Research and Lecture notes

In the previous lecture we introduced the signature of a bounded variation path as the formal series

If now , the iterated integrals can only be defined as Young integrals when . In this lecture, we are going to derive some estimates that allow to define the signature of some (not all) paths with a finite variation when . These estimates are due to Terry Lyons in his seminal paper and this is where the rough paths theory really begins.

For that can be writen as

we define

It is quite easy to check that for

Let . For , we denote

where is the set of subdivisions of the interval . Observe that for , in general

Actually from the Chen’s relations we have

It follows that needs not to be the -variation of .
The first major result of rough paths theory is the following estimate:

Proposition: Let . There exists a constant , depending only on , such that for every and ,

By , we of course mean . Some remarks are in order before we prove the result. If , then the estimate becomes

which is immediately checked because

We can also observe that for , the estimate is easy to obtain because

So, all the work is to prove the estimate when . The proof is split into two lemmas. The first one is a binomial inequality which is actually quite difficult to prove:

Lemma: For , , and ,

Proof: See Lemma 2.2.2 in the article by Lyons or this proof for the sharp constant

The second one is a lemma that actually already was essentially proved in the Lecture on Young’s integral, but which was not explicitly stated.

Lemma: Let . Let us assume that:

Then, for all ,


Proof:
See the proof of the Young-Loeve estimate or Lemma 6.2 in the book by Friz-Victoir

We can now turn to the proof of the main result.

Proof:
Let us denote

We claim that is a control. Indeed for , we have from Holder’s inequality

It is clear that for some constant which is small enough, we have for ,

Let us now consider

From the Chen’s relations, for ,

Therefore,






On the other hand, we have

We deduce from the previous lemma that

with . The general case is dealt by induction. The details are let to the reader

In this lecture we introduce the central notion of the signature of a path which is a convenient way to encode all the algebraic information on the path which is relevant to study differential equations driven by . The motivation for the definition of the signature comes from formal manipulations on Taylor series.

Let us consider a differential equation

where the ‘s are smooth vector fields on .

If is a function, by the change of variable formula,

Now, a new application of the change of variable formula to leads to

We can continue this procedure to get after steps

for some remainder term , where we used the notations:

• 
• If is a word with length ,

If we let , assuming (which is by the way true for small enough if the ‘s are analytic), we are led to the formal expansion formula:

This shows, at least at the formal level, that all the information given by on is contained in the iterated integrals .

Let be the non commutative algebra over of the formal series with indeterminates, that is the set of series

Definition: Let . The signature of (or Chen’s series) is the formal series:

As we are going to see in the next few lectures, the signature is a fascinating algebraic object. At the source of the numerous properties of the signature lie the following so-called Chen’s relations

Lemma: Let . For any word and any ,

where we used the convention that if is a word with length 0, then .

Proof: It follows readily by induction on by noticing that

To avoid heavy notations, it will be convenient to denote

This notation actually reflects a natural algebra isomorphism between and . With this notation, observe that the signature writes then

and that the Chen’s relations become

The Chen’s relations imply the following flow property for the signature:

Proposition: Let . For any ,

Proof: Indeed,

In the previous lecture we defined the Young’s integral when and with . The integral path has then a bounded -variation. Now, if is a Lipschitz map, then the integral, is only defined when , that is for . With this in mind, it is apparent that Young’s integration should be useful to solve differential equations driven by continuous paths with bounded -variation for . If , then the Young’s integral is of no help and the rough paths theory later explained is the correct one.

The basic existence and uniqueness result is the following. Throughout this lecture, we assume that .

Theorem: Let and let be a Lipschitz continuous map, that is there exists a constant such that for every ,

For every , there is a unique solution to the differential equation:

Moreover .

Proof: The proof is of course based again of the fixed point theorem. Let and consider the map going from the space into itself, which is defined by

By using basic estimates on the Young’s integrals, we deduce that



If is small enough, then , which means that is a contraction of the Banach space endowed with the norm .

