Feller’s Test for Finite-Time Explosion

Explosion and Infinite Endpoints

A locally well-behaved stochastic differential equation can have a solution only up to a random lifetime: the solution may leave every bounded region in finite time. For a one-dimensional diffusion, Feller’s test detects whether an infinite endpoint is reachable in finite time by combining two quantities. The scale function measures the direction and likelihood of travel toward the endpoint; the speed measure accounts for the time spent along the way.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a standard Brownian motion $B_t$. Consider $dX_t=\mu(X_t)\,dt+\sigma(X_t)\,dB_t$ on an open interval $I=(\ell,r)$, where $r=+\infty$ or $\ell=-\infty$. Assume $\mu$ and $\sigma$ are continuous, locally Lipschitz, and $\sigma(x)>0$ on $I$. The solution is the local strong solution, defined up to its lifetime, with the stochastic integral understood locally: after stopping on any compact subinterval, its progressively measurable integrand $\sigma(X_t)$ satisfies $\mathbb{E}\int_0^T \sigma^2(X_s)\,ds<\infty$. The lifetime is the increasing limit of the exit times from compact subintervals exhausting $I$. Explosion at $+\infty$ means that this lifetime is finite and $X_t\to+\infty$ as it is approached; explosion at $-\infty$ is defined analogously.

Definition: Fix a reference point $c\in I$. A scale density, scale function, and speed density are $$ p(x)=\exp\!\left(-\int_c^x\frac{2\mu(z)}{\sigma^2(z)}\,dz\right),\qquad s(x)=\int_c^x p(y)\,dy,\qquad m(x)=\frac{2}{\sigma^2(x)p(x)}. $$ The speed measure has density $m(x)$ with respect to Lebesgue measure. Multiplying $p$, and hence $s'$, by a positive constant changes the normalization but not the endpoint tests below.

The endpoints of $I$ are limits of these functions: for example, $s(+\infty)=\lim_{x\to+\infty}s(x)$, which may be infinite. The test concerns reachability of the endpoint, not whether every sample path explodes. In particular, an accessible endpoint can be reached with positive probability while other paths leave through the opposite side or continue for all time.

Feller’s Scale-and-Speed Test

Theorem (Feller’s test for an infinite endpoint): For an interior starting point $x$, the probability of explosion to $+\infty$ in finite time is positive if and only if $$ s(+\infty)<\infty \quad\text{and}\quad \int_c^{+\infty}\bigl(s(+\infty)-s(y)\bigr)m(y)\,dy<\infty. $$ If either condition fails, the probability of explosion to $+\infty$ is zero. For the left endpoint, the corresponding criterion is $$ s(-\infty)>-\infty \quad\text{and}\quad \int_{-\infty}^{c}\bigl(s(y)-s(-\infty)\bigr)m(y)\,dy<\infty. $$ The choice of finite reference point in the integral does not affect convergence.

The integral is an endpoint test: changing its finite lower limit changes only a finite-interval contribution, which is finite under the stated regularity and nondegeneracy assumptions. The two conditions have distinct roles. Finite scale at the endpoint is necessary for a path to have positive probability of moving there rather than being pulled away in scale. The speed integral then tests whether the time needed to approach it can be finite.

Proof of the finite-integral direction. We give the bounded-interval calculation that explains the scale-and-speed expression. Choose $a<x$, and for $b>x$ let $\tau_{a,b}$ be the first exit time from $(a,b)$. The standard exit-probability formula on a bounded interval is $\mathbb{P}_y(X_{\tau_{a,b}}=b)=(s(y)-s(a))/(s(b)-s(a))$. The standard Green formula for the expected occupation time before exit has kernel $$ G_{a,b}(x,y)= \frac{(s(x\wedge y)-s(a))(s(b)-s(x\vee y))}{s(b)-s(a)}, \qquad \mathbb{E}_x\!\int_0^{\tau_{a,b}}g(X_t)\,dt =\int_a^b G_{a,b}(x,y)g(y)m(y)\,dy $$ for nonnegative measurable $g$. These bounded-interval formulas are the standard results on exit probabilities and mean exit times.

Write $E_b=\{X_{\tau_{a,b}}=b\}$. By the Markov property, conditional on the process still being inside $(a,b)$ at time $t$, the probability of exiting at $b$ is $(s(X_t)-s(a))/(s(b)-s(a))$. Integrating this conditional probability over time gives $$ \mathbb{E}_x[\tau_{a,b}\mathbf{1}_{E_b}] =\int_a^b G_{a,b}(x,y) \frac{s(y)-s(a)}{s(b)-s(a)}m(y)\,dy. $$ Suppose $s(+\infty)<\infty$. Then $\mathbb{P}_x(E_b)$ tends to $(s(x)-s(a))/(s(+\infty)-s(a))>0$ as $b\to+\infty$. On $y\ge x$, the displayed integrand is bounded above by a finite constant, independent of sufficiently large $b$, times $(s(+\infty)-s(y))m(y)$: this follows by substituting the formula for $G_{a,b}$, noting $s(b)-s(y)\le s(+\infty)-s(y)$, and bounding the remaining scale factors. On the finite interval $[a,x]$, the integral stays bounded, since the coefficients and speed density are continuous there and the scale denominators stay bounded away from zero. Thus the assumed finite endpoint integral implies $\sup_{b\ge b_0}\mathbb{E}_x[\tau_{a,b}\mathbf{1}_{E_b}]<\infty$.

