Algebraic Structure of Stochastic Expansions and Efficient Simulation

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1 Algebraic Structure of Stochastic Expansions and Efficient Simulation Anke Wiese Heriot Watt University, Edinburgh, UK SSSC2012: ICMAT, Madrid November 5-9, 2012 Supported by EMS, IMA, LMS Work in collaboration with: Simon J.A. Malham (Proc. Roy. Soc. A, 2009), Kurusch Ebrahimi Fard, Alexander Lundervold, Simon J.A. Malham, Hans Munthe-Kaas (Proc. Roy. Soc. A, 2012)

2 Set-up Suppose solution y of stochastic differential equation in R N given by t d t y t = y 0 + V 0 (y τ )dτ + V i (y τ )dw i (τ) 0 0 i=1 where (W 1,...W d ) standard Wiener process, smooth vector fields V i = N j=1 Vj i y j on R N, i = 0,...,d, initial condition y 0 R N. For notational convenience, W 0 (τ) = τ.

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.

4 Itô s Lemma for Stratonovich Integrals Itô s Lemma: Suppose f smooth. Then f(y t ) =f(y 0 )++ =f(y 0 )+ d i=0 d i=0 t 0 t 0 N j=1 V j i (y τ) yj f(y τ ) } {{ } =V i f y τ V i f y τ dw i (τ), dw i (τ) where V i f = N j=1 V j i y j f.

5 Stochastic Taylor Series (Stratonovich Form) y t = y 0 + d i=0 t + = w 0 dw i (τ)v i y 0 + J w (t)v w y 0, i,j=0 t d W i (τ)dw j (τ) V i V j y 0 0 } {{ } =J ij (t) where w = (a 1,..., a n ) word from alphabet {0, 1,..., d} (including the empty word), and V w = V a1 V a2... V an, V i V j = N k=1 V k i yk V j. Stochastic Taylor flow (Lyons signature) ϕ t = w J w V w.

6 Principal Idea Suppose f is invertible function with real series expansion f(x) = c k x k (c k constant). Construct new series k=0 ψ t = f(ϕ t ) Truncate to finite series ˆψ t. c k ϕ k t. k=0 New approximation to flow map and to solution y t ˆϕ t = f 1 (ˆψ t ), ŷ t = ˆϕ t y 0. Mean-square error in this approximation (ϕ t ˆϕ t ) y 0 2 L 2. Aim: Find f, equivalently (c k ) k, such that error is smaller than error for corresponding truncated Taylor series expansion.

7 Exponential Lie Series Example: f = logarithmic function: where ϕ t = exp(ψ t ), ψ t = lnϕ t = d J i (t)v i + i<j i=0 1 2 (J ij J ji )(t)[v i,v j ]+. exponential Lie series, Chen-Strichartz formula (Magnus 1954, Chen 1957, Strichartz )

8 Accuracy: Exponential Lie Series Theorem: A(modified)exponentialLieseriesoforder 1 2, order1andorder11 2 is more accurate in the mean-square sense than the corresponding truncated Taylor series, if the diffusion vector fields V 1,..., V d commute. They do not need to commute with the drift vector field V 0. (Castell & Gaines 1995, Malham & W. 2008) Remark: Counterexample for non-commuting vector fields (Lord, Malham & W. 2008).

9 Sinhlog Series Theorem: The solution series constructed by taking the hyperbolic sine of the logarithm of the stochastic Taylor flow-map, ψ t = sinhlog(ϕ t ), when truncated, generates an approximation, ˆϕ t = expsinh 1 (ˆψ t ) ˆψ t + id+ ˆψ t 2, that is for all vector fields always at least as accurate as the corresponding truncated stochastic Taylor series expansion in the meansquare sense. (Malham & W. 2009)

10 Principal Idea Stochastic Taylor flow ϕ t = w J w V w. (after rearrangement). f(ϕ t ) = k ϕ k=0c k t = w F(w)V w Composition of vector fields is concatenation product, V w1 V w2 = V w1 w 2. Itó s Product Formula: Multiplication of multiple Wiener integrals is a shuffle product. e.g. J 12 J 34 = J J J J J J Shuffle product J u J v u v.

