Rough Paths and the Gap Between Deterministic and Stochastic Di¤Erential Equations

Rough paths and the Gap Between Deterministic and Stochastic Di¤erential Equations

P. K. Friz

TU Berlin and WIAS

December 2009

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 1/22 Outline

I Donsker’sinvariance principle and Brownian motion II Itô integration and SDEs III Doss-Sussman’sODE approach to SDEs IV More on ODEs: Euler estimates V Smart ODE limits: rough di¤erential equations VI Brownian motion as a rough path VII SDEs driven by non-Brownian noise VIII Rough path spaces IX Donsker’sprinciple revisited

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 2/22 Limit is a probability measure on C [0, 1] , called Wiener measure W.

Brownian motion (BM) (Bt ) is a stochastic process with law W . Straight-forward extension to Rd -valued case In particular, a d-dimensional Brownian motion is just an ensemble of d independent Brownian motions, say

1 d Bt = Bt , ... , Bt .

Donsker’sinvariance principle

Let (ξi ) be an IID sequence of zero-mean, unit-variance random variables. [Donsker ’52] shows that the rescaled, piecewise-linearly-connected, random-walk

(n) 1 W = ξ + + ξ + (nt [nt]) ξ t n1/2 1 [tn] [nt]+1 converges weakly in the space of continuous functions on [0, 1].

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 3/22 Brownian motion (BM) (Bt ) is a stochastic process with law W . Straight-forward extension to Rd -valued case In particular, a d-dimensional Brownian motion is just an ensemble of d independent Brownian motions, say

1 d Bt = Bt , ... , Bt .

Donsker’sinvariance principle

Let (ξi ) be an IID sequence of zero-mean, unit-variance random variables. [Donsker ’52] shows that the rescaled, piecewise-linearly-connected, random-walk

(n) 1 W = ξ + + ξ + (nt [nt]) ξ t n1/2 1 [tn] [nt]+1 converges weakly in the space of continuous functions on [0, 1]. Limit is a probability measure on C [0, 1] , called Wiener measure W.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 3/22 Straight-forward extension to Rd -valued case In particular, a d-dimensional Brownian motion is just an ensemble of d independent Brownian motions, say

1 d Bt = Bt , ... , Bt .

Donsker’sinvariance principle

Let (ξi ) be an IID sequence of zero-mean, unit-variance random variables. [Donsker ’52] shows that the rescaled, piecewise-linearly-connected, random-walk

(n) 1 W = ξ + + ξ + (nt [nt]) ξ t n1/2 1 [tn] [nt]+1 converges weakly in the space of continuous functions on [0, 1]. Limit is a probability measure on C [0, 1] , called Wiener measure W.

Brownian motion (BM) (Bt ) is a stochastic process with law W .

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 3/22 In particular, a d-dimensional Brownian motion is just an ensemble of d independent Brownian motions, say

1 d Bt = Bt , ... , Bt .

Donsker’sinvariance principle

Let (ξi ) be an IID sequence of zero-mean, unit-variance random variables. [Donsker ’52] shows that the rescaled, piecewise-linearly-connected, random-walk

(n) 1 W = ξ + + ξ + (nt [nt]) ξ t n1/2 1 [tn] [nt]+1 converges weakly in the space of continuous functions on [0, 1]. Limit is a probability measure on C [0, 1] , called Wiener measure W.

Brownian motion (BM) (Bt ) is a stochastic process with law W . Straight-forward extension to Rd -valued case

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 3/22 Donsker’sinvariance principle

Let (ξi ) be an IID sequence of zero-mean, unit-variance random variables. [Donsker ’52] shows that the rescaled, piecewise-linearly-connected, random-walk

(n) 1 W = ξ + + ξ + (nt [nt]) ξ t n1/2 1 [tn] [nt]+1 converges weakly in the space of continuous functions on [0, 1]. Limit is a probability measure on C [0, 1] , called Wiener measure W.

1 d Bt = Bt , ... , Bt .

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 3/22 (ii): continuous 0-mean Gaussian process with covariance

E (Bs Bt ) = min (s, t) s, t [0, 1] 8 2 2 (iii): Markov process with generator L 1 ∂ in the sense that = 2 ∂x 2

E [f (x + Bt )] x 1 Lf = f 00 f nice t ! 2 8 Again, straight-forward extension to Rd -valued case

Brownian motion: alternative characterizations

(i): continuous martingale such that B2 t is also a martingale t

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 4/22 2 (iii): Markov process with generator L 1 ∂ in the sense that = 2 ∂x 2

E [f (x + Bt )] x 1 Lf = f 00 f nice t ! 2 8 Again, straight-forward extension to Rd -valued case

Brownian motion: alternative characterizations

(i): continuous martingale such that B2 t is also a martingale t (ii): continuous 0-mean Gaussian process with covariance

E (Bs Bt ) = min (s, t) s, t [0, 1] 8 2

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 4/22 Again, straight-forward extension to Rd -valued case

Brownian motion: alternative characterizations

(i): continuous martingale such that B2 t is also a martingale t (ii): continuous 0-mean Gaussian process with covariance

E (Bs Bt ) = min (s, t) s, t [0, 1] 8 2 2 (iii): Markov process with generator L 1 ∂ in the sense that = 2 ∂x 2

