An introduction to rough paths

An introduction to rough paths Antoine LEJAY INRIA, Nancy, France From works from T. Lyons, Z. Qian, P. Friz, N. Victoir, M. Gubinelli, D. Feyel, A. de la Pradelle, A.M. Davie,... SPDE semester Isaac Newton Institute, Cambridge Rough Paths, SPDEs and Related Topics Workshop April 21

Rough paths theory? Define as a natural extension of the smooth cases the integral y t = f (x s ) dx s and the controlled differential equations z t = z + g(z s ) dx s when x α-hölder continuous path with values in R d (or an infinite dimensional space) 2

Young integrals x C α (α-hölder paths), y C β : How to define s y r dx r? Define J(s, t) := y s (x t x s ) J(s, r, t) := J(s, t) J(s, r) J(r, t) = (y r y s )(x t x r ) n 1 J(s, t Π) = J(ti n, ti+1), n Π = {t i } i=1,...,n partition of [s, t] i=1 J(s, t Π) J(s, t Π \ {t k }) = J(t k 1, t k, t k+1 ) H β (y)h α (x)(t k+1 t k ) α+β. 3

Choose t k s.t. (t k+1 t k 1 ) 2(t s) n (such a k always exists if n 2), J(s, t Π) J(s, t Π \ {t k }) 2 α+β H α (x)h β (y) (t s)α+β n α+β For θ := α + β and provided that θ > 1 J(s, t Π) y s (x t x s ) C(θ)H α (x)h β (y)(t s) θ ζ(θ) With a bit more analysis, if {Π n } is a family of partitions of [s, t] with mesh(π n ), exists I(s, t) = s y r dx r := lim n J(s, t Π n ) 4

Properties of the Young integrals Young integral I(s, t) = s y r dx r exists as soon as θ := α + β > 1 Satisfies for s r t T, ( ) I(s, r) + I(r, t) = I(s, t) (Chasles relation) ( ) I(s, t) y s (x t x s ) C(t s) θ May be reduced to a function of a single variable I(t) := I(, t) since from ( ), I(s, t) = I(t) I(s) is unique among all the function satisfying ( ) and ( ) t I(t) C α (follows from ( ) since x C α ) 5

Application to integrals If x C α, f C γ and y t = f (x t ) then y C γα and the condition α + γα > 1 implies that even for γ = 1, α > 1/2. It is possible to define I(t) := f (x s) dx s provided that α(1 + γ) > 1. x C α (t I(t)) C α is continuous w.r.t. x. 6

Controlled Differential Equations Define z n C α such that zt n+1 For s < t T : z n+1 = z + g(z n s ) dx s CH α(z n ) γ H α(x)(t s) α(γ+1) { }}{ t t zs n+1 g(zr n ) dx r g(zs n )(x t x s ) s + C( z γ + T αγ H α (z n ) γ ) H α (x)(t s) α C( z γ + T αγ H α (z n ) γ )H α (x)(t s) α = uniform small time estimate of H α (z n ) = uniform estimate of H α (z n ) 7

Controlled differential equations (ctd) Existence of a solution if g C γ, γ(α + 1) > 1 Uniqueness if g C 1+γ, γ(α + 1) > 1 Continuity of x C α z C α if g C 1+γ Convergence of the Euler scheme, flow properties,... The hypotheses of g are sharp (see work from A.M. Davie) 8

Beware on the notion of continuity x n smooth paths in R 2 converging in C α to x, α > 1/2 Π n = {ti n } family of partitions of [, T ], mesh(π n ) φ arbitrary smooth function = construction of y n = (x n + loops) using Π n y n Area ϕ t n k+1 ϕ t n k x n y n converges to x uniformly y n converges to x in C β, β 1/2 but not in C α, α > 1/2 (because the radius of the ball are of order φ(t) φ(s) and then of order t s) 9

f = f 1 (x) dx 1 + f 2 (x) dx 2 differential form f (x s ) dx s := f 1 (x s ) dx 1 s + f 2 (x s ) dx 2 s Using the properties of the integral and the Green-Riemann formula, for F = x2 f 1 x1 f 2, T n 1 n f (ys n ) dys n k+1 n 1 = f (xs n ) dxs n + F (z) dz k= tk n k= Loop(tk n,tn k+1 ) T f (x n s ) dx n s }{{} n T f (x s ) dx s n 1 + F (xt n k )(φ(tk+1) n φ(tk n )) k=1 }{{} n T F (x s ) dφ s 1