The fixed point of , let us say , is the unique solution to the differential equation:

By considering then a subdivision

such that , we obtain a unique solution to the differential equation:

As for the bounded variation case, the solution of a Young's differential equation is a function of the initial condition,

Proposition: Let and let be a Lipschitz continuous map. Let be the flow of the equation

Then for every , the map is and the Jacobian is the unique solution of the matrix linear equation

As we already mentioned it before, solutions of Young’s differential equations are continuous with respect to the driving path in the -variation topology

Theorem: Let and let be a Lipschitz and bounded continuous map such that for every ,

Let be the solution of the differential equation:

If converges to in -variation, then converges in -variation to the solution of the differential equation:

Proof: Let . We have





Thus, if is such that , we obtain

In the very same way, provided , we get

Let us fix and pick a sequence such that . Since , for with big enough, we have

We deduce that for ,

and




For with and big enough, we have

which implies

In this lecture we define the Young‘s integral when and with . The cornerstone is the following Young-Loeve estimate.

Theorem: Let and . Consider now with . The following estimate holds: for ,

Proof: For , let us define

We have for ,

As a consequence, we get

Let now . We claim that is a control. The continuity and the vanishing on the diagonal are obvious to check, so we just need to justify the superadditivity. Let , we have from Holder’s inequality,




We have then

For , consider then the control

Define now

If is such that , we can find a such that , . Indeed, the continuity of forces the existence of a such that . We obtain therefore

which implies by maximization,

By iterating times this inequality, we obtain



It is now clear that:






We conclude

and thus

Sending , finishes the proof

It is remarkable that the Young-Loeve estimate only involves and . As a consequence, we obtain the following result whose proof is let to the reader:

Proposition: Let and with . Let us assume that there exists a sequence such that in and a sequence such that in , then for every , converges to a limit that we call the Young’s integral of against on the interval and denote .
The integral does not depend of the sequences and and the following estimate holds: for ,

The closure of in is and we know that . It is therefore obvious to extend the Young’s integral for every and with and the Young-Loeve estimate still holds

From this estimate, we easily see that for and with the sequence of Riemann sums

will converge to when the mesh of the subdivision goes to 0. We record for later use the following estimate on the Young’s integral, which is also an easy consequence of the Young-Loeve estimate (see Theorem 6.8 in the book for further details).

Proposition: Let and with . The integral path is continuous with a finite -variation and we have

Our next goal in this course is to define an integral that can be used to integrate rougher paths than bounded variation. As we are going to see, Young’s integration theory allows to define as soon as has finite -variation and and has a finite -variation with . This integral is simply is a limit of Riemann sums as for the Riemann-Stiletjes integral. In this lecture we present some basic properties of the space of continuous paths with a finite -variation. We present these results for valued paths but most of the results extend without difficulties to paths valued in metric spaces (see chapter 5 in the book by Friz-Victoir).

Definition. A path is said to be of finite -variation, if the -variation of on , which is defined as

is finite. The space of continuous paths with a finite -variation will be denoted by .

The notion of -variation is only interesting when .

Proposition: Let be a continuous path of finite -variation with . Then, is constant.

Proof: We have for ,

Since is continuous, it is also uniformly continuous on . By taking a sequence of subdivisions whose mesh tends to 0, we deduce then that

so that is constant
The following proposition is immediate:

Proposition: Let , be a continuous path. If then

As a consequence

We remind that a continuous map that vanishes on the diagonal is called a control f if for all ,

Proposition: Let . Then is a control such that for every ,

Proof: It is immediate that

so we focus on the proof that is a control. If and , then . As a consequence, we obtain

thus

The proof of the continuity is left to the reader (see also Proposition 5.8 in the book by Friz-Victoir)

In the following sense, is the minimal control of a path .

Proposition: Let and let be a control such that for ,

then

Proof: We have

The next result shows that the set of continuous paths with bounded -variation is a Banach space.