E [f (x + Bt )] x 1 Lf = f 00 f nice t ! 2 8

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 4/22 Brownian motion: alternative characterizations

(i): continuous martingale such that B2 t is also a martingale t (ii): continuous 0-mean Gaussian process with covariance

E (Bs Bt ) = min (s, t) s, t [0, 1] 8 2 2 (iii): Markov process with generator L 1 ∂ in the sense that = 2 ∂x 2

E [f (x + Bt )] x 1 Lf = f 00 f nice t ! 2 8 Again, straight-forward extension to Rd -valued case

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 4/22 How to de…ne integration against Brownian motion? Itô’sbrilliant idea: with some help and intuition from martingale theory, 1 f (t, ω) dBt (ω) Z0 can be de…ned for reasonable non-anticipating f : start with simple integrands and complete with isometry

Itô integration

Fact: Typical sample paths of Brownian motion, t Bt (ω), have in…nite variation on every interval. 7!

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 5/22 t 1 2 Example: Bs dBs = B t ... 2nd order calculus! 0 2 t Fact: Itô-integrals have left-point Riemann-sum approximations. R De…ne Stratonovich-integration via mid-point Riemann-sum t 1 2 approximations = Bs ∂Bs = B (1st order calculus!) ) 0 2 t R

Itô integration

Fact: Typical sample paths of Brownian motion, t Bt (ω), have in…nite variation on every interval. 7! How to de…ne integration against Brownian motion? Itô’sbrilliant idea: with some help and intuition from martingale theory, 1 f (t, ω) dBt (ω) Z0 can be de…ned for reasonable non-anticipating f : start with simple integrands and complete with isometry

1 2 1 2 E f (t, ω) dBt (ω) = E f (t, ω) dt . "Z0 # Z0

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 5/22 Fact: Itô-integrals have left-point Riemann-sum approximations. De…ne Stratonovich-integration via mid-point Riemann-sum t 1 2 approximations = Bs ∂Bs = B (1st order calculus!) ) 0 2 t R

Itô integration

1 2 1 2 E f (t, ω) dBt (ω) = E f (t, ω) dt . "Z0 # Z0 t 1 2 Example: Bs dBs = B t ... 2nd order calculus! 0 2 t R

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 5/22 De…ne Stratonovich-integration via mid-point Riemann-sum t 1 2 approximations = Bs ∂Bs = B (1st order calculus!) ) 0 2 t R

Itô integration

1 2 1 2 E f (t, ω) dBt (ω) = E f (t, ω) dt . "Z0 # Z0 t 1 2 Example: Bs dBs = B t ... 2nd order calculus! 0 2 t Fact: Itô-integrals have left-point Riemann-sum approximations. R

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 5/22 Itô integration

1 2 1 2 E f (t, ω) dBt (ω) = E f (t, ω) dt . "Z0 # Z0 t 1 2 Example: Bs dBs = B t ... 2nd order calculus! 0 2 t Fact: Itô-integrals have left-point Riemann-sum approximations. R De…ne Stratonovich-integration via mid-point Riemann-sum t 1 2 approximations = Bs ∂Bs = B (1st order calculus!) ) 0 2 t P. K. Friz (TU Berlin and WIAS) R rough paths, gap ODE/SDEs December2009 5/22 e Let V0, ... , Vd be a collection of nice vector …elds on R

A solution (process) y = yt (ω) to

d i dy = V0 (y) dt + ∑ Vi (y) ∂B i=1 is, by de…nition, a solution to corresponding integral equation.

At the price of modifying the drift vector …eld V0 we can switch to Itô formulation (∂B dB) ! Existence, uniqueness by …xpoint arguments.

For simplicity only: from here on V0 = 0.

Stochastic di¤erential equations

Let B be a d-dimensional Brownian motion.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 6/22 A solution (process) y = yt (ω) to

d i dy = V0 (y) dt + ∑ Vi (y) ∂B i=1 is, by de…nition, a solution to corresponding integral equation.

At the price of modifying the drift vector …eld V0 we can switch to Itô formulation (∂B dB) ! Existence, uniqueness by …xpoint arguments.

For simplicity only: from here on V0 = 0.

Stochastic di¤erential equations

Let B be a d-dimensional Brownian motion. e Let V0, ... , Vd be a collection of nice vector …elds on R

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 6/22 At the price of modifying the drift vector …eld V0 we can switch to Itô formulation (∂B dB) ! Existence, uniqueness by …xpoint arguments.

For simplicity only: from here on V0 = 0.

Stochastic di¤erential equations

Let B be a d-dimensional Brownian motion. e Let V0, ... , Vd be a collection of nice vector …elds on R

A solution (process) y = yt (ω) to

d i dy = V0 (y) dt + ∑ Vi (y) ∂B i=1 is, by de…nition, a solution to corresponding integral equation.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 6/22 Existence, uniqueness by …xpoint arguments.

For simplicity only: from here on V0 = 0.

Stochastic di¤erential equations

Let B be a d-dimensional Brownian motion. e Let V0, ... , Vd be a collection of nice vector …elds on R

A solution (process) y = yt (ω) to

d i dy = V0 (y) dt + ∑ Vi (y) ∂B i=1 is, by de…nition, a solution to corresponding integral equation.

