hp-dg timestepping for time-singular parabolic PDEs and VIs

Transcription

1 hp-dg timestepping for time-singular parabolic PDEs and VIs Ch. Schwab ( SAM, ETH Zürich ) (joint with O. Reichmann) ACMAC / Heraklion, Crete September 26-28, 2011 September 27, 2011

2 Motivation Well-posedness Application to FBM models dg Timestepping Examples Generalizations Conclusions Ch. Schwab September 27, 2011 p. 2

3 FBM Definition H (0, 1) Hurst parameter. A continuous Gaussian process W H on the probability space (Ω,A,F,P) with W H (0) = 0, W H (t) being a random variable with vanishing mean for all t 0 and E[W(t) H W H (s)] = 1 2 (t2h +s 2H t s 2H ), s,t 0, is called fractional Brownian motion (fbm). H = 1/2: Brownian motion. Ch. Schwab September 27, 2011 p. 3

4 FBM: Backward equations Consider the following problem: let the evolution of the underlying S under a pricing measure Q be given by ds(t) = rs(t)dt+σs(t)dw H (t), S(0) = S 0 > 0, 0 t T, where r,σ > 0 are constants. Then the price P(t) of a contingent claim at time t with payoff g(x) is given by P(t) = e r(t t) Ẽ Q [g(s T ) F H t ], Ẽ Q quasi-conditional expectation under the measure Q. Proof. See [Benth 03] or [Necula 02]. Ch. Schwab September 27, 2011 p. 4

5 FBM: Backward equations Under suitable integrability and smoothness assumptions holds the following analog of the Feynman-Kac theorem ([Benth 03], [Elliot et al. 03]). Theorem Let u(t, s) denote the solution of the backward equation t u+hσ 2 t γ s 2 ss u+rs s u ru = 0 }{{} A(t)u u(t,s) = g(s), s R + where γ = 2H 1, H (0,1) (so that γ ( 1,1)). Then P(t) = u(t,s(t)). Ch. Schwab September 27, 2011 p. 5

6 Ex. 1: Time-inhomogeneous Diffusion (t γ b,t γ σ,0), b R d,γ ( 1,1),σ R d d >0. A BS (t)ϕ(x) = t γ b ϕ(x)+t γ1 2 tr[σd2 ϕ(x)]. where A is defined by for γ ] 1,1[. t u t γ Au = f on J D, (1) A := γ +1 2 u(0) = u 0, (2) n j,k=1 a j,k (x), x j x k Ch. Schwab September 27, 2011 p. 6

7 Ex. 2: Time-inhomogeneous Jump-Diffusion (d = 1) (0,t δα 1 σ 2,k(t,z)) z t δ k(t,z) = t δα 1e M z 1+α, σ > 0, δ (0,1), M > 1, α (0,2) and t (0,T). A(t) of X(t) reads A(t) = A BS (t)+a J (t), ( A J (t)ϕ(x) = ϕ(x+z) ϕ(x) zϕ (x) ) k(t,z)dz, A BS (t)ϕ(x) = t δα 11 2 σ2 ϕ (x). Ch. Schwab September 27, 2011 p. 7

8 Ex. 2: Time-inhomogeneous Jump-Diffusion (d = 1) Weak formulation: Parametric bilinear forms a(, ), a J (, ) and a BS (, ) for ϕ(x),ψ(x) C 0 (Rd ) are given for 0 < t < T by Questions: a(t;ϕ,ψ) = a Diff (t;ϕ,ψ) +a J (t;ϕ,ψ), a J (t;ϕ,ψ) = (A J (t)ϕ,ψ), a Diff (t;ϕ,ψ) = (A BS (t)ϕ,ψ). meaning of (, )? H < 1/2 = γ = 2H 1 < 0: a Diff (t;ϕ,ϕ) as t 0. Ch. Schwab September 27, 2011 p. 8

9 Abstract forward problem For a bounded Lipschitz domain D R d, d 1 and a finite time interval I := (0, T), consider forward parabolic problem: find u such that t u t γ Au = f on I D, u(0) = g, A L(V,V ) nondegenerate, elliptic in V H H V. Ch. Schwab September 27, 2011 p. 9

