The previous two posts described the behaviour of standard Brownian motion under stochastic changes of time and equivalent changes of measure. I now demonstrate some applications of these ideas to the study of stochastic differential equations (SDEs). Surprisingly strong results can be obtained and, in many cases, it is possible to prove existence and uniqueness of solutions to SDEs without imposing any continuity constraints on the coefficients. This is in contrast to most standard existence and uniqueness results for both ordinary and stochastic differential equations, where conditions such as Lipschitz continuity is required. For example, consider the following SDE for measurable coefficients and a Brownian motion B
If a is nonzero, is locally integrable and b/a is bounded then we can show that this has weak solutions satisfying uniqueness in law for any specified initial distribution of X. The idea is to start with X being a standard Brownian motion and apply a change of time to obtain a solution to (1) in the case where the drift term b is zero. Then, a Girsanov transformation can be used to change to a measure under which X satisfies the SDE for nonzero drift b. As these steps are invertible, every solution can be obtained from a Brownian motion in this way, which uniquely determines the distribution of X.
A standard example demonstrating the concept of weak solutions and uniqueness in law is provided by Tanaka’s SDE
with the initial condition . Here, sgn(x) is defined to be 1 for and -1 for . As any solution to this is a local martingale with quadratic variation
Lévy’s characterization implies that X is a standard Brownian motion. This completely determines the distribution of X, meaning that uniqueness in law is satisfied. However, there are many different solutions to (2). In particular, whenever X is a solution then –X will be another one. So, even though all solutions have the same distribution, pathwise uniqueness fails. Next, it is possible to construct a filtered probability space and Brownian motion B such that (2) has a solution. Simply let X be a Brownian motion and set which, again using Lévy’s characterization, implies that B is a Brownian motion. Solutions such as this, which are defined on some filtered probability space rather than an initial specified space are known as weak solutions. In fact, it is not hard to demonstrate that (2) does not have a solution on the filtration generated by B. As is invariant under replacing X by –X, it follows that the sets are X-measurable and invariant under changing the sign of X. In particular, X itself will not be -measurable, so no solutions to (2) can exist on such a probability space. If weak solutions to an SDE exist, then they can be constructed on an enlargement of the underlying filtered probability space but not, in general, on the original space.
Changes of Time
Consider the 1-dimensional SDE
for measurable and a Brownian motion B. We suppose that X and B are defined on some filtered probability space . As X is a continuous local martingale, it is a time-changed Brownian motion. That is, there exists a Brownian motion W adapted to some other filtration such that (possibly requiring an enlargement of the probability space). Furthermore, so, by the independent increments property, W and are independent. It is possible to express X entirely in terms of and W. First, the quadratic variation of X is,
If it is assumed that a is never zero, then a change of variables together with the identity gives the following
Therefore, is the unique time at which the strictly increasing process hits the value t. This shows that any solution X to SDE (3) can be written as
for a Brownian motion W independent of . So, the distribution of X satisfying (3) is uniquely determined by the distribution of .
We can also try going in the opposite direction. That is, if W is a Brownian motion independent from a random variable , then does the process defined by (4) solve our SDE? As long as is almost surely finite for each time t and , the answer is yes. In this case, let be the filtration generated by and W, and define the stopping times . Then,
is a local martingale with quadratic variation . So, is a local martingale. Consider the continuous time change , and . Then, B and are -local martingales and, by Lévy’s characterization, B is a Brownian motion. Applying the time change to the stochastic integral gives
and (3) is indeed satisfied.
Under some fairly weak conditions on a, we have shown that the SDE (3) has a solution for a Brownian motion B defined on some filtered probability space. As demonstrated by Tanaka’s SDE above, there isn’t necessarily a solution defined on any given space containing a Brownian motion B. This kind of solution to a stochastic differential equation is called a weak solution.
Definition 1 Consider the n-dimensional stochastic differential equation
(i=1,2,…,n) for a standard m-dimensional Brownian motion and measurable functions .
