THE EXPLOSION TIME IN STOCHASTIC DIFFERENTIAL EQUATIONS WITH SMALL DIFFUSION

PABLO GROISMAN AND JULIO D. ROSSI

Abstract. We consider solutions of a one dimensional stochastic differential equations that explode in finite time. We prove that, under suitable hypotheses, the explosion time converges almost surely to the one of the ODE governed by the drift term when the diffusion coefficient goes to zero.

1. Introduction

Explosions in one dimensional ODEs is a very well known phenomena. Let u(t) be the solution of

(1.1)  u̇ = b(u),  u(0) = x_{0}.

If b > 0, there exists a finite time T such that lim_{t↗T} u(t) = +∞ if and only if ∫^{∞} 1/b < +∞. In this case we have an explicit formula for the explosion time T in terms of b and x_{0},

(1.2)  T = ∫_{x_{0}}^{∞} \frac{1}{b(s)} ds.

On the other hand, let us consider the stochastic differential equation

(1.3)  dX = b(X) dt + σ(X) dW,  X(0) = x_{0} > 0,

where b and σ are smooth positive functions and W is a (one dimensional) Wiener process defined on a given probability space (Ω, ℙ).

As happens with (1.1), solutions of (1.3) may explode in finite time, that is, trajectories may diverge to infinity as t goes to some finite time S that in general depends on the particular sample path.

This phenomena has been considered, for example, in fatigue cracking (fatigue failures in solid materials) with b and σ of power type, [5]. In this case the explosion time corresponds to the time of ultimate damage or fatigue failure in the material.

Key words and phrases. Explosion, stochastic differential equations.

2000 Mathematics Subject Classification: 60H10, 60G17, 34F05.

The Feller Test for explosions (see [3, 4]) gives a precise description in terms of b, σ and x_{0} of whether explosions in finite time occur with probability zero, positive or one. For example, if b and σ behave like powers at infinity, i.e., b(s) ∼ s^{p}, σ(s) ∼ s^{q} as s → ∞, applying the Feller test one obtains that solutions to (3.1) explode with probability one if and only if p > 2q − 1 and p > 1.

There is no simple formula for the explosion time S as (1.2) (although there exists some expressions for a version of S that involve the scale function which can be found in [3]). Hence, to estimate S is a nontrivial task. In order to get information about the stochastic explosion time one can use adaptive numerical approximations like the ones described in [1] where the authors provide a numerical method that can be used to compute a convergent approximation of S.

In this article we find, by theoretical arguments, estimates on the explosion time S when the diffusion σ is small. That is, we look at (1.3) as a stochastic perturbation of the ODE (1.1). We prove that, under adequate hypotheses on b and σ, the stochastic explosion time, S = S(σ), converges to the deterministic one, T, almost surely when σ goes to zero. This means that the stochastic explosion times converge to a constant, T given by (1.2), that can be explicitly computed.

In the statement of the theorem we use the Stratonovich integral since the proofs are simpler. This is not a restriction thanks to the well known conversion formula (see below). We consider a family of SDE

(1.4)  dX = b(X) dt + σ(X, ε) ∘ dW,  X(0) = x_{0} > 0,

where ε > 0 is a parameter and σ(·, ε) → 0 as ε → 0. We introduce a function H: ℝ × ℝ_{+} × ℝ_{+} → ℝ (ℝ_{+} = [0, +∞)) defined in this way: let ϕ = ϕ_{ε}(t, x) the flux associated to the ODE

(1.5)  ẏ = σ(y, ε),  y(0) = x.

We assume that σ(·, ε) is globally Lipschitz and smooth and therefore ϕ_{ε} is globally defined. Then we define

H(s, x, ε) = \frac{b(ϕ_{ε}(s, x))σ(x, ε)}{σ(ϕ_{ε}(s, x), ε)}.

Theorem 1. Assume

(1) b > 0 in ℝ_{+} and σ(·, ε) > 0 has continuous bounded derivatives in ℝ_{+},

(2) Given s ∈ ℝ, there exists g_{s} ∈ L^{1}(ℝ_{+}) such that \frac{1}{H(s,x,ε)} ≤ g_{s}(x) for every x ∈ ℝ_{+},

(3) H(s, x, ε) ≥ H(t, x, ε) if s ≥ t,

(4) lim_{ε→0} H(s, x, ε) = b(x), then for almost every ω the (strong) solution of (1.4) explodes in finite time S_{ε}(ω) for every ε > 0 and

(1.6)  lim_{ε→0} S_{ε}(ω) = T.