10 Abstract forward problem Function Spaces: D R d bdd., Lipschitz, J = (0,T), T <. X := H 1 t γ/2 (J;V ) L 2 t γ/2 (J;V) = ( H 1 t γ/2 (J) V ) ( L 2 t γ/2 (J) V ), Y := L 2 t γ/2 (J;V) = L 2 t γ/2 (J) V, X (0 := {w X : w(0, ) = 0 in V }, X 0) := {w X : w(t, ) = 0 in V }, e.g. V := H 1 0(D), V = H 1 (D), L 2 t γ/2 (J) = C (J) L 2 t γ/2(j), H 1 t γ/2 (J) = C (J) H 1 t γ/2(j). Ch. Schwab September 27, 2011 p. 10

11 Well-posedness Possible approaches Semigroup approach: problems w. degeneracy in time. Lax-Milgram Lemma Works applied to space variable pointwise, but not uniformly in time Fails applied to space-time formulation inf sup or Babuska-Brezzi-conditions space-time variational formulation Ch. Schwab September 27, 2011 p. 11

12 BB-Conditions X, Y Hilbert spaces, B(, ) : X Y R bilinear form. BB-conditions: inf sup B(u, v) c stab > 0, (3) 0 u X 0 v Y u X v Y and 0 v Y : supb(u,v) > 0 (4) u X sup 0 u X,0 v Y B(u, v) u X v Y <. (5) Ch. Schwab September 27, 2011 p. 12

13 BB-Conditions Theorem Let B(, ) : X Y R satisfy (3)-(5). Then the problem: given f Y, find u X such that B(u,v) = f(v), v Y, (6) admits a unique variational solution u X with u X 1 c stab f Y. B(u,v) = T 0 (( u,v)+tγ a(u,v)) dt, a(u,v) = (Au,v), g = 0 (Homogeneous terminal conditions) Ch. Schwab September 27, 2011 p. 13

14 BB-Conditions For V := H0 1 (D) and L 2 t γ/2 (I) := C (0,1) L 2 t γ/2(i), u 2 L 2 t γ/2(i) := where X and Y given as B(u, v) inf sup 1 0 u X (0 0 v Y u X v Y 2 X := H 1 t γ/2 (I;V ) L 2 t γ/2 (I;V) = ( H 1 t γ/2 (I) V ) ( L 2 t γ/2 (I) V ), Y := L 2 t γ/2 (I;V) = L 2 t γ/2 (I) V, X (0 := {w X : w(0, ) = 0 in V }. I u 2 t γ dt. Ch. Schwab September 27, 2011 p. 14

15 Extensions Non-homogeneous terminal payoff functions. as natural BC s in (x, t) adjoint formulation B (u,v) = T for u Y, v X 0). 0 ( v,u)+t γ a(u,v) = f(v)+(v(0),g) The following embedding holds (P. Grisvard) where Λ = A 1/2 with X C 0 (I,D(Λ 1 2 γ 2 )), D(Λ 0 ) = V and D(Λ 1 ) = V. Ch. Schwab September 27, 2011 p. 15

16 Example: FBM (Reichmann 11) We consider the backward equation Transformations: t u+hσ 2 t 2H 1 S 2 SS u+rs S u ru = 0 u(t,s) = g(s). Time-reversal: t T τ Localization from R + to bounded domain /Weighted spaces Ch. Schwab September 27, 2011 p. 16

17 Localized forward formulation where B(v, w) = B(v,w) = f(v), T 0 (( v,w)+a(τ;v,w)) dτ a(τ;v,w) = σ2 2 (T τ)γ ( y v(τ,y), y w(τ,y)), f(v) = B( g, v) X := H 1 (T τ) γ/2 (I;V ) L 2 (T τ) γ/2 (I;V), Y := L 2 (T τ) γ/2 (I;V), V := H 1 0(D). Ch. Schwab September 27, 2011 p. 17