11 Hopf Algebra Structure - Recoding the Signature Representation of stochastic Taylor flow as ϕ t = w w w in Hopf algebra given by tensor product of (algebras of) real series of words with shuffle product on left and concatenation product on right. Given f(x) = k=0 c kx k, ) k f(ϕ t ) = ( c k w w k 0 w = ( ) c k u 1... u k w. w k=0 w=u 1...u }{{ k } =F(w)

12 Convolution Shuffle Algebra Convolution algebra of linear endomorphisms of Hopf algebra (with shuffle product) K 1 K 2 (w) = K 1 (v) K 2 (v ). Representation vv =w f(ϕ t ) = w F(w) w, where F is the linear endomorphism F = k c k id k =: f (id). Important: Action of f encoded by linear endomorphism f (id).

13 Generalization Construction through expansion about ǫ R f (X) = (X ǫν) k, k=0 where ν is the unit in the convolution shuffle algebra (ν sends non-empty words to 0 and the empty word to itself). For an endomorphism X define the sinhlog endomorphism by Thus where sinhlog (X) = 1 2( X X ( 1) ). sinhlog (id) = 1 2 ( id id ( 1) ) = 1 2 S (a 1...a n ) = ( 1) n a n...a 1 is the antipode (signed reverse). ( id S ),

14 Sinhlog Endomorphism Series representation sinhlog (id) = J 1 2 J J 3 +( 1) k+11 2 J k +, where J := ν id (augmented ideal projector). The compositional inverse of sinhlog is given by sinhlog 1 (X) = X + ( X 2 +ν ) (1/2). Similar, the coshlog endomorphism is coshlog (X) = 1 2( X +X ( 1) ).

15 Sinhlog Series On computational interval [t n,t n+1 ]: sinhlog(ϕ t ) = d J i (t n,t n+1 )V i + i=0 + d i,k,k=0 d i,j=0 1 2( Jij J ji ) (tn,t n+1 )V ij 1 2( Jijk +J kji ) (tn,t n+1 )V ijk +...

16 Inner Product of Endomorphism Measure accuracy by mean-square error: for stochastic processes X t = u X(u)V u(y 0 ) and Y t = v Y(v)V v(y 0 ) X t,y t = E [ X ty ] t = V u(y 0 )E [ X(u)Y(v) ] V v (y 0 ), u,v where denotes matrix transpose. Corresponding inner product for endomorphisms X and Y: Write X(w) = u X w,uu with real-valued coefficients X w,u. Define matrix X = ( X w,u )w,u. Define matrix W = ( Ww,u )w,u, where W w,u = E [ w u ]. Define matrix V = ( Vw,u )w,u, where V w,u = V wv u (inner product). X,Y := tr ( XWY V ). Note that the definition of the inner product depends on the V; all our subsequent results hold independently of V.

17 Orthogonality of Sinhlog and Coshlog id = sinhlog (id)+coshlog (id) For any n, consider the restriction of the following endomorphisms on the subspace S n generated by words of length n. Then 1. S, S = S,S = id,id ; 2. sinhlog (id),coshlog (id) = 0; 3. id 2 = sinhlog (id) 2 + coshlog (id) 2. Proof is based on the following identity (see Malham & W. 2009): For any pair u,v, we have E[J u J v ] = E(u v) E ( ( S u) ( S v) ) = E[J S u J S v ]. This implies property 1. Property 2 follows using the bilinearity of the inner product: sinhlog (id),coshlog (id) = 1 2 (id S), 1 2 (id+s) Property 3 now immediate. = 1 4( id,id S,S ) = 0.