For any probability measure on , a weak solution to SDE (5) with initial distribution is a continuous adapted process defined on some filtered probability space such that and (5) is satisfied for some -Brownian motion B.
The solution to (5) is said to be unique in law if, for any probability measure on , all weak solutions X with have the same distribution.
Using these definitions, the argument above can be completed to get the following existence and uniqueness results for solutions to the original SDE (3).
Theorem 2 Suppose that is a measurable and nonzero function such that is locally integrable. Then, SDE (3) has weak solutions satisfying uniqueness in law.
Proof: The argument given above proves uniqueness in law. Also, setting for a random variable and independent Brownian motion W, the above construction provides a weak solution with initial value , provided that the process is finite and increases to infinity as . To show that this is indeed the case, consider the following identity
where , which holds for any continuous function and continuous semimartingale M. As F is twice continuously differentiable with , this is just Ito’s lemma. A straightforward application of the monotone class theorem extends (6) to all bounded measurable f and, then, by monotone convergence to all nonnegative and locally integrable f.
In particular, if as above, then and setting in (6) shows that is almost surely finite. As f is nonnegative, F will also be nonnegative. So, the local martingale has the following bound
In the event that , L is bounded below and, by martingale convergence, exists. In that case, converges to a finite limit as . This has zero probability unless F is constant due to the recurrence of Brownian motion (it hits every real value at arbitrarily large times). Furthermore, is strictly positive, so F is strictly convex and, in particular, is not constant. So, we have shown that , and the construction of weak solutions given above applies here. ⬜
Changes of Measure
In the previous section, time-changes were applied to construct weak solutions to (3). Alternatively, changes of measure can be used to transform the drift term of stochastic differential equations. Consider the following n-dimensional SDE
(i=1,2,…,n) for a standard m-dimensional Brownian motion and measurable functions . Suppose that a weak solution has been constructed on some filtered probability space . Choosing any bounded measurable (j=1,2,…,m) the idea is to apply a Girsanov transformation to obtain a new measure under which B gains drift c(t,X). That is
for a -Brownian motion . Then, looking from the perspective of , X is a solution to
where . That is, a Girsanov transformation can be used to change the drift of the SDE.
Although this idea is simple enough, there are technical problems to be overcome. To construct the Girsanov transformation, we first define the local martingales
If U is a uniformly integrable martingale then would have the required property. Unfortunately, this condition is not satisfied in many cases. Consider, for example, the simple case where and c is a nonzero constant. Then, B is a 1-dimensional Brownian motion and is recurrent (hits every value at arbitrarily large times) with probability 1. However, under the transformed measure , B will be a Brownian motion plus a constant nonzero drift, and tends to plus or minus infinity with probability one, hence is not recurrent. So, the desired measure cannot be equivalent to .
The solution is to construct locally. That is, for each time , construct a measure such that (8) holds over the interval [0,T]. By Novikov’s criterion, the stopped process is a uniformly integrable martingale (equivalently, U is a martingale). Indeed, is bounded, so is integrable. Then, define the measure on by .
Note that if and then, applying the martingale property for U,
So, . We would like to imply the existence of a measure satisfying for all times . For such a result to hold, it is necessary to restrict consideration to certain special filtered probability spaces so that Kolmogorov’s extension theorem can be used.
Lemma 3 For some , let be the space of continuous functions from to , and let Y be the coordinate process,
(). Furthermore, let be the sigma algebra on generated by and set .
Suppose that, for each , is a probability measure on and that whenever . Then, there is a unique measure on satisfying for all T.
The superscript 0 is used here to signify that the filtration is not complete. That is, we do not add all zero probability sets of to the filtration (no probability measure has been defined yet, so this wouldn’t even make any sense). This means that care should be taken when applying results from stochastic calculus, as completeness is often assumed. However, it is only really needed to guarantee cadlag versions of martingales and stochastic integrals, which we do not need to be too concerned about for the moment.