18 discontinuous Galerkin ( dg ) timestepping M = {I m } M m=1, partition of J = (0,T), r NM 0 dg orders. dg-timestepping: U V r (M;X) := {u : J V : u Im P rm (I m,v),m = 1 : M} such that for all v V r (M;V) B dg (U,v) = F dg (v), where M B dg (U,v) = (U,v) H dt+ I m + m=1 M m=1 I m t γ a(u,v)dt M ([U] m 1,v m 1 + ) H +(U 0 +,v+ 0 ) H m=2 F dg (v) = (u 0,v + 0 ) H + M m=1 I m (f(t),v) V Vdt Ch. Schwab September 27, 2011 p. 18

19 discontinuous Galerkin ( dg ) timestepping 1. Assume: Ex. 0 < α β < such that v V : α v 2 V V Av,v V β v 2 V then shall show that for a suitable dg Interpolant I holds ( u U L 2 C 1+ α ) u Iu t γ/2(i;v) β L 2 γ/2(i;v), t Ch. Schwab September 27, 2011 p. 19

20 discontinuous Galerkin ( dg ) timestepping 2. dg-interpolanti: Let I = (0,1). For u L 2 (I;V) continuous at t = 1 define Π r u P r (I,X), r 1, via the (Π r u u,q) H dt = 0, q P r 1 (I;V) and I Π r u(1 ) = u(1 ) V. Ch. Schwab September 27, 2011 p. 20

21 discontinuous Galerkin ( dg ) timestepping 3. Quasioptimality : u X (0 exact solution, U V r (M,V) time-semidiscrete dg solution. Let Iu V r (M,V) denote the dg interpolant of u be defined Then I m M : Iu Im = Π rm I m (u Im ). u U L 2 t γ/2(i;v) C for C > 0 independent of M and of r. ( 1+ α ) u Iu β L 2 γ/2(i;v), t Ch. Schwab September 27, 2011 p. 21

22 discontinuous Galerkin ( dg ) timestepping 4. scaled local dg interpolation error bounds Let I = (a,b), k = b a, r N 0 and u H s 0+1 for s 0 N 0. Then ex. C > 0 s.t. u Π r I u 2 L 2 (I,V) C Γ(r +1 s) max{1,r} 2 Γ(r +1+s) for any k > 0, r N 0, 0 s min{r,s 0 }, s real. ( ) k 2(s+1) u 2 H 2 s+1 (I,V), Ch. Schwab September 27, 2011 p. 22

23 discontinuous Galerkin ( dg ) timestepping 5. Solution regularity Assume u 0 H θ for 0 θ 1, g suff. regular A(t) t γ A, γ < 1 ( analytic case). Then u satisfies for 0 < a b < min(1,t) and for every k N u (k) (t) 2 H s t γ/2((a,b),v ) Cd 2(k+s) Γ(2k +3)a 2(k+s)+θ. Ch. Schwab September 27, 2011 p. 23

24 discontinuous Galerkin ( dg ) timestepping 6. Geometric Timesteps/ linear order vector: A geometric time partition M n,σ = {I m } n+1 m=1 with grading factor σ (0,1) and n+1 time steps I m, m = 1,2,...,n+1 is given by the nodes t 0 = 0, t m = Tσ n+1 m, 1 m n+1. Time step size: k m = I m = t m t m 1 k m = λt m 1, λ = 1 σ σ, 2 m M = n+1. Ch. Schwab September 27, 2011 p. 24

25 discontinuous Galerkin ( dg ) Timestepping 6. Geometric Timesteps/ linear order vector: A polynomial degree vector r = {r m } n+1 m=1 is called linear with slope ν 0 on the geometric partition M n,σ of (0,T) if r 1 = 0 and r m = νm for 2 m n+1. (Schötzau Diss. ETH 1999/ Schötzau and CS SINUM 2000). Ch. Schwab September 27, 2011 p. 25

26 discontinuous Galerkin ( dg ) timestepping 7. global dg interpolation error bounds There exist C,d > 0 such that I m M n,σ, 2 m n+1 and s m = α m r m with arbitrary α m (0,1), u Π rm I m u ( ) 2 L 2 Cσ(n m+2)θ γ (µd) 2αm(1 α m) 1 αm rm t γ/2(im,x) (1+α m ) 1+αm where µ = max{1,λ} and λ = 1 σ σ. C,d > 0 only depend on u 0 H θ and on γ ( 1,1). Ch. Schwab September 27, 2011 p. 26