On the continuity of integrals: conclusion Given x C α and φ C 1, it is possible to construct a family y n of paths in C α such that y n x uniformly n y n x in n Cβ, β 1/2 but T f (y n s ) dy n s n T f (x s ) dx s + T F (x s ) dφ s This phenomenon arises only for x C α ([, T ]; R d ), d 2. 11

Linear Controlled Differential Equations C matrix, x smooth path in R ( ) z t = z exp(cx t ) ( ) = dz t = z t C dx t Extension by continuity of ( ) to x C([, T ]; R) using ( ) that does not depend on the regularity of x The non-linear case of z t = f (z t ) dx t may be treated as well when x t R: z t = Φ(z, x t ) with Φ(a, t) flow of y t = a + f (y s) ds, (Doss-Sussmann theory) 12

x smooth paths with value in R 2 a c i = 1, 2, C i H = b (a, b, c) R 3 How to solve dz t = z t C 1 dxt 1 + z t C 2 dxt 2? Set A(s, t) = 1 2 s (x 1 r x 1 s ) dx 2 r 1 2 s (x 2 r x 2 s ) dx 1 r A(s, t) =A(s, r) + A(r, t) A(s, r) xr A(r, t) ( ) + 1 2 x s,r x r,t xs }{{} A(s, t) area of the triangle (x s, x r, x t ) xt xs 1 2 (xr xs) (xt xr ) xt 13

Set [C 1, C 2 ] = C 1 C 2 C 2 C 1 (Lie brackets) and z t = z exp(c 1 x 1,t + C 2 x 2,t + [C 1, C 2 ]A(, t)) Z s,t := exp(c 1 x 1 s,t + C 2 x 2 s,t + [C 1, C 2 ]A(s, t)) Remark that if P, Q, R H then P Q H and P QR = = exp(p ) exp(q) = exp (P + Q + 12 ) [P, Q], P, Q H (truncated version of the Baker-Campbell-Hausdorff-Dynkin formula) Deduce that ( ) = Z s,t = Z s,r Z r,t and z solution to ( ) dz t = z t C 1 dx 1 t + z t C 2 dx 2 t since A(s, t) C(t s) 2 14

Extension to α-hölder continuous paths, α > 1/2 x C α, α > 1/2 Define A(s, t) as a Young integral: A(s, t) C(t s) 2α = ( ) has a solution z C α for x C α. I : x z is continuous in C α, α > 1/2. Using the previous example, A(s, t y n ) n = I(y n ) n A(s, t x) + φ(t) φ(s). u the solution to du t = u t C 1 dx 1 t + u t C 2 dx 2 t + u t [C 1, C 2 ] dφ t. The map I is not continuous when the convergence holds in C β, β 1/2. 15

For x C α, α > 1/2, Time to take a breath... We have seen how to construct the integral s f (x r) dx r from the approximation J(s, t) = f (x s )(x t x s ) and the inequality J(s, t) J(s, r) J(r, t) C(t s) θ, θ > 1. We have seen how to define solutions to z t = z + f (z s) dx s from the definition of the integral. 16

Time to take a breath... We have seen that x f (x r) dx r and x z are continuous when the convergence holds in C α but not in C β, β < 1/2, even if each term of the sequence belongs to C α, α > 1/2. Our counter-examples endow the importance of the area enclosed between the path and its chord for any times. The theory of rough paths allows one to consider α < 1/2 and to deal with our counter-examples. To go deep in the theory of rough paths, we are only missing a proper algebraic setting. 17

An algebraic structure T (R 2 ) := R R d (R d R d ) truncated tensor algebra (addition and tensor product, keeping only the tensor products between two terms) T 1 (R 2 ) := { a T (R 2 ) a = 1 + a 1 + a 2, a 1 R d, a 2 (R d ) 2 M(d) } {e i } d i=1 the canonical basis of Rd For a, b in T 1 (R d ) a b = (1 + a 1 + a 2 ) (1 + b 1 + b 2 ) = 1 + a 1 + b 1 + a 2 + b 2 + a 1 b 1 For a T 1 (R d ), a 1 := 1 a 1 a 2 a 1 a 1 satisfies a 1 a = 1. 18

Rough paths An α-hölder continuous rough path of order 2 is a continuous path d d x t = }{{} 1 + xt 1,i e i + x 2,i,j t e i e j, R i=1 i,j=1 } {{ } x 1 R d }{{} x 2 R d R d M(d d) living in T 1 (R d ) satisfying { } x 1 x α := sup max s,t x 2 s<t T (t s), s,t < + α (t s) α with x s,t = x 1 s x t Def.: x lives above the path x 1 or is a lift of x 1 19