Moreover, if in addition H verifies

(5) For every s ∈ ℝ, there exists f_{s} ∈ L^{1}(ℝ_{+}) such that \frac{∂}{∂ε} \frac{1}{H(s,x,ε)} ≤ f_{s}(x) for every x ∈ ℝ_{+} and 0 < ε < ε_{0},

then S_{ε}(ω) is Lipschitz continuous at ε = 0 almost surely, that is, there exist a random variable C = C(ω) such that with total probability

|T − S_{ε}(ω)| ≤ Cε.

Remark 1.1. If a SDE is given in Itô form, we can apply this result thanks to the conversion formula. In fact, X(t) solves dX = f(X)dt + g(X)dW if and only if it solves (3.1) with b = f − \frac{1}{2}σ′σ, σ = g. In this case we obtain that also b depends on ε, however the result still holds and the proof is similar.

Remark 1.2. If b(x)/σ(x, ε) is increasing in x then the monotonicity of H(s, x, ε) in s, hypothesis (3), holds.

Remark 1.3. If H(s, x, ε) is increasing (or decreasing) in ε then we can get rid of hypothesis (2), using the Monotone Convergence Theorem instead of the Dominated Convergence Theorem in the proof.

2. Some simple examples

In this section we consider some simple examples to illustrate the main ideas used in the proof of Theorem 1 and the principal features of the problem.

The main idea is to change variables in order to transform the SDE into a random differential equation. Then we obtain bounds for the explosion time by using sub and supersolutions given by ODEs.

Example 1: additive noise. Let u(t) be the solution of (1.1) with b increasing and ∫^{∞} 1/b < +∞. Let X be a solution of the Itô SDE

dX = b(X)dt + εdW, X(0) = x_{0}.

Let Z = X − εW, then Z solves

dZ = dX − εdW = b(Z + εW)dt, Z(0) = x_{0}.

This gives a non-autonomous ODE for each ω such that W (·, ω) is continuous,

(2.1)  Ż_{ω}(t) = b(Z_{ω}(t) + εW (t, ω)),  Z_{ω}(0) = x_{0}.

In this equation ω is regarded as a parameter.

Given M > 0, we consider \overline{z} and \underline{z} the solutions of

˙\overline{z}(t) = b(\overline{z}(t) + εM), \overline{z}(0) = x_{0}

and

˙\underline{z}(t) = b(\underline{z}(t) − εM), \underline{z}(0) = x_{0}.

These solutions explode in finite time given by

\overline{T}_{ε} = ∫_{x_{0}}^{∞} \frac{1}{b(s + εM)} ds, \underline{T}_{ε} = ∫_{x_{0}}^{∞} \frac{1}{b(s − εM)} ds,

respectively. Since b is increasing, by the Monotone Convergence Theorem we get

(2.2)  lim_{ε→0} \overline{T}_{ε} = lim_{ε→0} \underline{T}_{ε} = T.

Let

A_{M} = {ω: W (·, ω) is continuous and max_{0≤t≤T +1} |W (·, ω)| ≤ M} .

For ω ∈ A_{M}, \overline{z} and \underline{z} are super and subsolutions of (2.1) for 0 < t < T + 1. Using (2.2), a comparison argument gives

\underline{z}(t) ≤ Z_{ω}(t) ≤ \overline{z}(t),

as long as all of them are defined. Hence, for ω ∈ A_{M},

\overline{T}_{ε} ≤ S_{ε}(ω) ≤ \underline{T}_{ε}.

Therefore, by (2.2),

lim_{ε→0} S_{ε}(ω) = T.

As

ℙ( ⋃_{M=1}^{∞} A_{M}) = 1

we get the desired result.

Example 2: multiplicative noise. Let u(t) be as in Example 1. Let X be the solution of the Itô SDE

dX = b(X)dt + εXdW, X(0) = x_{0}.

As in the preceding example, we want to get an ODE for each ω. To do that, let Z = Xe^{−εW}. Using Itô’s rule we get that Z solves

dZ = (e^{−εW} b(Ze^{εW}) + ε^{2}Z) dt, Z(0) = x_{0}.

As before, this gives a non-autonomous ODE for each ω such that W (·, ω) is continuous,

(2.3) Ż_{ω}(t) = e^{−εW (t,ω)}b(Z_{ω}(t)e^{εW (t,ω)}) + ε^{2}Z_{ω}(t),  Z_{ω}(0) = x_{0}.

Given M > 0, we consider \overline{z} and \underline{z} the solutions of

˙\overline{z}(t) = e^{εM}b(\overline{z}(t)e^{εM}) + ε^{2}\overline{z}(t), \overline{z}(0) = x_{0}

and

˙\underline{z}(t) = e^{−εM}b(\underline{z}(t)e^{−εM}) + ε^{2}\underline{z}(t), \underline{z}(0) = x_{0}.

