((\mathbb{E}\int_0^T H_s^2ds)^{1/2}\); the Ito isometry gives convergence of their integrals in \(L^2\), and conditional expectation is a contraction in \(L^2\). Passing to the limit preserves the conditional-expectation identity. The theorem's process is precisely this integral, so it is a true square-integrable martingale on \([0,T]\). \(\square\)
Worked Polynomial Examples
Worked Example: A Fourth-Power Test for Brownian Motion
Let \(X_t=B_t\), so \(\mu(y)=0\) and \(\sigma(y)=1\). Take the polynomial \(f(y)=y^4\). Its derivatives are \(f'(y)=4y^3\) and \(f''(y)=12y^2\). Therefore the diffusion generator is \(\mathcal{L}f(y)=0\cdot4y^3+\tfrac12(1)^2(12y^2)=6y^2\).
For \(\tau_n=\inf\{t:|B_t|\ge n\}\), Ito's formula explicitly reads $$ f'(B_s)=4B_s^3,\qquad f''(B_s)=12B_s^2,\qquad d(B_t^4)=4B_t^3\,dB_t+\tfrac12(12B_t^2)\,dt. $$ Thus \(B_t^4-\int_0^t6B_s^2\,ds\) is a local martingale, with stopped stochastic integrand \(\mathbf{1}_{\{s\le\tau_n\}}4B_s^3\). On \([0,T]\) its squared integral has expectation at most \(16n^6T\), so each stopped integral is in \(\mathcal{L}^2\).
Check: the Gaussian moment identities give \(\mathbb{E}[B_t^4]=3t^2\) and \(\mathbb{E}[6B_s^2]=6s\). Hence \(\mathbb{E}\int_0^t6B_s^2ds=\int_0^t6s\,ds=3t^2=\mathbb{E}[B_t^4]\), as the martingale identity requires. This check also verifies the generator's factor of \(1/2\).
Worked Example: A Fourth-Power Test for an Ornstein-Uhlenbeck Process
Consider \(dX_t=-X_t\,dt+2\,dB_t\), \(X_0=0\). Here \(\mu(y)=-y\), \(\sigma(y)=2\), and for \(f(y)=y^4\), \(f'(y)=4y^3\), \(f''(y)=12y^2\). The generator is \(\mathcal{L}f(y)=(-y)(4y^3)+\tfrac12(2)^2(12y^2)=-4y^4+24y^2\).
In Ito's formula the first-derivative drift term is \((-X_s)4X_s^3=-4X_s^4\), and the second-derivative correction is \(\tfrac12(12X_s^2)(2)^2ds=24X_s^2ds\). The stochastic integrand is \(f'(X_s)\sigma(X_s)=8X_s^3\). Stopping at \(\tau_n=\inf\{t:|X_t|\ge n\}\) makes this integrand bounded by \(8n^3\), so its stopped integral is in \(\mathcal{L}^2\) on every finite interval. This proves that \(y^4\) has extended generator \(-4y^4+24y^2\).
Check: the explicit solution is \(X_t=2\int_0^t e^{-(t-s)}dB_s\), a centered Gaussian random variable. Its variance is \(v_t=4\int_0^t e^{-2(t-s)}ds=2(1-e^{-2t})\), so \(v'_t=4e^{-2t}=4-2v_t\). The Gaussian fourth moment gives \(\mathbb{E}[X_t^4]=3v_t^2\), and therefore \(\frac{d}{dt}\mathbb{E}[X_t^4]=6v_t(4-2v_t)=24v_t-12v_t^2\). The expected generator is \(\mathbb{E}[-4X_t^4+24X_t^2]=-12v_t^2+24v_t\), exactly the same expression. In particular, the squared diffusion coefficient is \(4\), not \(2\); omitting that square would give an incorrect generator and variance.