Prop.: If x is a α-hölder continuous rough path of order 2 and φ C 2α (R d R d ), then x + φ is a α-hölder continuous rough path of order 2. Prop.: If α > 1/2 and x C α ([, T ]; R d ), then d x t = 1 + x t + e i e j (xr i x) i dxr j i,j=1 is the only possible α-hölder continuous rough path living above x. Yet there are an infinite number of β-hölder continuous rough paths living above x as soon as β 1/2. 2

For this, check that d x s,t = 1 + x t s + e i e j (xr i xs) i dxs j i,j=1 s }{{} CH α (x)(t s) 2α (Young integral) Note that 1 2,i,j (xs,t x 2,j,i s,t ) = A(s, t (x i, x j )) 2 1 2,i,j (xs,t + x 2,j,i s,t ) = 1 2 2 x s,tx i s,t. j For P such that 1 + P T 1 (R d ), Hence log(x t ) = x t + log(1 + P ) = P 1 2 P P 1 i<j d (e i e j e j e i )A(, t (x i, x j )) 21

Integral along a rough path f = (f 1,..., f d ), f i C 1+γ (R d ; R), x : [, T ] R d smooth enough s f (x r ) dx r = f (x s )(x t x s )+ f (x s ) (x r x s ) dx r +... s }{{} R d R d For α > 1/3 and x C α consider a α-hölder continuous rough path x of order 2 living above x use as an approximation of s f (x r) dx r the quantity J(s, t) = f (x s )xs,t 1 + f (x s )xs,t 2 22

Check that J(s, r, t) := J(s, t) J(s, r) J(r, t) = f (x s ) xs,r 1 xr,t 1 (f (x r ) f (x s )) x }{{}}{{} r,t ( f 1 (x r ) f (x s ))xr,t 2 since x s,t = x s,r x t r,t f (x s + τxs,r 1 )xs,r 1 dτ = J(s, r, t) C x 1+γ α (t s) α(2+γ) = existence of an integral I(x) = f (x s) dx s C α provided that α(2 + γ) > 1 Proof: Use the same arguments as in the beginning (passing from J(s, r, t) to I(x, t) using J(s, t) with the sewing lemma since J(s, t) is an almost rough path) 23

Properties of rough integrals If φ C 2α (R d R d ), 1/3 < α < 1/2, f (x s ) d(x s + φ s ) = f (x s ) dx s + f (x s ) dφ s A class of rough paths x (geometric rough paths) may be identified as a limit of a family of smooth paths x n lifted as rough paths x n of order 2, in the sense that I(x n ) converges to I(x). counter-examples on the continuity of Young integrals = example of the continuity of I w.r.t. rough paths of order 2. 24

Rough Differential Equations (RDE) g is a vector field: g( ) L(R d ; R m ) consider the RDE z t = z + g(z s ) dx s For g C 1+γ, 1/3 < α 1, α(2 + γ) > 1, Existence of a solution if g C 1+γ, g bounded, but not necessarily uniqueness. The solution z may be lifted as a α-hölder continuous rough path z of order 2. When φ C 2+α (R d R d ), then the solution to z t = z + g(z s) d(x s + φ s ) is solution to z t = z + g(z s) dx s + g g(z s) dφ s. 25

Rough Differential Equations (ctd) If in addition g C 2+γ, α(2 + γ) > 1, g and 2 g bounded, Uniqueness of the solution z and the rough path z Local Lipschitz continuity of the map x z w.r.t. α. Convergence of the Euler scheme defined by e k+1 = e k + g(e k )x 1 t n k,tn k+1 + g g(e k)x 2 t n k,tn k+1 when mesh{t n i } n. 26

Summary and extensions The iterated integrals up to order 1/α is sufficient to define properly f (x s ) dx s and RDE as natural extension to the smooth case when x C α. The theory of rough paths is a deterministic theory with roots in control theory, differential geometry, theory of differential equations and integration, with a rich algebraic setting. However, the construction of a rough path lying above a given path may require the use of a probabilistic tools, as well as notions from various fields (sub-riemanniann geometry, quantum field theory, Hopf algebra,...). 27

The continuity with respect to the smooth case is one of the main feature of the theory of rough paths, which means that the integrals are as intuitive and natural as possible. The theory of rough paths as well as some algebraic and numerical approaches on ODE provide us with a set of ideas that could be used both for extension (SPDEs and application to infinite dimensional setting) and simulation A lot of directions, mainly toward SPDEs (to get rid of the Brownian/white noise setting for example), remain largely to explore. 28