At the price of modifying the drift vector …eld V0 we can switch to Itô formulation (∂B dB) !

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 6/22 For simplicity only: from here on V0 = 0.

Stochastic di¤erential equations

Let B be a d-dimensional Brownian motion. e Let V0, ... , Vd be a collection of nice vector …elds on R

A solution (process) y = yt (ω) to

d i dy = V0 (y) dt + ∑ Vi (y) ∂B i=1 is, by de…nition, a solution to corresponding integral equation.

At the price of modifying the drift vector …eld V0 we can switch to Itô formulation (∂B dB) ! Existence, uniqueness by …xpoint arguments.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 6/22 Stochastic di¤erential equations

Let B be a d-dimensional Brownian motion. e Let V0, ... , Vd be a collection of nice vector …elds on R

A solution (process) y = yt (ω) to

d i dy = V0 (y) dt + ∑ Vi (y) ∂B i=1 is, by de…nition, a solution to corresponding integral equation.

At the price of modifying the drift vector …eld V0 we can switch to Itô formulation (∂B dB) ! Existence, uniqueness by …xpoint arguments.

For simplicity only: from here on V0 = 0.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 6/22 Let V be a nice vector …eld on Re Aim: …nd solution to SDE

dy = V (y) ∂dB.

Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus. This is an ODE solution method for SDEs.

Bene…t: solution depends in a robust way on B and y0.

A drift V0(y)dt can be incorporated (‡ow decomposition) ... but this method fails when d > 1.

The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 Aim: …nd solution to SDE

dy = V (y) ∂dB.

Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus. This is an ODE solution method for SDEs.

Bene…t: solution depends in a robust way on B and y0.

A drift V0(y)dt can be incorporated (‡ow decomposition) ... but this method fails when d > 1.

The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1. Let V be a nice vector …eld on Re

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus. This is an ODE solution method for SDEs.

Bene…t: solution depends in a robust way on B and y0.

A drift V0(y)dt can be incorporated (‡ow decomposition) ... but this method fails when d > 1.

The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1. Let V be a nice vector …eld on Re Aim: …nd solution to SDE

dy = V (y) ∂dB.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 This is an ODE solution method for SDEs.

Bene…t: solution depends in a robust way on B and y0.

A drift V0(y)dt can be incorporated (‡ow decomposition) ... but this method fails when d > 1.

The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1. Let V be a nice vector …eld on Re Aim: …nd solution to SDE

dy = V (y) ∂dB.

Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 Bene…t: solution depends in a robust way on B and y0.

A drift V0(y)dt can be incorporated (‡ow decomposition) ... but this method fails when d > 1.

The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1. Let V be a nice vector …eld on Re Aim: …nd solution to SDE

dy = V (y) ∂dB.

Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus. This is an ODE solution method for SDEs.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 A drift V0(y)dt can be incorporated (‡ow decomposition) ... but this method fails when d > 1.

The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1. Let V be a nice vector …eld on Re Aim: …nd solution to SDE

dy = V (y) ∂dB.

Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus. This is an ODE solution method for SDEs.

Bene…t: solution depends in a robust way on B and y0.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 ... but this method fails when d > 1.

The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1. Let V be a nice vector …eld on Re Aim: …nd solution to SDE

dy = V (y) ∂dB.

Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus. This is an ODE solution method for SDEs.

Bene…t: solution depends in a robust way on B and y0.

A drift V0(y)dt can be incorporated (‡ow decomposition)

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 The Doss-Sussman approach

Let B be a d-dimensional Brownian motion, d = 1. Let V be a nice vector …eld on Re Aim: …nd solution to SDE

dy = V (y) ∂dB.

Let etV be the solution ‡ow to the ODE z˙ = V (z) . Then

Bt (ω)V y (t, ω) := e y0

is the SDE solution. Proof: First order calculus. This is an ODE solution method for SDEs.

Bene…t: solution depends in a robust way on B and y0.

A drift V0(y)dt can be incorporated (‡ow decomposition) ... but this method fails when d > 1.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 7/22 Let us now look at such di¤erential equations when B is replaced by some path x C 1 [0, 1] , Rd ; that is 2 d ( ) :y ˙t = ∑ Vi (yt ) x˙t i=1 This is a classical setup in system control theory ...

input signal x = output signal y ) ... and in our case the system response is modelled by ODE ( ). How would one simulate ( ) on a computer?

More on ODEs: Euler estimates

So far, we have been interested in stochastic di¤erential equations of the type d i dyt = ∑ Vi (yt ) ∂Bt i=1

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 8/22 This is a classical setup in system control theory ...

input signal x = output signal y ) ... and in our case the system response is modelled by ODE ( ). How would one simulate ( ) on a computer?

More on ODEs: Euler estimates

So far, we have been interested in stochastic di¤erential equations of the type d i dyt = ∑ Vi (yt ) ∂Bt i=1 Let us now look at such di¤erential equations when B is replaced by some path x C 1 [0, 1] , Rd ; that is 2 d ( ) :y ˙t = ∑ Vi (yt ) x˙t i=1

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 8/22 ... and in our case the system response is modelled by ODE ( ). How would one simulate ( ) on a computer?

