In the previous post it was shown how the existence and uniqueness of solutions to stochastic differential equations with Lipschitz continuous coefficients follows from the basic properties of stochastic integration. However, in many applications, it is necessary to weaken this condition a bit. For example, consider the following SDE for a process X
where Z is a given semimartingale and are fixed real numbers. The function has derivative which, for , is bounded on bounded subsets of the reals. It follows that f is Lipschitz continuous on such bounded sets. However, the derivative of f diverges to infinity as x goes to infinity, so f is not globally Lipschitz continuous. Similarly, if then f is Lipschitz continuous on compact subsets of , but not globally Lipschitz. To be more widely applicable, the results of the previous post need to be extended to include such locally Lipschitz continuous coefficients.
In fact, uniqueness of solutions to SDEs with locally Lipschitz continuous coefficients follows from the global Lipschitz case. However, solutions need only exist up to a possible explosion time. This is demonstrated by the following simple non-stochastic differential equation
For initial value , this has the solution , which explodes at time .
As in the previous post, let us consider SDEs written in the following integral form
(1) |
where is a semimartingale and is a cadlag adapted process. The problem, then, is to find a solution for the process up to a possible explosion time.
The coefficients in (1) assign a predictable and -integrable process to each n-dimensional cadlag adapted process . In most cases, will just be a function of but, more generally, it can depend on the path of X before time t. Let us consider coefficients defined on some open subset U of , and we assume that . Furthermore, let denote the cadlag and adapted processes X such that and are in U at all times t. Then, the properties required for are as follows.
- (P1) is a map from to the set of predictable and -integrable processes.
- (P2) For each compact subset , there is a constant such that
whenever , for all times and .
The local Lipschitz property, (P2), ensures that at any time t only depends on the values of X prior to this time. In particular, if is a stopping time and X is a cadlag process defined on the closed stochastic interval such that for all , then the pre-stopped process
will be in (at least, it is when restricted to ). This means that is well-defined on the interval , so equation (1) makes sense on , regardless of the values of for . Then, for any stopping time , the following can be used to precisely define what is meant by a solution to the SDE on the closed and open intervals and respectively.
- A solution X to the SDE (1) on is a cadlag, adapted process such that for all and such that (1) holds on .
- A process X is a solution to the SDE (1) on if there exists a sequence of stopping times such that and there exist solutions to (1) on which agree with X on the open intervals .
Here, as always, processes are considered to be equal whenever they agree up to evanescence.
The main existence and uniqueness result up to a possible explosion time is as follows.
Theorem 1 Suppose that satisfies properties (P1), (P2) above. Then, there is a unique stopping time and solution X to the SDE (1) on the interval , which does not extend to a solution on any larger interval with and .
Furthermore, if Y is any other solution to (1) on an interval for a stopping time , then and Y=X on .
The following result states what happens to the solution X at time , which should make clear why it is not possible to extend to a solution on a larger interval. That is, whenever is finite then as it is approached, X either does not converge to a limit in U or, simply, jumps out of U.
Theorem 2 Assuming the conditions of Theorem 1 then, with probability 1, precisely one of the following statements holds for the explosion time .
- .
- and, for any , is not contained in a compact of subset of U.
- and exists in U, but
is not in U.
In many situations, Z and N are continuous processes or U is the whole of , so the third case cannot arise. So, at the explosion time, X must either diverge to infinity or get arbitrarily close to the edge of the set U. In particular, if , then
whenever .
Now, we can look at a condition for non-explosion. That is, under what conditions is the explosion time for the SDE guaranteed to be infinite, so that a unique solution exists and is defined at all times. Consider the simple non-stochastic SDE where and is an increasing function. The solution to this satisfies . The explosion time is , so a necessary and sufficient condition for non-explosion in this case is for this integral to be infinite. More generally, this gives a sufficient, but not necessary, condition for non-explosion of stochastic differential equations.
Theorem 3 Assume the conditions of Theorem 1 and that . Suppose furthermore, that there is an increasing function such that
for all , and that for . Then, the explosion time is almost surely infinite.
In particular, if the coefficients are locally Lipschitz continuous with no more than linear growth, then global existence and uniqueness of solutions is guaranteed.
Proofs
The results above for locally Lipschitz coefficients can be implied from the case for globally Lipschitz coefficients studied in the previous post. The idea is to extend from compact sets to the whole of as a globally Lipschitz continuous function. Throughout this section we assume that satisfy properties (P1) and (P2) above.
Lemma 4 Let S be a compact subset of U. Then, there exist maps such that the following hold.
- for all .
- (P2′) There is a constant K such that
for all and all times .
Proof: If is a Lipschitz continuous function equal to 1 on S, and zero outside some compact set R with then, using property (P2) that is Lipschitz continuous and -integrable when restricted to processes in R, the following is easily seen to satisfy the required properties
Constructing such a function f is not hard. For any write which, by compactness of S, will be strictly bounded below by some outside U. If then on R and zero elsewhere is Lipschitz continuous. ⬜
This allows us to apply the results for globally Lipschitz continuous coefficients here, and show that locally, solutions to the SDE exist and are unique.