Theorem: Let . The space endowed with the norm is a Banach space.

Proof: The proof is identical to the case , so we let the careful reader check the details

Again, the set of smooth paths is not dense in for the -variation convergence topology. The closure of the set of smooth paths in the -variation norm shall be denoted by . We have the following characterization of paths in .

Proposition: Let . if and only if

Proof: See Theorem 5.31 in the book by Friz-Victoir

The following corollary shall often be used in the sequel:

Corollary: If , then .

Proof: Let whose mesh is less than . We have



As a consequence, we obtain

Let and let be a Lipschitz continuous map. In order to analyse the solution of the differential equation,

and make the geometry enter into the scene, it is convenient to see as a collection of vector fields , where the ‘s are the columns of the matrix . The differential equation then of course writes

Generally speaking, a vector field on is a map

A vector field can be seen as a differential operator acting on differentiable functions as follows:

We note that is a derivation, that is for ,

For this reason we often use the differential notation for vector fields and write:

Using this action of vector fields on functions, the change of variable formula for solutions of differential equations takes a particularly concise form:

Proposition: Let be a solution of a differential equation that writes

then for any function ,

Let be a Lipschitz vector field on . For any , the differential equation

has a unique solution . By time homogeneity of the equation, the flow of this equation satisfies

and therefore is a one parameter group of diffeomorphisms . This group is generated by in the sense that for every ,

For these reasons, we write . Let us now assume that is a Lipschitz vector field on . If is a diffeomorphism, the pull-back of the vector field by the map is the vector field defined by the chain rule,
. In particular, if is another Lipschitz vector field on , then for every , we have a vector field . The Lie bracket between and is then defined as

It is computed that

Observe that the Lie bracket obviously satisfies and the so-called Jacobi identity that is:

What the Lie bracket really quantifies is the lack of commutativity of the respective flows generated by and .

Lemma: Let be two Lipschitz vector fields on . Then, if and only if for every ,


Proof: This is a classical result in differential geometry, so we only give one part the proof. From the very definition of the Lie bracket and the multiplicativity of the flow, we see that if and only if for every , . Now, suppose that . Let be the solution of the equation

Since , we obtain that is also a solution of the equation. By uniqueness of solutions, we obtain that

As a conclusion,

If we consider a differential equation

as we will see it throughout this class, the Lie brackets play an important role in understanding the geometry of the set of solutions. The easiest result in that direction is the following:

Proposition: Let and let be Lipschitz vector fields on . Assume that for every , , then the solution of the differential equation

can be represented as

Proof: Let

Since the flows generated by the ‘s are commuting, we get that

The change of variable formula for bounded variation paths implies then that is a solution and we conclude by uniqueness

In this lecture we establish the basic existence and uniqueness results concerning differential equations driven by bounded variation paths and prove the continuity in the 1-variation topology of the solution of an equation with respect to the driving signal.

Theorem: Let and let be a Lipschitz continuous map, that is there exists a constant such that for every ,

For every , there is a unique solution to the differential equation:

Moreover .

Proof: The proof is a classical application of the fixed point theorem. Let and consider the map going from the space of continuous functions into itself, which is defined by

By using estimates on Riemann-Stieltjes integrals, we deduce that



If is small enough, then , which means that is a contraction that admits a unique fixed point . This is the unique solution to the differential equation:

By considering then a subdivision

such that , we obtain a unique solution to the differential equation:

The solution of a differential equation is a continuous function of the initial condition, more precisely we have the following estimate:

Proposition: Let and let be a Lipschitz continuous map such that for every ,

If and are the solutions of the differential equations:

and

then the following estimate holds:

Proof: We have

and conclude by Gronwall’s lemma

This continuity can be understood in terms of flows. Let and let be a Lipschitz map. Denote by , , , the unique solution of the equation

The previous proposition shows that for a fixed , the map is Lipschitz continuous. The set is called the flow of the equation.
Under more regularity assumptions on , the map is even and the Jacobian map solves a linear equation.

