The Hopf algebraic structure of stochastic expansions and efficient simulation

Transcription

1 The Hopf algebraic structure of stochastic expansions and efficient simulation Kurusch Ebrahimi Fard, Alexander Lundervold, Simon J.A. Malham, Hans Munthe Kaas and Anke Wiese York: 15th October 2012 KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

2 Introduction SDE for y t R N in Stratonovich form y t = y 0 + d i=0 t 0 V i (y τ ) dw i τ. Driven by a d-dimensional Wiener process (W 1,..., W d ); Governed by drift V 0 and diffusion vector fields V 1,..., V d ; Use convention W 0 t t; Assume t R + lies in the interval of existence. Focus on solution series and strong simulation. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

3 Accuracy: Problem Background: Solutions only in exceptional cases given explicitly. Typically, approximations are based on the stochastic Taylor expansion, truncated to include the necessary terms to achieve the desired order (e.g. Euler scheme, Milstein scheme). Question: Are there other series such that any truncation generates an approximation that is always at least as accurate as the corresponding truncated stochastic Taylor series, independent of the vector fields and to all orders? We will call such approximations efficient integrators. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

4 Flowmap For the system of equations dy dt = V (y) we can define the flowmap ϕ t : y 0 y t. Chain rule = d dt f (y) = V (y) yf (y) } {{ } V f y d dt (f ϕ t y 0 ) = V f ϕ t y 0 d dt (f ϕ t) = V f ϕ t f ϕ t = exp(tv ) f KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

5 Flowmap for SDEs Itô lemma for Stratonovich integrals = f y t = f y 0 + d t i=0 0 V i f y τ dw i τ. where V i f = V i y f. Hence for flowmap we have f ϕ t = f id + d i=0 t 0 V i f ϕ τ dw i τ. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

6 Flowmap series Non-autonomous linear functional differential equation for f t f ϕ t = set up the formal iterative procedure (with f (0) t = f ): f (n+1) t = f + d i=0 t 0 V i f (n) τ dw i τ, Iterating = f ϕ t = f + d ( t i=0 dwt i 0 ) } {{ } J i (t) V i f + d ( t τ1 ) dwτ i 2 dw j τ 1 V i V j f } {{ }} {{ } i,j=0 J ij (t) V ij KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

7 Stochastic Taylor expansion Stochastic Taylor expansion for the flowmap: ϕ t = J w (t) V w. w A w = a 1... a n are words from alphabet A {0, 1,..., d}; Scalar random variables J w (t) t 0 τn 1 dw a 1 τ n 0 dw a n τ 1 ; Partial differential operators V w V a1 V an. Encodes all stochastic and geometric information of system. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

8 Basic Strategy Is as follows: 1 Construct the series σ t = f (ϕ t ); 2 Truncate the series σ t to ˆσ t according to a grading; 3 Compute ˆϕ t = f 1 (ˆσ t ) = numerical scheme. For example: Stochastic Taylor series implies ϕ t = ˆϕ t = J w V w + J w V w g(w) n J w V w + g(w) n+1 g(w) n g(w)=n+1 Ē(J w )V w KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

9 Example: Castell Gaines method Exponential Lie series ψ t = log ϕ t given by ψ t = (ϕ t id) 1 2 (ϕ t id) (ϕ t id) 3 + d 1 = J i V i + 2 (J ij J ji )[V i, V j ] +. i=0 i>j Truncate, then across [t n, t n+1 ], we have d ( ˆψ tn,t n+1 = W i (t n, t n+1 ) ) V i + Â ij (t n, t n+1 ) [V i, V j ]. i=0 i>j Then ŷ tn+1 exp( ˆψ tn,t n+1 ) ŷ tn. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

10 Example: Castell Gaines method recovery For each path simulated W i (t n, t n+1 ) and Âij(t n, t n+1 ) are fixed constants = ˆψ tn,t n+1 an autonomous vector field. Thus, for τ [0, 1] and with u(0) = ŷ tn, solve u (τ) = ˆψ tn,t n+1 u(τ). Using a suitable high order ODE integrator generates u(1) ŷ tn+1. Castell & Gaines: this method is asymptotically efficient in the sense of N. Newton. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

11 Hopf algebra representation Stochastic Taylor series for the flowmap: ϕ = w w, w lies in the product algebra (over K = R): K A sh K A co. See Reutenauer (u x)(v y) = (u v) (xy), KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

