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. 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 . 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 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 R 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 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 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 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 t 1 2 Example: Bs dBs = B t ..