Locally Lipschitz Coefficients with Linear Growth

From Local Control to a Global Solution

The global Lipschitz existence theorem applies when the drift and diffusion coefficients have one Lipschitz constant valid on the entire state space. Many useful coefficients are only Lipschitz on bounded sets: their local Lipschitz constants may grow as the state moves farther away. Local control alone permits a solution only up to the time it leaves each bounded set. The additional linear growth condition prevents those exit times from accumulating at a finite time.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a one-dimensional Brownian motion \(B_t\) relative to the filtration. Consider an \(\mathbb{R}^d\)-valued process with initial condition \(\xi\), where \(\xi\) is \(\mathcal{F}_0\)-measurable and \(\mathbb{E}|\xi|^2<\infty\). For a fixed finite horizon \(T\), the equation is

where \(\mu\) and \(\sigma\) take values in \(\mathbb{R}^d\). The drift is \(\mu\); the diffusion coefficient is \(\sigma\). An integral against \(B_t\) is an Itô integral. Its integrand is progressively measurable and belongs to \(\mathcal{L}^2\) on \([0,T]\), meaning \(\mathbb{E}\int_0^T |H_s|^2\,ds<\infty\). When an integrand is only locally square-integrable, the integral is understood after stopping where its accumulated square is bounded.

Truncating the Coefficients

Let \(p_R\) be projection onto the closed ball of radius \(R\) centred at the origin: \(p_R(x)=x\) if \(|x|\le R\), and \(p_R(x)=Rx/|x|\) if \(|x|>R\). Projection onto a closed ball is 1-Lipschitz and satisfies \(|p_R(x)|\le |x|\). Define the truncated coefficients by \(\mu_R(t,x)=\mu(t,p_R(x))\) and \(\sigma_R(t,x)=\sigma(t,p_R(x))\). They agree with the original coefficients when \(|x|\le R\). They are globally Lipschitz: applying the local Lipschitz bound on the ball of radius \(R\) and then the projection bound gives

They also retain the same linear growth bound, since \(|p_R(x)|\le|x|\). Thus the earlier global Lipschitz existence and uniqueness theorem applies to each truncated equation with the same initial condition \(\xi\). Denote its strong solution by \(X^R\), and define its exit time by \(\tau_R=\inf\{t\ge0:|X^R_t|\ge R\}\), with the infimum of the empty set equal to infinity. If \(|\xi|\ge R\), this definition gives \(\tau_R=0\).

Consistency Before Exit

Proof. Let \(\rho\) be the first time either solution reaches radius \(R\). Before \(\rho\), both processes lie in the ball of radius \(R\); there the two truncated coefficient pairs agree with the original pair. The original coefficients are Lipschitz on this ball with constant \(L_R\). Apply the standard \(L^2\) maximal inequality for stochastic integrals and Cauchy–Schwarz to the difference of the two stopped integral equations. For \(0\le t\le T\), these estimates give a finite constant \(A_{R,T}\) such that \[ \mathbb{E}\!\left[\sup_{0\le u\le t}|X^R_{u\wedge\rho}-X^S_{u\wedge\rho}|^2\right] \le A_{R,T}\int_0^t \mathbb{E}\!\left[\sup_{0\le v\le s}|X^R_{v\wedge\rho}-X^S_{v\wedge\rho}|^2\right]ds. \] Both processes have the same initial condition. Gronwall’s inequality therefore makes the left side zero for every \(t\le T\). The stopped paths agree almost surely. By continuity, they reach the boundary of the radius-\(R\) ball at the same time, if either reaches it. Hence the exit times from that ball coincide. \(\square\)

This comparison is local: it uses the Lipschitz constant on the ball being considered, not a global Lipschitz constant for the original coefficients. For any fixed outcome with finite \(|\xi|\), the comparison applies to all sufficiently large radii. The solutions can consequently be joined consistently up to their successive exit times. To obtain a solution on all finite horizons, it remains to show that those exit times do not accumulate in finite time.

Uniform Moment Control Rules Out Explosion

We use the a priori second-moment estimate established for globally Lipschitz SDEs with linear growth. In the form needed here, it says that for each finite \(T\) there is a constant \(K_T<\infty\), depending on \(T\), the growth bound, and \(\mathbb{E}|\xi|^2\), but not on \(R\), such that

The uniformity follows because every truncated equation has the same linear growth bound. The estimate is the standard a priori moment bound for SDEs; the required stochastic-integral maximal inequality is a standard result, and its proof is beyond this tutorial.

Proof. First control the exit probability. If \(\tau_R\le T\), continuity implies that \(\sup_{0\le t\le T}|X^R_t|\ge R\). Markov’s inequality and the uniform moment bound give

Thus the probability of exit by time \(T\) tends to zero as \(R\) tends to infinity. For almost every outcome, \(|\xi|\) is finite, so all sufficiently large radii exceed \(|\xi|\). The consistency proposition makes the truncated solutions agree up to their nested exit times for those radii. If their limiting exit time were at most \(T\), then \(\tau_R\le T\) for every sufficiently large \(R\). The probability of that event is at most \(\mathbb{P}(\tau_R\le T)\) for each such \(R\), and hence is zero by the bound \(K_T/R^2\). The exit times therefore tend to infinity almost surely. Joining the consistent solutions gives a continuous adapted process on every finite horizon, satisfying the original SDE there. Its drift and diffusion integrals are well-defined locally by stopping at bounded exit times; the linear growth estimate and the uniform moment bound give square-integrability on each finite horizon.

For uniqueness, let \(X\) and \(\widetilde X\) be two solutions driven by the same \(B_t\) with the same initial condition. Stop both when either reaches radius \(R\). On the ball of radius \(R\), the coefficients are Lipschitz. The same stopped difference estimate used in the proposition, followed by Gronwall’s inequality, shows that the stopped processes agree almost surely. Each continuous solution has bounded paths on \([0,T]\), so the probability that either exits every sufficiently large ball by time \(T\) is zero. Letting \(R\) increase proves \(X_t=\widetilde X_t\) for all \(t\le T\) almost surely. Since \(T\) is arbitrary, pathwise uniqueness holds on every finite horizon. The construction gives a strong solution, adapted to the given filtration and driven by its given Brownian motion; it is not merely an assertion of existence in law. \(\square\)

What the Argument Does—and Does Not—Assume

The local Lipschitz condition ensures that solutions of the truncated equations cannot separate before leaving a bounded region. The linear growth condition supplies a moment bound uniform over all truncation levels. Together, these facts give local uniqueness and prevent finite-time escape. Neither condition is a claim that every equation lacking it has no solution: they are hypotheses that make this existence-and-uniqueness argument work.

A common mistake is to apply the global Lipschitz theorem directly to the original coefficients because each bounded region has its own Lipschitz constant. This is invalid: a constant \(L_R\) that depends on \(R\) is not one global Lipschitz constant. The correction is to apply the theorem to each truncated equation, compare those solutions only before exit, and then use a bound uniform in \(R\) to show that exits do not obstruct a global solution.

Another error is to infer non-explosion from consistency alone. Consistency says that solutions can be joined up to their exit times; it does not say those times tend to infinity. The estimate \(\mathbb{P}(\tau_R\le T)\le K_T/R^2\) is the separate step that rules out finite-time explosion.