The first few lectures are essentially reminders of undergraduate real analysis materials. We will cover some aspects of the theory of differential equations driven by continuous paths with bounded variation. The point is to fix some notations that will be used throughout the course and to stress the importance of the topology of convergence in 1-variation if we are interested in stability results for solutions with respect to the driving signal.
If , we will denote by , the set of subdivisions of the interval , that is can be written
Definition: A continuous path is said to have a bounded variation on , if the 1-variation of on , which is defined as
is finite. The space of continuous bounded variation paths , will be denoted by .
is not a norm, because constant functions have a zero 1-variation, but it is obviously a semi-norm. If is continuously differentiable on , it is easily seen (Exercise !) that
Proposition: Let . The function is additive, i.e for ,
and controls in the sense that for ,
The function is moreover continuous and non decreasing.
Proof: If and , then . As a consequence, we obtain
Let now :
Let . By the triangle inequality, we have
Taking the of yields
which completes the proof. The proof of the continuity and monoticity of is let to the reader
This control of the path by the 1-variation norm is an illustration of the notion of controlled path which is very useful in rough paths theory.
Definition: A map is called superadditive if for all ,
If, in adition, is continuous and , we call a control. We say that a path is controlled by a control , if there exists a constant , such that for every ,
Obviously, Lipschitz functions have a bounded variation. The converse is of course not true: has a bounded variation on but is not Lipschitz. However, any continuous path with bounded variation is the reparametrization of a Lipschitz path in the following sense.
Proposition: Let . There exist a Lipschitz function , and a continuous and non-decreasing function such that .
Proof: We assume and consider
It is continuous and non decreasing. There exists a function such that because implies . We have then, for ,
The next result shows that the set of continuous paths with bounded variation is a Banach space.
Theorem: The space endowed with the norm is a Banach space.
Proof: Let be a Cauchy sequence. It is clear that
Thus, converges uniformly to a continuous path . We need to prove that has a bounded variation. Let
be a a subdivision of . There is , such that , thus
Thus, we have
For approximations purposes, it is important to observe that the set of smooth paths is not dense in for the 1-variation convergence topology. The closure of the set of smooth paths in the 1-variation norm, which shall be denoted by is the set of absolutely continuous paths.
Proposition: Let . Then, if and only if there exists such that,
Proof: First, let us assume that
for some . Since smooth paths are dense in , we can find a sequence in such that . Define then,
This implies that . Conversely, if , there exists a sequence of smooth paths that converges in the 1-variation topology to . Each can be written as,
We still have
so that converges to some in . It is then clear that
Exercise: Let . Show that is the limit in 1-variation of piecewise linear interpolations if and only if .
Let be a piecewise continuous path and . It is well-known that we can integrate against by using the Riemann–Stieltjes integral which is a natural extension of the Riemann integral. The idea is to use the Riemann sums
where . It is easy to prove that, when the mesh of the subdivision goes to 0, the Riemann sums converge to a limit which is independent from the sequence of subdivisions that was chosen. The limit is then denoted and called the Riemann-Stieltjes integral of against . Since has a bounded variation, it is easy to see that, more generally,
with would also converge to . If
is an absolutely continuous path, then it is not difficult to prove that we have
where the integral on the right hand side is understood in Riemann’s sense.
Thus, by taking the limit when the mesh of the subdivision goes to 0, we obtain the estimate
where is the notation for the Riemann-Stieltjes integral of against the bounded variation path . We can also estimate the Riemann-Stieltjes integral in the 1-variation distance. We collect the following estimate for later use:
Proposition: Let be a piecewise continuous path and . We have
The Riemann-Stieltjes satisfies the usual rules of calculus, for instance the integration by parts formula takes the following form
Proposition: Let and .
We also have the following change of variable formula:
Proposition: Let and let be a map. We have
Proof: From the mean value theorem
with . The result is then obtained by taking the limit when the mesh of the subdivision goes to 0
We finally state a classical analysis lemma, Gronwall’s lemma, which provides a wonderful tool to estimate solutions of differential equations.
Proposition: Let and let be a bounded measurable function. If,
for some , then
Proof: Iterating the inequality
times, we get
where is a remainder term that goes to 0 when . Observing that
and sending to finishes the proof