Worked Example: A Cubic Test for Geometric Brownian Motion
Let \(dX_t=\mu X_t\,dt+\sigma X_t\,dB_t\), \(X_0=x\), with constants \(\mu,\sigma\). For \(f(y)=y^3\), the derivatives are \(f'(y)=3y^2\) and \(f''(y)=6y\). Thus \(\mathcal{L}f(y)=\mu y(3y^2)+\tfrac12\sigma^2y^2(6y)=3(\mu+\sigma^2)y^3\).
The Ito drift term is \(3\mu X_s^3ds\), and the correction term is \(\tfrac12(6X_s)(\sigma X_s)^2ds=3\sigma^2X_s^3ds\). The stochastic integrand is \(3\sigma X_s^3\). It may be unbounded, but on stopping at \(\tau_n=\inf\{t:|X_t|\ge n\}\), its absolute value is at most \(3|\sigma|n^3\). The stopped integral is consequently in \(\mathcal{L}^2\) on finite time intervals, and the cubic test has the stated extended generator.
Check: the explicit solution is \(X_t=x\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\). To verify it, apply Ito's formula to \(F(t,z)=x\exp((\mu-\tfrac12\sigma^2)t+\sigma z)\): \(F_t=(\mu-\tfrac12\sigma^2)F\), \(F_z=\sigma F\), and \(F_{zz}=\sigma^2F\). The correction \(\tfrac12F_{zz}dt\) makes the drift \(\mu Fdt\), while the diffusion is \(\sigma FdB_t\). Using the normal exponential moment, \(\mathbb{E}[X_t^3]=x^3\exp(3\mu t+3\sigma^2t)\). Its derivative is \(3(\mu+\sigma^2)\mathbb{E}[X_t^3]\), agreeing with the expected generator. If \(x=0\), both sides are zero, so the check includes that degenerate initial value.
What Localization Does—and Does Not—Give
In each example, the polynomial and its derivatives are unbounded on the whole real line. Localization lets Ito's formula be applied with bounded stopped integrands; increasing the stopping regions then establishes the local martingale property. This is exactly the purpose of the extended generator: it accommodates functions beyond a semigroup generator's domain without claiming global bounds that are not available.
There is an important distinction between the two martingale conclusions. A local martingale is a martingale after suitable stopping; removing the stops does not automatically preserve integrability or conditional expectations. In the examples above, the finite Gaussian moments show that the stochastic integrands are globally square-integrable on every finite interval, so the stronger true-martingale statement also holds. For a different diffusion or test function, that calculation must be made rather than inferred from the local argument.
Common mistake. A frequent error is to compute the generator of the Ornstein-Uhlenbeck example as \(-4y^4+12y^2\), using \(12y^2\) as the second-order term. The correct calculation is \(\tfrac12\sigma^2 f''=\tfrac12(2)^2(12y^2)=24y^2\). The factor \(\sigma^2\) is essential: the diffusion coefficient is \(2\), but the quadratic-variation contribution uses its square. A separate error is to call the resulting process a true martingale solely because it is a local martingale. The required global integrability check is distinct from localization.
The practical procedure is therefore: calculate the formal generator, stop the process before it leaves bounded sets, check the stopped stochastic integrand is in \(\mathcal{L}^2\), and then let the stopping regions increase. If a true martingale identity is needed, verify global integrability separately.
Check Your Understanding
For each question, specify whether the conclusion is local or requires a further integrability check.
- For Brownian motion and \(f(y)=y^6\), calculate \(f'\), \(f''\), and the extended generator.
- For \(dX_t=-X_tdt+2dB_t\), calculate the extended generator of \(f(y)=y^2\), showing the diffusion correction.
- Explain why stopping at \(\tau_n=\inf\{t:|X_t|\ge n\}\) makes \(\sigma(X_t)f'(X_t)\) bounded when the coefficients and \(f'\) are continuous.
- State the additional condition in the proposition that turns the local martingale into a square-integrable martingale on a finite interval.
- For geometric Brownian motion, verify the generator of \(f(y)=y^2\) and identify which term comes from the Ito correction.