18 Construction of Stochastic Approximations Function f with formal power series given; corresponding endomorphism f (id); truncate to include only words up to length n, denote this Apply compositional inverse error in approximation Π S n f (id); f 1 π S n f (id); id f 1 π S n f (id); measure error in norm defined on endomorphisms. compare with error in corresponding truncated Taylor approximation id id π S n id;

19 Efficiency A numerical approximation to the solution of an SDE is an efficient integrator if it generates a strong numerical integration scheme that is more accurate in the mean square sense than the corresponding truncated stochastic Taylor integration scheme of the same order (according to word length), independent of the governing vector fields and to all orders.

20 Efficiency of Sinhlog Integrator Theorem: The integrator based on the sinhlog endomorphism is efficient. Idea of Proof: Define P :=π S n sinhlog (id) Q :=π S n+1 sinhlog (id), where π S n+1 is projector on space generated by words of length n+1 or more. Error in approximation sinhlog 1 (P+Q) sinhlog 1 P = Q+ 1 2( P Q+Q P ) +O ( Q 2 ). Leading order term Q π Sn+1. Leading order term in truncated Taylor series is id π Sn+1, and id π Sn+1 = Q π Sn+1 + coshlog (id) π Sn+1 > Q π Sn+1.

21 New Class of Integrators Define the class of endomorphisms f (X;ǫ) = 1 2 (X ǫx ( 1) ). for any ǫ R. f (id,+1) is the sinhlog endomorphism; f (id, 1) is the coshlog endomorphism. Compositional inverse of f (X;ǫ) is f 1 (X;ǫ) = X + ( X 2 +ǫν ) (1/2).

22 Optimality of Sinhlog Integrator Theorem: For every ǫ > 0 the class of integrators f (id;ǫ) is efficient. When ǫ = 1, the error of the integrator f (id;1) realizes its smallest possible value compared to the error of the corresponding stochastic Taylor integrator.

23 Coshlog Integrator The coshlog endomorphism does not generate an efficient integrator. The compositional inverse of the coshlog endomorphism is Define as before Error in approximation coshlog 1 (X) = X + ( X 2 ν ) 1 2. P :=π S n coshlog (id) Q :=π S n+1 coshlog (id) coshlog 1 (P+Q) coshlog 1 P = Q+(J ( 1) π S n ) Q+... J ( 1) exists formally, but introduces terms of same order as remained in integrator: thus there is a from of order reduction.

24 Example 1: Linear 2 2 System Sinh Log Stochastic Taylor Magnus Number of sampled paths= log 10 (global error) time x 10 4

25 Example 1: Error Number of sampled paths= log 10 (global error) Sinh Log Stochastic Taylor Magnus time x 10 5

26 Example 1: Difference between Error and Stochastic Taylor Error 5 x 10 7 Sinh Log Exponential Lie Number of sampled paths= difference of global errors time x 10 4

27 Example 2: Global Error 1 Stochastic Taylor 1.5 Sinh Log 1.5 Number of sampled paths= log 10 (global error) log 10 (stepsize)

28 Conclusion Natural connection between stochastic approximations and convolution algebra of endomorphisms on Hopf shuffle algebra of words. Algebraic abstraction enables new class of efficient integrators. Within this class, the sinhlog integrator is optimal, that is the error of the integrator realizes its smallest value when compared with the error of the corresponding truncated stochastic Taylor integrator in the mean-square sense. Generalization to SDEs driven by Lévy processes and semimartingale algebra.

29 Future Directions Algebraic structure of BSDEs? Algebraic structure of SPDEs?

30 References Castell, F., Gaines, J An efficient approximation method for stochastic differential equations by means of the exponential Lie series. Math. Comp. Simulation 38, Chen, K. T Integration of paths, geometric invariants and a generalized Baker Hausdorff formula. Annals of Mathematics 65(1), Curry, C., Ebrahimi-Fard, K., Malham, S.J.A. and Wiese, A The algebra of semimartingales and efficient simulation of stochastic systems driven by Lévy processes. 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 expansions and Hopf algebras. Proc. R. Soc. A 465,

31 References Magnus, W On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7, Reutenauer, C Free Lie algebras, London Mathematical Society Monographs New Series 7, Oxford Science Publications