More on ODEs: Euler estimates

input signal x = output signal y )

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 8/22 How would one simulate ( ) on a computer?

More on ODEs: Euler estimates

input signal x = output signal y ) ... and in our case the system response is modelled by ODE ( ).

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 8/22 More on ODEs: Euler estimates

input signal x = output signal y ) ... and in our case the system response is modelled by ODE ( ). How would one simulate ( ) on a computer? P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 8/22 Step-2 Euler scheme: t t r i i j yt ys Vi (ys ) dx + Vi Vj (ys ) dx dx Zs Zs Zs

= (ys ,xs,t ) E with | {z } t t r d d d xs,t = dx, dx dx R R . s s s 2 Z Z Z Natural scaling assumption. For some α (0, 1], 2 t t r 1/2 i i j α dx dx dx c1 t s . s _ s s Z Z Z j j

[Okay for BM with α < 1/2 but keep x C 1 for now ...] 2

1 d 2,b e e More precisely: x C [0, 1] , R , V1, ..., Vd C (R , R ) 2 2 i dy = V (y) dx y˙ = Vi (y) x˙ () (Summation over repeated indices!) Usual Euler-scheme: t i yt ys Vi (ys ) dx Zs

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 9/22 Natural scaling assumption. For some α (0, 1], 2 t t r 1/2 i i j α dx dx dx c1 t s . s _ s s Z Z Z j j

[Okay for BM with α < 1/2 but keep x C 1 for now ...] 2

1 d 2,b e e More precisely: x C [0, 1] , R , V1, ..., Vd C (R , R ) 2 2 i dy = V (y) dx y˙ = Vi (y) x˙ () (Summation over repeated indices!) Usual Euler-scheme: t i yt ys Vi (ys ) dx Zs Step-2 Euler scheme: t t r i i j yt ys Vi (ys ) dx + Vi Vj (ys ) dx dx Zs Zs Zs

= (ys ,xs,t ) E with | {z } t t r d d d xs,t = dx, dx dx R R . s s s 2 Z Z Z

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 9/22 1 d 2,b e e More precisely: x C [0, 1] , R , V1, ..., Vd C (R , R ) 2 2 i dy = V (y) dx y˙ = Vi (y) x˙ () (Summation over repeated indices!) Usual Euler-scheme: t i yt ys Vi (ys ) dx Zs Step-2 Euler scheme: t t r i i j yt ys Vi (ys ) dx + Vi Vj (ys ) dx dx Zs Zs Zs

= (ys ,xs,t ) E with | {z } t t r d d d xs,t = dx, dx dx R R . s s s 2 Z Z Z Natural scaling assumption. For some α (0, 1], 2 t t r 1/2 i i j α dx dx dx c1 t s . s _ s s Z Z Z j j

[Okay for BM with α < 1/2 but keep x C 1 for now ...] P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs 2 December2009 9/22 Easy to see that α ys , xs,t c3 t s , c3 = c3 (c1) E j jα yt ys c4 t s , c4 = c4 (c1) . j j j j 1 d Take xn C [0, 1] , R with uniform bounds 2 t t r 1/2 i i j α sup dxn dxndxn c1 t s n s _ s s Z Z Z j j

s.t. xn + iterated integrals converge (pointwise) to

(1) (2) d d d x = x , x R R ; t t t 2 then call t x a (geometric) rough path. 7! t

Davie’sLemma: Error estimate on Step-2 Euler scheme θ yt ys ys , x c2 t s E s,t j j with θ = 3α > 1 = need α> 1/3 [Okay for BM ...]. The catch is uniformity )

c2 = c2 (c1) .... not c2 ( x˙ ) or c2 x j j∞ j jLip

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 10/22 1 d Take xn C [0, 1] , R with uniform bounds 2 t t r 1/2 i i j α sup dxn dxndxn c1 t s n s _ s s Z Z Z j j

s.t. xn + iterated integrals converge (pointwise) to

(1) (2) d d d x = x , x R R ; t t t 2 then call t x a (geometric) rough path. 7! t

Davie’sLemma: Error estimate on Step-2 Euler scheme θ yt ys ys , x c2 t s E s,t j j with θ = 3α > 1 = need α> 1/3 [Okay for BM ...]. The catch is uniformity )

c2 = c2 (c1) .... not c2 ( x˙ ) or c2 x j j∞ j jLip Easy to see that α ys , xs,t c3 t s , c3 = c3 (c1) E j jα yt ys c4 t s , c4 = c4 (c1) . j j j j

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 10/22 Davie’sLemma: Error estimate on Step-2 Euler scheme θ yt ys ys , x c2 t s E s,t j j with θ = 3α > 1 = need α> 1/3 [Okay for BM ...]. The catch is uniformity )

c2 = c2 (c1) .... not c2 ( x˙ ) or c2 x j j∞ j jLip Easy to see that α ys , xs,t c3 t s , c3 = c3 (c1) E j jα yt ys c4 t s , c4 = c4 (c1) . j j j j 1 d Take xn C [0, 1] , R with uniform bounds 2 t t r 1/2 i i j α sup dxn dxndxn c1 t s n s _ s s Z Z Z j j

s.t. xn + iterated integrals converge (pointwise) to

(1) (2) d d d x = x , x R R ; t t t 2 then call t xt a (geometric) rough path. P. K. Friz (TU Berlin and7! WIAS) rough paths, gap ODE/SDEs December2009 10/22 Error estimates for step-N Euler schemes [F-Victoir , JDE 07] RDE smoothness and Malliavin calculus [Cass-F, Annals of Math 09] Continuity of Φ as ‡ow of di¤eomorphisms [Caruana-F, JDE 08] Rough partial di¤erential equations [Caruana-F-Oberhauser ...]