These solutions explode in finite time given by

\overline{T}_{ε} = ∫_{x_{0}}^{∞} \frac{1}{e^{εM}b(se^{εM}) + ε^{2}s} ds, \underline{T}_{ε} = ∫_{x_{0}}^{∞} \frac{1}{e^{−εM}b(se^{−εM}) + ε^{2}s} ds,

respectively. We have

(2.4)  lim_{ε→0} \overline{T}_{ε} = lim_{ε→0} \underline{T}_{ε} = T.

Let A_{M} as before. Since b is increasing, for ω ∈ A_{M}, \overline{z} and \underline{z} are super and subsolutions of (2.3) for 0 < t < T + 1 and hence, using (2.4), we can compare their explosion times

\overline{T}_{ε} ≤ S_{ε}(ω) ≤ \underline{T}_{ε}.

Therefore

lim_{ε→0} S_{ε}(ω) = T.

and we get the desired result.

3. Proof of the main result

3.1. Pathwise solutions of the SDE. We want to apply the same ideas used in the previous examples, that is, to transform the SDE in a non-autonomous ODE where ω plays the role of a parameter. This is easier when the equation is understand in Stratonovich sense.

The study of pathwise solutions to stochastic differential equations via an appropriate reduction to an ODE was first done in [2, 6]. We refer to those works and to [3] for details.

Consider a solution of the Stratonovich SDE

(3.1)  dX = b(X)dt + σ(X) ∘ dW.

This solution may explode in finite time or may be globally defined. Let y be a solution of the ODE

(3.2)  ẏ = σ(y),  y(0) = x,

and let ϕ(t, x) the flux associated to (3.2) which is globally defined and has continuous derivatives, since σ is smooth and globally Lipschitz.

Consider Z_{ω} = Z_{ω}(t) the solution of the random differential equation

(3.3)  \begin{matrix}dZ_{ω}(t) = \frac{b(ϕ(W (t, ω), Z_{ω}(t)))}{ϕ_{x}(W (t, ω), Z_{ω}(t))} dt, \\ Z_{ω}(0) = x_{0}. \end{matrix}

Then X(t, ω) = ϕ(W (t, ω), Z_{ω}(t)) is a strong solution of (3.1) up to a possible explosion time S_{ε}. In fact, since (3.1) is interpreted in Stratonovich sense, we have

\begin{matrix}dX = ϕ_{t}(W, Z_{ω}) dW + ϕ_{x}(W, Z_{ω}) dZ_{ω} = σ(X) dW + b(X) dt, \\ X(0) = x_{0}. \end{matrix}

Note that the explosion time S_{ε}(ω) is the maximal existence time of (3.3) for each ω. We are going to use this fact to prove Theorem 1.

3.2. Proof of Theorem 1. First of all observe that assumptions (1) and (2) ensures on the one hand that solutions to (1.1),(1.4) are positive and on the other hand that solutions to (1.1) explodes in finite time T given by (1.2). Applying the Feller Test for explosions one can see that these hypotheses also ensure that (1.4) explodes in finite time with probability one. Nevertheless we are going to show this fact in the course of the proof.

For each ω such that W (·, ω) is continuous, consider the ODE

(3.4)  Ż_{ω}(t) = \frac{b(ϕ_{ε}(W (t, ω), Z_{ω}(t)))}{(ϕ_{ε})_{x}(W (t, ω), Z_{ω}(t))},  Z_{ω}(0) = x_{0}.

Here ϕ_{ε} is the flux associated to the ODE (1.5). The equation (3.4) can be written in terms of H as

(3.5)  Ż_{ω}(t) = H(W (t, ω), Z_{ω}(t), ε),  Z_{ω}(0) = x_{0}.

In fact, integrating (1.5) we get

∫_{x}^{ϕ_{ε}(t,x)} \frac{dτ}{σ(τ, ε)} = t.

Differentiating with respect to x we obtain

\frac{(ϕ_{ε})_{x}(t, x)}{σ(ϕ_{ε}(t, x), ε)} − \frac{1}{σ(x, ε)} = 0,

hence

(ϕ_{ε})_{x}(t, x) = \frac{σ(ϕ_{ε}(t, x), ε)}{σ(x, ε)}

and so

H(s, x, ε) = \frac{b(ϕ_{ε}(s, x))}{(ϕ_{ε})_{x}(s, x)}.

Given M > 0, we consider \overline{z} and \underline{z} the solutions of

˙\overline{z}(t) = H(M, \overline{z}(t), ε), \overline{z}(0) = x_{0}

and

˙\underline{z}(t) = H(−M, \underline{z}(t), ε), \underline{z}(0) = x_{0}.

These solutions explode in finite time given by

\overline{T}_{ε} = ∫_{x_{0}}^{∞} \frac{1}{H(M, x, ε)} dx, \underline{T}_{ε} = ∫_{x_{0}}^{∞} \frac{1}{H(−M, x, ε)} dx,

respectively. By assumption (2) we can apply the Dominated Convergence Theorem to get

(3.6)  lim_{ε→0} \overline{T}_{ε} = lim_{ε→0} \underline{T}_{ε} = T.