27 discontinuous Galerkin ( dg ) Timestepping Theorem : [Reichmann & CS 2011] Consider the time-inhomogeneous forward problem on J = (0,1) with initial data u 0 H θ for some θ (0,1] and right hand side g. Discretize in time using dgfem on geometric partition M n,σ. Then for all degree vectors r = {r m } n+1 m=1 with slope ν ν 0 > 0 the semidiscrete dgfem solution U obtained in V r (M n,σ,v) converges exponentially w.r. to N, No. of time-dofs : u U L 2 t γ/2(j;v) C(σ,ν 0)exp( bn 1/2 ) Ch. Schwab September 27, 2011 p. 27

28 Examples Example 1: Exponential Convergence for fbm d = sigma=0.7 sigma=0.75 sigma=0.8 sigma=0.85 sigma= L error Number of time steps Figure: Convergence rates for different parameters σ and γ = 0.5. Ch. Schwab September 27, 2011 p. 28

29 Examples Example 2: Exponential Convergence time-inhomogeneous Lévy-like additive process 10 2 sigma=0.1 sigma=0.2 sigma=0.3 sigma=0.4 sigma= L error Number of time steps Figure: Convergence rates for different parameters σ and γ = 0. Ch. Schwab September 27, 2011 p. 29

30 Examples Example 2 (cont d): Exponential Convergence time-inhomogeneous Lévy-like additive process sigma=0.1 sigma=0.2 sigma=0.3 sigma=0.4 sigma=0.5 L error Number of time steps Figure: Convergence rates for different parameters σ and γ = 0.9. Ch. Schwab September 27, 2011 p. 30

31 Examples Example 3: algebraic convergence FBM, u 0 (x) = (x x 0 ) +, d = 1, dg(0) L2 finite element error Linfinity finite element error O(N 2 ) log error log number of mesh points Figure: Convergence rate in the L and L 2 norm at maturity of a plain vanilla European contract using dg(0) timestepping and linear C 0 -FEM in space. Hurst parameter H = 0.1, γ = 2H 1 = 0.8, graded mesh M (time grading parameter β = 5). Ch. Schwab September 27, 2011 p. 31

32 Examples Example 4: FBM PVI in X ( American Put ) H=0.1 H=0.3 H=0.5 H=0.7 H=0.9 Payoff 0.3 Option price Moneyness Figure: Prices of American puts in the FBM market model using dg(0) timestepping and linear FE in space: H [0.1,...,0.9] Ch. Schwab September 27, 2011 p. 32

33 Generalization γ-pseudohomogeneous, Gevrey time dependence: [ A BS (t)ϕ(x) = a(t) b ϕ(x)+ 1 ] 2 tr[σd2 ϕ(x)] where 0 < c 1 a(t)/t γ c 2 < in (0,T], and there exist d > 0 such that for some δ 1 k N : d k a(t) dt k (t) dk+1 (k!) δ t γ k, 0 < t T. Then u U L 2 t γ/2(j;v) C(q,ν 0)exp( bn 1/(1+δ) ) Ch. Schwab September 27, 2011 p. 33

34 Conclusions Well-posedness for time-degenerate parabolic problems hp-dg timestepping: exponential convergence in number N of space problems. (wavelet discretizations in (log-)price space: bounded condition number of matrices, ind. of dim.) h-dg: optimal conv. rates for time-graded M. time-degenerate PVIs: stable formulation, opt. rate for dg(0). Ch. Schwab September 27, 2011 p. 34

35 Conclusions Well-posedness for time-degenerate parabolic problems hp-dg timestepping: exponential convergence in number N of space problems. (wavelet discretizations in (log-)price space: bounded condition number of matrices, ind. of dim.) h-dg: optimal conv. rates for time-graded M. time-degenerate PVIs: stable formulation, opt. rate for dg(0). Thank you!!! Ch. Schwab September 27, 2011 p. 34