12 Hopf algebra of words K A co : Product, concatenation: u v uv; Coproduct, deshuffle: w u v=w u v; Unit, counit and antipode S : a 1... a n ( 1) n a n... a 1. K A sh : Product, shuffle: u v u v; Coproduct, deconcatenation: w (w) = uv=w u v; Unit, counit and antipode same. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

13 Power series function Suppose we apply function to the flow-map ϕ: f (ϕ) = c k ϕ k k 0 k = c k w w k 0 w A = c k (u 1... u k ) (u 1... u k ) k 0 u 1,...,u k A = c k u 1... u k w w A k 0 w=u 1...u k = F(w) w w A KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

14 Power series function recoding In other words where f (ϕ) = F w = k 0 c k w A (F w) w, u 1... u k. u 1,...,u k w=u 1...u k = encode the action of f by F End(K A sh ). Note the stochastic Taylor series is F = id. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

15 Shuffle convolution algebra Embedding End(K A sh ) K A sh K A co given by X X(w) w, is an algebra homomorphism for the convolution product: X Y = (X Y ). w We henceforth denote H End(K A sh ). Unit in H is ν and the antipode S id = id S = ν. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

16 Functions of the identity F (id) = c k id k k 0 log (id) = J 1 2 J J 3 + ( 1)k+1 k J k + sinhlog (id) = 1 2 (id S) S = ν J + J 2 J 3 + sinhlog (id) = J 1 2 J J 3 + ( 1) k J k +, where J id ν is the augmented ideal projector, it is the identity on non-empty words and zero on the empty word: J k w = u 1... u k. w=u 1...u k where all the words in the decomposition must be non-empty. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

17 Generating endomorphisms from endomorphisms For any X H: f (X) = c k (X ɛ ν) k k 0 log ( 1) k+1 (X) = (X ν) k k k 1 exp 1 (X) = k! X k k 0 ( sinhlog (X) = ) 1 2 X X ( 1) ( coshlog (X) = ) 1 2 X + X ( 1) KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

18 Expectation endomorphism Let D A be the free monoid of words on D = {0, 11, 22,..., dd}. Definition (Expectation endomorphism) Linear map E H such that t n(w) E: w 2 d(w) n(w)! 1, w D, 0 1, w A \D. Here d(w) is the number of non-zero consecutive pairs in w from D, n(w) = d(w) + z(w) where z(w) is the number of zeros in w. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

19 Inner product Definition (Inner product) We define the inner product of X, Y H with respect to V to be ( ) X, Y H X(u) Y (v) (u, v). u,v A Ē The norm of an endomorphism X H is X H X, X 1/2. Motivation: if x t = X(w) V w (y 0 ) and y t = Y (w) V w (y 0 ), w A w A our definition is based on the L 2 -inner product x t, y t L 2 = Ē (x t, y t ). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

20 Graded class Definition (Grading map) This is the linear map g: K A sh Z + given by g: w w. Definition (Graded class) For a given n Z +, we set S n { w K A sh : g(w) = n } and S n S k and S n S k. k n A subset S A is of graded class if, for a given n Z +, S = S n, S = S n or S = S n. We denote by π S : K A sh S the canonical projection onto any graded class subspace S, k n KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

21 Reversing Lemma Lemma (M. & Wiese 2009) For any pair u, v A, we have E (u v) E ( ( S u) ( S v) ). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

22 Lemma: properties Lemma For A = {0, 1,..., d} and graded class subspace S = S n : 1 X, Y = S X, S Y ; 2 S, S = S, S = id, id ; 3 sinhlog (id), coshlog (id) = 0; 4 id 2 = sinhlog (id) 2 + coshlog (id) 2 ; 5 X, E Y = E X, E Y ; 6 E id, E id = E S, E S = E S, E S ; 7 E sinhlog (id), E coshlog (id) = 0; 8 S, J n = id, J n, KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

23 Accuracy measurement We measure the accuracy by r t y 0 L 2 where r t ϕ t ˆϕ t. A general stochastic integrator has the form îd f ( 1) π S n f (id). The error is R in the H-norm where R id îd. Definition (Pre-remainder) Q f (id) π S n f (id). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

24 Efficient integrator Definition (Efficient integrator) We will say that a numerical approximation to the solution of an SDE is an efficient integrator if it generates a strong scheme that is more accurate in the root mean square sense than the corresponding stochastic Taylor scheme of the same strong order, independent of the governing vector fields and to all orders: i.e. (id E) R 2 (id E) id 2. Sinhlog integrator of strong order n/2: P π S n sinhlog (id) = ( J 1 2 J ( 1)n+1 J n) π S n. Q = sinhlog (id) π S n+1. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