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

... and this "Itô-Lyons" map Φ is continuous in the above sense [Lyons 98]. Various useful extensions ...

Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 11/22 Error estimates for step-N Euler schemes [F-Victoir , JDE 07] RDE smoothness and Malliavin calculus [Cass-F, Annals of Math 09] Continuity of Φ as ‡ow of di¤eomorphisms [Caruana-F, JDE 08] Rough partial di¤erential equations [Caruana-F-Oberhauser ...]

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

... and this "Itô-Lyons" map Φ is continuous in the above sense [Lyons 98]. Various useful extensions ...

Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

Various useful extensions ...

Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

... and this "Itô-Lyons" map Φ is continuous in the above sense [Lyons 98].

Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

... and this "Itô-Lyons" map Φ is continuous in the above sense [Lyons 98]. Various useful extensions ...

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 11/22 RDE smoothness and Malliavin calculus [Cass-F, Annals of Math 09] Continuity of Φ as ‡ow of di¤eomorphisms [Caruana-F, JDE 08] Rough partial di¤erential equations [Caruana-F-Oberhauser ...]

Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

... and this "Itô-Lyons" map Φ is continuous in the above sense [Lyons 98]. Various useful extensions ... Error estimates for step-N Euler schemes [F-Victoir , JDE 07]

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 11/22 Continuity of Φ as ‡ow of di¤eomorphisms [Caruana-F, JDE 08] Rough partial di¤erential equations [Caruana-F-Oberhauser ...]

Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 11/22 Rough partial di¤erential equations [Caruana-F-Oberhauser ...]

Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

... and this "Itô-Lyons" map Φ is continuous in the above sense [Lyons 98]. Various useful extensions ... Error estimates for step-N Euler schemes [F-Victoir , JDE 07] RDE smoothness and Malliavin calculus [Cass-F, Annals of Math 09] Continuity of Φ as ‡ow of di¤eomorphisms [Caruana-F, JDE 08]

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 11/22 Apply Davie’slemma: get yn with uniform Hölder bound c4. f g

Arzela–Ascoli = yn : n 1 has limit points ... call them RDE) solutionsf g

More regularity + a bit work = ! RDE solution y Φ (x; y0) and write ) 9 dy = V (y) dx

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 11/22 As example, consider

y˙ = V1 (y) + V2 (y) dy = V1 (y) dt + V2 (y) dt; () we immediately get the (splitting) result

1 V 1 V 1 V 1 V V +V e n 2 e n 1 e n 2 e n 1 e 1 2 ! where etW denotes the solution ‡ow to z˙ = W (z). Indeed, it su¢ ces to approximation the diagonal t (t, t) by a 1/n - step function 7! This approximation converges with uniform 1-Hölder (i.e. Lipschitz) bounds

Examples of RDEs

ODEs: For a smooth driving signal, RDEs are just ODEs. Even here, continuity statements are powerful.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 12/22 Indeed, it su¢ ces to approximation the diagonal t (t, t) by a 1/n - step function 7! This approximation converges with uniform 1-Hölder (i.e. Lipschitz) bounds

Examples of RDEs

ODEs: For a smooth driving signal, RDEs are just ODEs. Even here, continuity statements are powerful. As example, consider

y˙ = V1 (y) + V2 (y) dy = V1 (y) dt + V2 (y) dt; () we immediately get the (splitting) result

1 V 1 V 1 V 1 V V +V e n 2 e n 1 e n 2 e n 1 e 1 2 ! where etW denotes the solution ‡ow to z˙ = W (z).

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 12/22 This approximation converges with uniform 1-Hölder (i.e. Lipschitz) bounds

Examples of RDEs

ODEs: For a smooth driving signal, RDEs are just ODEs. Even here, continuity statements are powerful. As example, consider

y˙ = V1 (y) + V2 (y) dy = V1 (y) dt + V2 (y) dt; () we immediately get the (splitting) result

1 V 1 V 1 V 1 V V +V e n 2 e n 1 e n 2 e n 1 e 1 2 ! where etW denotes the solution ‡ow to z˙ = W (z). Indeed, it su¢ ces to approximation the diagonal t (t, t) by a 1/n - step function 7!