Proof: Let be the space of all functions , with coordinate process and filtration . Under the inclusion , induces a measure on . Since for all , these measures are consistent. So, Kolmogorov’s extension theorem says that there exists a measure on such that . By construction, has a continuous modification over the bounded intervals so, taking the limit , has a continuous modification, on . This defines a measurable map , inducing the required measure on . By definition, is uniquely determined on the algebra generating so, by the monotone class theorem, this uniquely determines . ⬜
We can always restrict consideration to spaces of continuous functions when looking for weak solutions of SDEs.
Proof: Let be a weak solution defined on some filtered probability space , with respect to the continuous m-dimensional Brownian motion B. Then, is a continuous (m+n)-dimensional process. Taking d=m+n, and letting be as in Lemma 3, Z defines a map from to and we can define to be the induced measure on . The coordinate process of can be written as , where and . So, has the same joint distribution as and is a weak solution to the SDE with respect to the Brownian motion . ⬜
Putting this together enable us to apply the required Girsanov transformation locally, transforming the drift of our SDE.
Lemma 5 Let be a weak solution to the SDE (7) for an m-dimensional Brownian motion B, and defined on the filtered probability space , where are as in Lemma 3. For bounded measurable functions , let U be the martingale defined by (10).
Proof: As shown above, U is a martingale and the measures satisfy the condition for . So, Lemma 4 gives a unique measure such that as required.
Next, by the theory of Girsanov transformations, the stopped processes are -local martingales, so is a -local martingale. Furthermore, as is equivalent to over finite time horizons, the covariations agree under the and measures so, by Lévy’s characterization, is a -Brownian motion. Finally, the SDE (9) follows from substituting expression (8) for into (7). ⬜
We can now prove weak existence and uniqueness in law for SDEs with transformed drift term.
Theorem 6 Let be measurable functions (i=1,…,n, j=1,…,m) such that are bounded. Setting , consider the SDE
- (11) has a weak solution for any given initial distribution for if and only if (7) does.
- (11) satisfies uniqueness in law for any given initial distribution for if and only if (7) does.
Suppose that X is a weak solution to (7) with respect to an m-dimensional Brownian motion B under probability measure . Then, the transformed measure defined by Lemma 5 makes into a Brownian motion, with respect to which X satisfies (7). Therefore, under , X is a weak solution to (11). Also, so X has the same initial distribution under and . This shows that weak existence for (7) implies weak existence for (11). The converse statement follows by exchanging the roles of and , and replacing by .
Now suppose that (11) satisfies uniqueness in law for solutions with initial distribution , and let X be a weak solution to (7) with , with respect to m-dimensional Brownian motion B under a measure . As above, X is a weak solution to (11) under the transformed measure with the same initial distribution as under . So, by uniqueness in law, the distribution of X under is uniquely determined. Also, can be replaced by for bounded measurable functions satisfying without affecting the conclusion of the theorem. So, by orthogonal projection onto the row-space of , we can suppose that for some . Furthermore, it is not hard to see that can be constructed as a measurable function from . So, the local martingale M defined by (10) can be written as
This expresses M, and hence U, entirely in terms of the process X under measure , which is uniquely determined due to uniqueness in law of (11). This uniquely defines the measure , since
for all and . So, (7) also satisfies uniqueness in law for the given initial distribution of X. The converse statement follows by exchanging the roles of and and replacing by . ⬜
Finally, this result allows us to prove weak existence and uniqueness in law for the SDE mentioned at the top of the post.
Theorem 7 Let and be measurable functions such that a is nonzero, is locally integrable and is bounded. Then, the SDE
(for Brownian motion B) has weak solutions satisying uniqueness in law.
Proof: Theorem 2 guarantees weak existence and uniqueness in law for the SDE (3) with zero drift term. Then, is bounded and measurable, so Theorem 6 extends this to give weak existence and uniqueness in law for SDE (12) with drift . ⬜