Proposition: Let and let be a Lipschitz continuous map. Let be the flow of the equation

Then for every , the map is and the Jacobian is the unique solution of the matrix linear equation
,
where the ‘s denote the columns of the matrix .

Proof: We refer to the Chapter 4 in the book by Friz-Victoir

We finally turn to the important estimate showing that solutions of differential equations are continuous with respect to the driving path in the 1-variation topology

Theorem: Let and let be a Lipschitz and bounded continuous map such that for every ,

If and are the solutions of the differential equations:

and

then the following estimate holds:

Proof: We first give an estimate in the supremum topology. It is easily seen that the assumptions imply

From Gronwall’s lemma, we deduce that

Now, we also have for any ,

This implies,

and yields the conclusion

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.

We have




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

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 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

thus

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,

We have

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 us consider a differential equation that writes

where the ‘s are vector fields on and where the driving signal is a continuous bounded variation path. If the vector fields are Lipschitz continuous then, for any fixed initial condition, there is a unique solution to the equation. We can see this solution as a function of the driving signal . It is an important question to understand for which topology, this function is continuous.

A simple example shows that the topology of uniform convergence is not the correct one here. Indeed, let us consider the differential equation



where

A straightforward computation shows that . When , converges uniformly to 0 whereas, of course, does not converge to 0. In this framework , a correct topology is given by the topology of convergence in 1-variation on compact sets. To fix the ideas, let us work on the interval . The distance in 1-variation between two continuous bounded variation paths is given by

where the supremum is taken over all the subdivisions

It is then a fact that is going to be proved in this class that if the ‘s are bounded and if is a sequence of bounded variation paths that converges in 1-variation to a continuous path with bounded variation, then the solutions of the differential equations

converge in 1-variation to the solution of

This type of continuity result suggests to use a topology in -variation, , to try to extend the map to a larger class of driving signals . More precisely, for , let us denote by the closure of the set of continuous with bounded variation paths with respect to the distance in -variation which is given by

We will then prove the following result:

Proposition: Let . If is a sequence of bounded variation paths that converges in -variation to a path , then the solutions of the differential equations

converge in -variation to some . Moreover is the solution of the differential equation

where the integrals are understood in the sense of Young’s integration.

The value is really a treshold: The result is simply false for . The main idea of the rough paths theory is to introduce a much stronger topology than the convergence in -variation. This topology, that we now explain, is related to the continuity of lifts of paths in free nilpotent Lie groups.

Let be the free -step nilpotent Lie group with generators . If is continuous with bounded variation, the solution of the equation

is called the lift of in . For , let us
denote the closure of the set of lifted paths with respect to the distance in -variation which is given by

where denotes the Carnot-Caratheodory distance on the group . This is a distance that will be explained in details later. Its main property is that it is homogeneous with respect to the natural dilation of .

Consider now the map which associates to a continuous with bounded variation path
the continuous path with bounded variation that solves the ordinary differential equation

It is clear that there exists a unique map from the set of continuous with bounded variation lifted paths onto the set of continuous with bounded variation lifted paths which makes the following diagram commutative

The fundamental theorem of Lyons is the following:

Theorem: If , then in the topology of -variation, there exists a continuous extension of from into .

In particular, we can now give a sense to differential equations driven by some continuous paths with finite -variation, for any . Indeed, let which is continuous with a fnite -variation and assume that there exists whose projection onto is . The projection onto of is then understood as being a solution of

An important example of application is given by the case where the driving signal is a Brownian motion . Brownian motion has a -finite variation for any and, as we will see, admits a canonical lift in . As a conclusion, we can consider in the rough paths sense, solutions to the equation

It turns out that this notion of solution is exactly equivalent to solutions that are obtained by using the Stratonovitch integration theory. Therefore, the theory of stochastic differential equations appears as a very special case of the rough paths theory !