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 12/22 Examples of RDEs

ODEs: For a smooth driving signal, RDEs are just ODEs. Even here, continuity statements are powerful. As example, consider

y˙ = V1 (y) + V2 (y) dy = V1 (y) dt + V2 (y) dt; () we immediately get the (splitting) result

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 12/22 Di¤erential equations driven by pure area: 0 0 t t x , 7! t 0 t 0 is the limit (with uniform 1/2-Hölder bounds ...) of the highly oscillatory 1 2 2 xn (t) = n exp 2πin t C = R . 2 Given two vector …elds V = (V1,V2) the RDE solution dy = V (y) dx (1) models the e¤ective behaviour of the highly oscillatory ODE

dy n = V (y n) dxn as n ∞. ! In fact, the RDE solution of (1) solves the ODE

y˙ = [V1, V2](y)

where [V1, V2] is the Lie bracket of V1 and V2.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 13/22 Stochastic di¤erential equations: Let B be d-dimensional Brownian motion. Since B (ω) / C 1 careful interpretation of the stochastic di¤erential equation 2

dy = V (y) ∂B is necessary (Itô-theory). De…ne enhanced Brownian motion

t Bt (ω) = Bt , Bs ∂Bs 0 Z where ∂ indicates (Stratonovich) stochastic integration. Then P [B is a geometric rough path] = 1.

In fact, martingale arguments shows that B (ω) is the limit of piecewise linear approximations (with uniform (1/2 ε)-Hölder bounds ...). RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution ...

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 14/22 Key to understanding: view B as level-N rough path; [F-Oberhauser, JFA 09] By rough path continuity, this would not happen if, for some α (1/3, 1/2], 2 t t r 1/2 (n) (n) (n) α dBt dBs dBs C (ω) t s . s _ s s Z Z Z j j

Caution: topology matters. Possible that, uniformly in t,

t t (n) (n) (n) Bt , Bs dBs Bt , Bs ∂Bs 0 ! 0 Z Z while DE solutions converge to the "wrong" limit.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 15/22 By rough path continuity, this would not happen if, for some α (1/3, 1/2], 2 t t r 1/2 (n) (n) (n) α dBt dBs dBs C (ω) t s . s _ s s Z Z Z j j

Caution: topology matters. Possible that, uniformly in t,

t t (n) (n) (n) Bt , Bs dBs Bt , Bs ∂Bs 0 ! 0 Z Z while DE solutions converge to the "wrong" limit. Key to understanding: view B as level-N rough path; [F-Oberhauser, JFA 09]

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 15/22 Caution: topology matters. Possible that, uniformly in t,

t t (n) (n) (n) Bt , Bs dBs Bt , Bs ∂Bs 0 ! 0 Z Z while DE solutions converge to the "wrong" limit. Key to understanding: view B as level-N rough path; [F-Oberhauser, JFA 09] By rough path continuity, this would not happen if, for some α (1/3, 1/2], 2 t t r 1/2 (n) (n) (n) α dBt dBs dBs C (ω) t s . s _ s s Z Z Z j j

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 15/22 SDE theory with (semi)martingale noise follows Itô’sapproach and is well-known SDE theory with Gaussian noise is: previous to rough path theory restricted to special examples (e.g. fractional Brownian motion) SDE theory with Markovian noise: previous to rough path theory, hardly anything Thanks to rough path theory: large and natural classes of the above processes can be lifted to rough paths with resulting path-by-path stochastic di¤erential equations.

Di¤erential equations with non-Brownian noise

Recall BM = martingale, Gaussian, Markov

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 16/22 SDE theory with Gaussian noise is: previous to rough path theory restricted to special examples (e.g. fractional Brownian motion) SDE theory with Markovian noise: previous to rough path theory, hardly anything Thanks to rough path theory: large and natural classes of the above processes can be lifted to rough paths with resulting path-by-path stochastic di¤erential equations.

Di¤erential equations with non-Brownian noise

Recall BM = martingale, Gaussian, Markov SDE theory with (semi)martingale noise follows Itô’sapproach and is well-known

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 16/22 SDE theory with Markovian noise: previous to rough path theory, hardly anything Thanks to rough path theory: large and natural classes of the above processes can be lifted to rough paths with resulting path-by-path stochastic di¤erential equations.

Di¤erential equations with non-Brownian noise

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 16/22 Thanks to rough path theory: large and natural classes of the above processes can be lifted to rough paths with resulting path-by-path stochastic di¤erential equations.

Di¤erential equations with non-Brownian noise

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 16/22 Di¤erential equations with non-Brownian noise

Recall BM = martingale, Gaussian, Markov SDE theory with (semi)martingale noise follows Itô’sapproach and is well-known SDE theory with Gaussian noise is: previous to rough path theory restricted to special examples (e.g. fractional Brownian motion) SDE theory with Markovian noise: previous to rough path theory, hardly anything Thanks to rough path theory: large and natural classes of the above processes can be lifted to rough paths with resulting path-by-path stochastic di¤erential equations.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 16/22 d d d The (vector) space R R R with basis 1, bi , bjk ; 1 i, j, k d has (truncated tensor) algebra structure; e.g. 2b1 (4 3b2) = 8b1 6b12 d d d x T1 := 1 R R and (T1, , 1) is a Lie group s,t 2 f g d d d T1 = exp R R Non-linear key identity [Chen ’37] x x = x , 0 s t u 1. s,t t,u s,u

Rough path spaces

1 d For x C [0, 1] , R , x0 = 0, de…ne generalized increments 2 t t r d d d xs,t = 1, dx, dx dx R R R , 0 s t 1 s s s 2 Z Z Z

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 17/22 d d d x T1 := 1 R R and (T1, , 1) is a Lie group s,t 2 f g d d d T1 = exp R R Non-linear key identity [Chen ’37] x x = x , 0 s t u 1. s,t t,u s,u