Take an increasing sequence $b_n\to+\infty$. The events $E_{b_n}$ decrease, and their intersection has positive probability. On this intersection the exit times $\tau_{a,b_n}$ increase. Fatou’s lemma and the uniform expectation bound imply that their limit is finite almost surely on the intersection. Continuity of paths then forces $X_t\to+\infty$ at that finite limiting time. This proves positive probability of explosion under the two conditions. The converse—that failure of either condition rules out finite-time access to the endpoint—is the converse part of the classical Feller accessibility theorem for one-dimensional diffusions; its full boundary-classification proof is beyond this tutorial. Applying the same argument with the scale direction reversed gives the stated left-endpoint test. $\square$

Worked Applications

Worked Example: Brownian Motion with Positive Constant Drift

Consider $dX_t=\mu\,dt+dB_t$ on $\mathbb{R}$, with $\mu>0$, and test whether $+\infty$ is reachable in finite time. Here $\sigma(x)=1$, so $$ p(x)=e^{-2\mu(x-c)},\qquad s(+\infty)-s(y)=\int_y^\infty e^{-2\mu(z-c)}\,dz =\frac{e^{-2\mu(y-c)}}{2\mu}, \qquad m(y)=2e^{2\mu(y-c)}. $$ The scale endpoint is finite because the exponential is integrable. Multiplying the tail scale by the speed density gives $(s(+\infty)-s(y))m(y)=1/\mu$. Consequently the endpoint integral is $\int_c^\infty (1/\mu)\,dy=\infty$, and Feller’s test says there is no finite-time explosion to $+\infty$.

Check: the solution is $X_t=x+\mu t+B_t$, which is finite for every finite $t$ because Brownian paths are continuous. This independently confirms the test.

Worked Example: A Superlinear Drift

Consider $dX_t=X_t^2\,dt+dB_t$ on $\mathbb{R}$, starting at $x>0$. Test the right endpoint using any fixed $c$. The scale density is $p(y)=\exp(-\tfrac23(y^3-c^3))$, so $s(+\infty)<\infty$. For $y\ge1$, the tail satisfies $$ s(+\infty)-s(y) =e^{2c^3/3}\int_y^\infty e^{-2z^3/3}\,dz \le e^{2c^3/3}\frac{e^{-2y^3/3}}{2y^2}. $$ Indeed, $z^2\ge y^2$ for $z\ge y$, and integrating $2z^2 e^{-2z^3/3}$ from $y$ to infinity gives $e^{-2y^3/3}$. Since $m(y)=2e^{2(y^3-c^3)/3}$, it follows that $(s(+\infty)-s(y))m(y)\le 1/y^2$. The integral of this bound over $[1,\infty)$ is finite, and the contribution on any finite interval is finite by continuity. Both Feller conditions hold, so explosion to $+\infty$ has positive probability.

Check: the drift $x^2$ is positive for $x>0$, and the scale-and-speed calculation identifies the finite-time accessibility that a drift-only ODE intuition would suggest. The conclusion is positive probability, not certainty: Brownian fluctuations still affect individual paths.

Worked Example: Multiplicative Noise with Quadratic Drift

Consider $dX_t=X_t^2\,dt+X_t\,dB_t$ on $(0,\infty)$, with $X_0>0$. The coefficients are locally Lipschitz and the diffusion coefficient is positive throughout this state interval. Here $$ p(x)=e^{-2(x-c)},\qquad s(+\infty)-s(y)=\frac12e^{-2(y-c)},\qquad m(y)=\frac{2e^{2(y-c)}}{y^2}. $$ Their product is exactly $1/y^2$. Thus $s(+\infty)<\infty$ and $\int_c^\infty (s(+\infty)-s(y))m(y)\,dy=\int_c^\infty y^{-2}\,dy=1/c$ when $c>0$. The right endpoint is accessible in finite time with positive probability.

Check: the scale tail times speed density is $y^{-2}$, not $2y^{-2}$: the factors $\tfrac12$ and $2$ cancel. Its integral from $c>0$ is exactly $1/c$, confirming convergence and the required strict positivity of the reference point in this evaluation.

Interpretation and Common Pitfalls

The test matters because local existence and uniqueness do not by themselves provide a global solution. On each compact subinterval, the coefficients in the examples are regular; explosion is a question about what happens as the solution approaches an unbounded endpoint. Scale and speed turn that asymptotic question into an integral test.

Common mistake: checking only whether scale is finite. In the constant-drift example, $s(+\infty)<\infty$, but the speed-weighted tail integral diverges. Concluding that the process explodes from scale finiteness alone would be wrong. The second condition is essential: it tests the time scale, not just the directional probability.

Another distinction: accessibility means a positive probability of reaching the endpoint in finite time from the interior, not almost-sure explosion. For the right endpoint, failure of either condition gives zero probability of explosion there; success of both does not assert that every path explodes there. The left endpoint has its own test, and behavior at one endpoint does not replace checking the other.

Takeaway: For a right infinite endpoint, check scale finiteness first, then integrate the scale tail against speed. For a left infinite endpoint, use the scale distance from the left endpoint and integrate toward it. Both parts are needed to decide finite-time accessibility.

Check Your Understanding

Use the definitions and endpoint test above to answer the following.

For $dX_t=\mu\,dt+dB_t$ with $\mu>0$, which part of the right-endpoint test fails?
Why does a finite scale limit alone not establish finite-time explosion?
For $dX_t=X_t^2\,dt+X_t\,dB_t$ on $(0,\infty)$, compute the scale-tail times speed density.
State the two conditions for accessibility of the left infinite endpoint.
Does accessibility imply explosion almost surely? Explain what the test establishes instead.

Feller’s Test for Finite-Time Explosion

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