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