Rough path spaces

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 17/22 Non-linear key identity [Chen ’37] x x = x , 0 s t u 1. s,t t,u s,u

Rough path spaces

1 d For x C [0, 1] , R , x0 = 0, de…ne generalized increments 2 t t r d d d xs,t = 1, dx, dx dx R R R , 0 s t 1 s s s 2 Z Z Z d d d The (vector) space R R R with basis 1, bi , bjk ; 1 i, j, k d has (truncated tensor) algebra structure; e.g. 2b1 (4 3b2) = 8b1 6b12 d d d x T1 := 1 R R and (T1, , 1) is a Lie group s,t 2 f g d d d T1 = exp R R

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 17/22 Rough path spaces

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 17/22 2 Dilation δλ (1 + v + M) = 1 + λv + λ M Carnot-Carathoedory norm: 1 + v + M v M 1/2 v Anti (M) 1/2 k kCC j j _ j j j j _ j j

geometric interpretation of log xs,t : path- and area-increment For d = 2, G isomorphic to the 3-dimensional Heisenberg group Familiar concepts (scalar product, norm) generalize

x := x de…nes a G-valued path (which lifts x) and x = x 1 x t 0,t s,t s t

Actually, x G := exp Rd so (d) since (1st order calculus!) s,t 2 t r 1 t t Sym dx dx = dx dx s s 2 s s Z Z Z Z G = exp Rd so (d) is (a realization of) step-2 nilpotent Lie group with d generators

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 18/22 2 Dilation δλ (1 + v + M) = 1 + λv + λ M Carnot-Carathoedory norm: 1 + v + M v M 1/2 v Anti (M) 1/2 k kCC j j _ j j j j _ j j

For d = 2, G isomorphic to the 3-dimensional Heisenberg group Familiar concepts (scalar product, norm) generalize

x := x de…nes a G-valued path (which lifts x) and x = x 1 x t 0,t s,t s t

Familiar concepts (scalar product, norm) generalize

x := x de…nes a G-valued path (which lifts x) and x = x 1 x t 0,t s,t s t

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 18/22 2 Dilation δλ (1 + v + M) = 1 + λv + λ M Carnot-Carathoedory norm: 1 + v + M v M 1/2 v Anti (M) 1/2 k kCC j j _ j j j j _ j j x := x de…nes a G-valued path (which lifts x) and x = x 1 x t 0,t s,t s t

Actually, x G := exp Rd so (d) since (1st order calculus!) s,t 2 t r 1 t t Sym dx dx = dx dx s s 2 s s Z Z Z Z G = exp Rd so (d) is (a realization of) step-2 nilpotent Lie group with d generators geometric interpretation of log xs,t : path- and area-increment For d = 2, G isomorphic to the 3-dimensional Heisenberg group Familiar concepts (scalar product, norm) generalize

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 18/22 Carnot-Carathoedory norm: 1 + v + M v M 1/2 v Anti (M) 1/2 k kCC j j _ j j j j _ j j x := x de…nes a G-valued path (which lifts x) and x = x 1 x t 0,t s,t s t

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 18/22 x := x de…nes a G-valued path (which lifts x) and x = x 1 x t 0,t s,t s t

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 18/22 Actually, x G := exp Rd so (d) since (1st order calculus!) s,t 2 t r 1 t t Sym dx dx = dx dx s s 2 s s Z Z Z Z G = exp Rd so (d) is (a realization of) step-2 nilpotent Lie group with d generators geometric interpretation of log xs,t : path- and area-increment For d = 2, G isomorphic to the 3-dimensional Heisenberg group Familiar concepts (scalar product, norm) generalize 2 Dilation δλ (1 + v + M) = 1 + λv + λ M Carnot-Carathoedory norm: 1 + v + M v M 1/2 v Anti (M) 1/2 k kCC j j _ j j j j _ j j x := x de…nes a G-valued path (which lifts x) and x = x 1 x t 0,t s,t s t

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 18/22 The space of all (α-Hölder, geometric) rough paths [previously introduced as pointwise limits of C 1-paths + iterated integrals subject to uniform α-Hölder bounds] is precisely

dCC (xs , xt ) x C ([0, 1] , G ) : sup α < ∞ 2 0 s

Very convenient! E.g. to show rough path regularity of Bt (ω) ...

Recall our assumption in Davie’slemma:

t t r 1/2 α dx dx dx c1 t s s _ s s Z Z Z j j

... this says precisely that t x is a Hölder continuous path, with 7! t exponent α, in the space G with Carnot-Caratheodory metric

1 dCC (x , x ) := x x = x . s t s t CC s,t CC

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 19/22 Very convenient! E.g. to show rough path regularity of Bt (ω) ...