Let A_{M} be as in the examples. Since H(s, x, ε) is increasing in the s variable, for any ω ∈ A_{M}, \overline{z} and \underline{z} are super and subsolutions of (3.5) for 0 < t < T + 1. Using this fact and (3.6), their explosion times can be compared. Since X(t) = ϕ_{ε}(W (t), Z_{ω}(t)) and ϕ_{ε} is globally defined, the explosion times of X and Z_{ω} coincide a.s. Then we obtain

\overline{T}_{ε} ≤ S_{ε}(ω) ≤ \underline{T}_{ε}.

Therefore

lim_{ε→0} S_{ε}(ω) = T.

As

ℙ( ⋃_{M=1}^{∞} A_{M}) = 1,

we have proved (1.6). It rest to show the Lipschitz continuity. To do this observe that the Taylor expansion of 1/H(±M, x, ε) at ε = 0 gives

|S_{ε}(ω) − T | ≤ |∫_{x_{0}}^{∞} \frac{1}{H(±M, x, ε)} dx − T |
= |∫_{x_{0}}^{∞} \frac{1}{b(x)} dx + ∫_{x_{0}}^{∞} ε\frac{∂}{∂ε} \frac{1}{H(±M, x, η_{ε})} dx − T |
= |∫_{x_{0}}^{∞} ε\frac{∂}{∂ε} \frac{1}{H(±M, x, η_{ε})} dx|
≤ ε |∫_{x_{0}}^{∞} f_{M}(x) dx|
≤ Cε.

This completes the proof. □

4. More Examples

In this section we present two additional examples where the result can be applied.

Example 3: unbounded diffusion. Let u(t) be the solution of (1.1) and consider the SDE

dX = b(X)dt + εσ(X) ∘ dW, X(0) = x_{0},

with

b(x) ∼ x^{p}, σ(x) ∼ x^{q}, 0 < q < 1 < p,

for large x and bounded below away from zero.

In this case we have

ϕ_{ε}(t, x) ∼ (\frac{x^{1−q}}{1 − q} + εt)^{\frac{1}{1−q}} ,

for x large and t > 0. Hence, the behavior of H(s, x, ε) at infinity is given by

H(s, x, ε) ∼ ϕ_{ε}(t, x) ∼ (\frac{x^{1−q}}{1 − q} + εt)^{\frac{p−q}{1−q}} x^{q} ∼ Cx^{p}.

From these expressions it is easy to check hypotheses (1),(2) and (4). If we assume (3), then we can apply our theorem to get S_{ε} → T almost surely. Note that (3) holds if we take, for example, b(x) = (1 + |x|)^{p}, σ(x) = ε(1 + |x|)^{q}. In this case (5) also holds and so S_{ε} is Lipschitz at ε = 0 almost surely.

Example 4: bounded diffusion. In this last example we consider

dX = b(X)dt + εσ(X) ∘ dW, X(0) = x_{0},

with a bounded σ, 0 < c_{1} ≤ σ ≤ C_{2} and b such that ∫^{∞} 1/b < +∞. We have

\frac{1}{H(s, x, ε)} ≤ g_{s}(x) := \frac{C}{b(x)}.

If we assume that (3) holds (the rest of the hypotheses can be easily checked) we obtain again that S_{ε} → T almost surely.

Acknowledgments We want to thank J. A. Langa for several interesting discussions. Supported by Universidad de Buenos Aires under grant X066, ANPCyT PICT No. 03-05009, Fundacion Antorchas and CONICET (Argentina).

References

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[2] H. Doss. Liens entre équations différentielles stochastiques et ordinaries. Ann. Inst. Henri Poincaré, 13(2):99–125, 1977.

[3] I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.

[4] H. P. McKean, Jr. Stochastic integrals. Probability and Mathematical Statistics, No. 5. Academic Press, New York, 1969.

[5] K. Sobczyk and B. F. Spencer, Jr. Random fatigue. Academic Press Inc., Boston, MA, 1992. From data to theory.

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P. Groisman Instituto de Cálculo, FCEyN, Universidad de Buenos Aires, Pabellón II, Ciudad Universitaria (1428), Buenos Aires, Argentina.

E-mail address: pgroisma@dm.uba.ar
Web page: http://mate.dm.uba.ar/∼pgroisma

J. D. Rossi Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 123, Madrid, Spain, on leave from Departamento de Matemática, FCEyN UBA (1428) Buenos Aires, Argentina.

E-mail address: jrossi@dm.uba.ar
Web page: http://mate.dm.uba.ar/∼jrossi