Lemma 5 Let V be a bounded open set with closure . Then, there exists a unique stopping time and solution X to (1) on such that for all and whenever .
Furthermore, if is any other stopping time and Y is a solution to (1) on then Y=X on .
Proof: Let be as in Lemma 4, and let be a solution to the SDE
As satisfies the global Lischitz constraint (P2′), this has a unique solution. Let the first time at which it exits V be,
Then, set on the interval . By construction, for all , so that on and X is a solution to (1) on this interval.
Now, suppose that is a stopping time and that Y is a solution to (1) on . If
then the stopped processes both satisfy the SDE
Again, by uniqueness for globally Lipschitz continuous coefficients, and X=Y on . However, note that, by the construction of , whenever . It follows that and Y=X on the interval .
Finally, suppose that for and whenever . If then the first condition gives the contradiction , so . Alternatively, if then the second condition gives the contradiction , so . Then, X=Y on , proving uniqueness. ⬜
This lemma can now be applied to obtain existence and uniqueness of a solution to the SDE up to an explosion time. The following result includes the conclusions of Theorems 1 and 2.
Lemma 6 There exists a unique stopping time and solution X to (1) on such that, almost surely, one of the following is satisfied.
- .
- and, for any , is not contained in a compact of subset of U.
- and exists in U, but
is not in U.
Furthermore, if is a stopping time and Y is any other solution to (1) on , then and Y=X on .
Proof: Choose any increasing sequence of bounded open sets with closure whose union is the whole of U. Let be stopping times and be solutions to (1) on as given by Lemma 5. Then on . If then we have the contradiction and . So, and on .
Now, let and define X on by for all r large enough that . By definition, this is a solution to (1) on . Furthermore, as was assumed to be in U, it will be in for large r, giving and, hence, as required.
Note that if for large r, then giving case 3 above. Alternatively, if for all r, then are not all contained in any of the and therefore will not be contained in any compact subset of U, giving case 2.
Now, suppose that is a stopping time and that Y is a solution to (1) on . By definition, there is a sequence of stopping times and solutions to (1) on such that on and . Then, by Lemma 5, on . In particular, if then, case 2 above gives the contradiction and, similarly, case 3 gives . So, and on .
Finally, suppose that Y also satisfies one of the three cases in the statement of the lemma at the time . Then, if , it is not possible to extend to a solution on the larger interval , giving almost surely. This proves uniqueness of and X, as required. ⬜
It only remains to give a proof of the condition for non-explosion given by Theorem 3, in the case where the domain is .
Proof: We are given the increasing function such that and for . Then, define
and,
This is an increasing concave function with .
If Y is any positive semimartingale, then see the proof of Lemma 3 in the previous post for the following inequality,
(2) |
In particular, take where X is a solution to (1) over an interval . Then, integration by parts gives
Here, I left the summations over i,j,k implicit. Combine this with inequality (2).
We have . Also, fixing a constant , we are free to replace by without changing any of the statements so far. Then, as long as . So, stopping at the first time at which , the integrands on the right hand side of the above inequality are all bounded by 1, and these integrals are bounded in probability up to any finite time for which . Therefore, over any finite time period for which , cannot go to infinity and, as , X cannot explode. As L is arbitrary, this shows that X cannot explode at any time. ⬜
Reading the start of the last proof, the first inequality doesn’t seem to follow from f(y) > y. This would imply the opposite inequality than what it suggested, surely?
Also I’m not sure how the second inequality in (2) is derived.
You are correct, of course. I updated the post to fix the proof, and also modified the condition for non-explosion. It is now a slightly stronger condition and, in fact, the condition that I previously used () was too weak, and there are counterexamples to this.
Hello, does your result says that the one dimensional CIR process starting from a positive point admit unique and strong solution?
No, it only shows that there is a unique solution up until it hits 0. This is because the SDE involves the square root of the process, and this is not locally Lipschitz continuous at 0. More advanced methods are required to handle this and, often, the existence of local times are used for the uniqueness proof.
Hi, can you provide a book/paper that covers what you discussed in the post for reference purpose? I find that it’s only covers the existence and uniqueness of a solution up to an explosion time in Protter’s “Stochastic Integration and Differential Equations”. But it doesn’t address the linear growth condition. Thanks!
I’ll have a look for references. The fact that explosion does not occur under linear growth is well known, although the precise statement and proof of theorem 3 is my own (I did make a mistake when I posted originally, but it is fixed now).
as say prof dr mircea orasanu and prof horia orasan u
I wonder how i can take your result of Theorem 3 as a reference. As, to be honest, there is no similar result available. And this result is very important in my paper.
Hi, thank you for your nice post. There is a little problem: that pre-stopped process seems not to be in D^(U) in general, unless X_{\tau-} is in U, right? But this can be fixed trivially and does not affect the subsequent proofs.