Recall our assumption in Davie’slemma:

t t r 1/2 α dx dx dx c1 t s s _ s s Z Z Z j j

... this says precisely that t x is a Hölder continuous path, with 7! t exponent α, in the space G with Carnot-Caratheodory metric

1 dCC (x , x ) := x x = x . s t s t CC s,t CC

The space of all (α-Hölder, geometric) rough paths [previously introduced as pointwise limits of C 1-paths + iterated integrals subject to uniform α-Hölder bounds] is precisely

dCC (xs , xt ) x C ([0, 1] , G ) : sup α < ∞ 2 0 s

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 19/22 Recall our assumption in Davie’slemma:

t t r 1/2 α dx dx dx c1 t s s _ s s Z Z Z j j

... this says precisely that t x is a Hölder continuous path, with 7! t exponent α, in the space G with Carnot-Caratheodory metric

1 dCC (x , x ) := x x = x . s t s t CC s,t CC

The space of all (α-Hölder, geometric) rough paths [previously introduced as pointwise limits of C 1-paths + iterated integrals subject to uniform α-Hölder bounds] is precisely

dCC (xs , xt ) x C ([0, 1] , G ) : sup α < ∞ 2 0 s

Very convenient! E.g. to show rough path regularity of Bt (ω) ...

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 19/22 Theorem [Lamperti ’62] Assuming …nite moments of all order, convergence holds in α-Hölder topology, any α < 1/2. Theorem [Breuillard-F-Huessman, Proc. AMS ’09] Under previous assumptions, convergence holds in α-Hölder rough path topology, α < 1/2. Corollary: Universal limit theorem for Markov chain approximations to stochastic di¤erential equations.

Donsker’stheorem revisited

Theorem [Donsker ’52] Under …nite second moment assumptions, renormalized random walk in Rd converges weakly (in sup-topology) to BM.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 20/22 Theorem [Breuillard-F-Huessman, Proc. AMS ’09] Under previous assumptions, convergence holds in α-Hölder rough path topology, α < 1/2. Corollary: Universal limit theorem for Markov chain approximations to stochastic di¤erential equations.

Donsker’stheorem revisited

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 20/22 Corollary: Universal limit theorem for Markov chain approximations to stochastic di¤erential equations.

Donsker’stheorem revisited

Theorem [Donsker ’52] Under …nite second moment assumptions, renormalized random walk in Rd converges weakly (in sup-topology) to BM. Theorem [Lamperti ’62] Assuming …nite moments of all order, convergence holds in α-Hölder topology, any α < 1/2. Theorem [Breuillard-F-Huessman, Proc. AMS ’09] Under previous assumptions, convergence holds in α-Hölder rough path topology, α < 1/2.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 20/22 Donsker’stheorem revisited

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 20/22 Convergence of f.d.d. follows from a CLT on free nilpotent groups. Remains to establish tightness in α-Hölder rough path topology: Boils down to showing

p < ∞ : E ξ ξ 4p = O k2p . 8 k 1 k kCC h i

Sketch of proof:

Random walk can be viewed as random walk (ξi ) on the step-2 free nilpotent group .

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 21/22 Remains to establish tightness in α-Hölder rough path topology: Boils down to showing

p < ∞ : E ξ ξ 4p = O k2p . 8 k 1 k kCC h i

Sketch of proof:

Random walk can be viewed as random walk (ξi ) on the step-2 free nilpotent group . Convergence of f.d.d. follows from a CLT on free nilpotent groups.

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 21/22 Boils down to showing

p < ∞ : E ξ ξ 4p = O k2p . 8 k 1 k kCC h i

Sketch of proof:

Random walk can be viewed as random walk (ξi ) on the step-2 free nilpotent group . Convergence of f.d.d. follows from a CLT on free nilpotent groups. Remains to establish tightness in α-Hölder rough path topology:

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 21/22 Sketch of proof:

p < ∞ : E ξ ξ 4p = O k2p . 8 k 1 k kCC h i

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 21/22 [Caruana, Levy, Lyons ’05]: Di¤erential equations driven by rough paths, St. Flours lecture ... google "st_‡our" and …nd Lyons’handwritten St. Flour notes [Friz, Victoir ’10]: Multidimensional stochastic processes as rough paths, Cambridge Univ. Press ... download the pdf from my homepage

References

If you want to read more about this:

[Lyons, Qian ’02]: System control and rough paths, Oxford Univ. Press

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 22/22 ... google "st_‡our" and …nd Lyons’handwritten St. Flour notes [Friz, Victoir ’10]: Multidimensional stochastic processes as rough paths, Cambridge Univ. Press ... download the pdf from my homepage

References

If you want to read more about this:

[Lyons, Qian ’02]: System control and rough paths, Oxford Univ. Press [Caruana, Levy, Lyons ’05]: Di¤erential equations driven by rough paths, St. Flours lecture

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 22/22 [Friz, Victoir ’10]: Multidimensional stochastic processes as rough paths, Cambridge Univ. Press ... download the pdf from my homepage

References

If you want to read more about this:

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 22/22 ... download the pdf from my homepage

References

If you want to read more about this:

[Lyons, Qian ’02]: System control and rough paths, Oxford Univ. Press [Caruana, Levy, Lyons ’05]: Di¤erential equations driven by rough paths, St. Flours lecture ... google "st_‡our" and …nd Lyons’handwritten St. Flour notes [Friz, Victoir ’10]: Multidimensional stochastic processes as rough paths, Cambridge Univ. Press

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 22/22 References

If you want to read more about this:

P. K. Friz (TU Berlin and WIAS) rough paths, gap ODE/SDEs December2009 22/22