25 Sinhlog pre-remainder and remainder h (X, +ν) = ( X 2 + ν ) (1/2) = ν X X 4 + R = id sinhlog ( 1) P = sinhlog ( 1) (P + Q) sinhlog ( 1) P = Q + h (P + Q, +ν) h (P, +ν) = Q + 2( 1 (P + Q) 2 P 2) + ( ) ( = Q + ) 1 2 P Q + Q P + O Q 2 = Q π Sn+1, since sinhlog (id) = J + : P Q = ( J π S n ) ( J πs n+1 ). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

26 Sinhlog efficient Reversing Lemma with S = S n+1 = and id 2 = Q 2 + coshlog (id) 2. (id E) id 2 = (id E) Q 2 + (id E) coshlog (id) 2. Indeed, for any ɛ: sinhlog ɛ (id) = J ( 1) k+1 J k + ɛj (n+1) k=2 id 2 = Q ɛ 2 + coshlog (id) 2 ɛ id S, J (n+1) ɛ 2 J (n+1) 2. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

27 Optimality Theorem f (X; ɛ) 1 2 (X ɛx ( 1) ). Theorem f ( 1) (X; ɛ) = X + h (X, ɛ ν), For every ɛ > 0 the class of integrators f (id; ɛ) is efficent. When ɛ = 1, the error of the integrator f (id; 1) realizes its smallest possible value compared to the error of the corresponding stochastic Taylor integrator. Thus a strong stochastic integrator based on the sinhlog endomorphism is optimally efficient within this class, to all orders. f (id; ɛ) = 1 2 (1 ɛ)ν (1 + ɛ)j 1 2 ɛ( J 2 J 3 + ). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

28 Optimality Theorem proof: step 1 h : (x, y) (x 2 + y) 1/2. h(x + q, y) h(x, y) = x (x 2 + y) 1/2 q +, P π S n f (id; ɛ) R = id îd = f ( 1) (P + Q) f ( 1) P = Q + h (P + Q, ɛ ν) h (P, ɛ ν). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

29 Optimality Theorem proof: step 2 R = Q + h (P + Q, ɛ ν) h (P, ɛ ν) = Q + ( (P + Q) 2 + ɛ ν ) (1/2) ( P 2 + ɛ ν ) (1/2) = Q (P 2 + ɛ ν) ( 1/2) ( P Q + Q P ) + O ( Q 2). P = 1 2 (1 ɛ) ν + O(J), (P 2 + ɛ ν) ( 1/2) = ( 1 2 (1 + ɛ)) 1 ν + O(J). R = Q + 1 ɛ Q + O(J Q) 1 + ɛ = ɛ Q π S n+1. KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

30 Optimality Theorem proof: step 3 On S n+1 : id 2 = R + id R, R + id R = R R, id R + id R, id R = R id ɛ (id ɛs), 1 (id ɛs) 1 + ɛ + id 1 1 (id ɛs), id (id ɛs) 1 + ɛ 1 + ɛ = R 2 1 ( + 1 ) 1 2ɛ id, S + ɛ 2 (1 + ɛ) 2 = R 2 + 2ɛ (1 + ɛ) 2 ( 1 + id, S ). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

31 Optimality Theorem proof: step 4 R = Q (P 2 ν) ( 1/2) ( P Q + Q P ) + O ( Q 2). P = ν J 2 + O ( J 3). (P 2 ν) (1/2) = ( 2 (P ν) (1/2) ν (P ν) + O( (P ν) 2)). = order reduction. R = Q + ( J ( 1) π S n ) Q + O ( J Q ). KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32

32 References Ebrahimi-Fard, K., Lundervold, A., Malham, S.J.A., Munthe Kaas, H. and Wiese, A Algebraic structure of stochastic expansions and efficient simulation, Proc. R. Soc. A. Lord, G., Malham, S.J.A. & Wiese, A Efficient strong integrators for linear stochastic systems, SIAM J. Numer. Anal. 46(6), Malham, S.J.A., Wiese, A Stochastic Lie group integrators, SIAM J. Sci. Comput. 30(2), Malham, S.J.A., Wiese, A Stochastic expansions and Hopf algebras. Proc. R. Soc. A 465, Malham, S.J.A., Wiese, A Lévy area logistic expansion and simulation, arxiv: v1 Newton, N.J. 1991, SIAM J. Appl. Math. 51, Reutenauer, C Free Lie algebras, London Mathematical Society Monographs New Series 7, Oxford Science Publications KEF, AL, SM, HMK & AW () Hopf Algebras/SDEs York: 15th October / 32