Explicit approximate Green s function for parabolic equations.

Transcription

1 Explicit approximate Green s function for parabolic equations. Anna Mazzucato 1 1 Department of Mathematics Penn State University MSRI Inverse Problems Seminar, September 17, 2010

2 Collaborators Victor Nistor (PSU Math ), Nick Costanzino (PSU Math, Postdoc), Wen Cheng (PSU Math, Graduate student), John Liechty (PSU Business School), MCMC methods, Radu Costantinescu (JPMorgan), financial models.

3 Set up Green s Function Known results Parabolic equations Solve the parabolic equation in R N : { t u Lu = g, t > 0, where L = i,j u(0) = h. a ij (x) i j + j b j (x) j + c(x), with smooth, bounded coefficients, and A = [a ij ] symmetric and positive definite ( L is strongly elliptic). Solution in term of Green s function or fundamental solution: u(x, t) = Gt L (x, y) f (y) dy.

4 Set up Green s Function Known results Parabolic equations Solve the parabolic equation in R N : { t u Lu = g, t > 0, where L = i,j u(0) = h. a ij (x) i j + j b j (x) j + c(x), with smooth, bounded coefficients, and A = [a ij ] symmetric and positive definite ( L is strongly elliptic). Solution in term of Green s function or fundamental solution: u(x, t) = Gt L (x, y) f (y) dy.

5 Set up Green s Function Known results Goal is to obtain approximations of the Green s function, explicitly computable and accurate to any order as t 0 +. Approximation obtained via elementary methods, easily implemented algorithmically. Method works for time-dependent coefficients (W. Cheng s PhD Thesis), and in certain free-boundary problems. Approximate solutions for semi-linear equations by fixed-point method (e.g. from non-linear Feynmann-Kac formula). Application to parameter estimation. E.g., reconstruct volatility from prices on contingent claims (V. Isakov for Black-Scholes)

6 Set up Green s Function Known results Goal is to obtain approximations of the Green s function, explicitly computable and accurate to any order as t 0 +. Approximation obtained via elementary methods, easily implemented algorithmically. Method works for time-dependent coefficients (W. Cheng s PhD Thesis), and in certain free-boundary problems. Approximate solutions for semi-linear equations by fixed-point method (e.g. from non-linear Feynmann-Kac formula). Application to parameter estimation. E.g., reconstruct volatility from prices on contingent claims (V. Isakov for Black-Scholes)

7 Set up Green s Function Known results Goal is to obtain approximations of the Green s function, explicitly computable and accurate to any order as t 0 +. Approximation obtained via elementary methods, easily implemented algorithmically. Method works for time-dependent coefficients (W. Cheng s PhD Thesis), and in certain free-boundary problems. Approximate solutions for semi-linear equations by fixed-point method (e.g. from non-linear Feynmann-Kac formula). Application to parameter estimation. E.g., reconstruct volatility from prices on contingent claims (V. Isakov for Black-Scholes)

8 Set up Green s Function Known results Applications/Extensions When b j (x) 0, t L is a Fokker-Planck or Forward Kolmogorov operator evolution of the p.d.f associated to the following stochastic process: dx = b(x) dt + A(X) dw (X), W Brownian Motion. (Statistical Mechanics, Probability,...). Can allow for certain singular coefficients, if bounded in the Varadhan metric A 1 (x) and the metric is of bounded geometry (curvature and its derivatives bounded). Example: Lu(x) = σ(x) x 2 2 x u(x) + 2r(x)(x x u(x) u(x)), on x > 0. The Varadhan metric is (σ(x) x 2 ) 2 dx 2.

9 Set up Green s Function Known results Applications/Extensions When b j (x) 0, t L is a Fokker-Planck or Forward Kolmogorov operator evolution of the p.d.f associated to the following stochastic process: dx = b(x) dt + A(X) dw (X), W Brownian Motion. (Statistical Mechanics, Probability,...). Can allow for certain singular coefficients, if bounded in the Varadhan metric A 1 (x) and the metric is of bounded geometry (curvature and its derivatives bounded). Example: Lu(x) = σ(x) x 2 2 x u(x) + 2r(x)(x x u(x) u(x)), on x > 0. The Varadhan metric is (σ(x) x 2 ) 2 dx 2.

10 Set up Green s Function Known results Well-posedness G L (x, y) is the Green s function for t L, i.e., the distributional kernel of the solution operator e t L. Use also notation T (x, y) to denote the kernel of operator T, if a smooth function. Error estimates sought in largest space where uniqueness holds exponentially weighted Sobolev space: W r,p a (R N ) := e a x W r,p (R N ) = {u : R N C, x α ( e a x u( ) ), L p (R N ), α r}, if r Z +, where x = (1 + x 2 ) 1/2, Such initial data arise in applications.

11 Set up Green s Function Known results Well-posedness G L (x, y) is the Green s function for t L, i.e., the distributional kernel of the solution operator e t L. Use also notation T (x, y) to denote the kernel of operator T, if a smooth function. Error estimates sought in largest space where uniqueness holds exponentially weighted Sobolev space: W r,p a (R N ) := e a x W r,p (R N ) = {u : R N C, x α ( e a x u( ) ), L p (R N ), α r}, if r Z +, where x = (1 + x 2 ) 1/2, Such initial data arise in applications.

12 Set up Green s Function Known results Heat Kernel short-time asymptotics Short-time asymptotic expansions well-known in literature: Geodesic flow (L = Laplace-Beltrami operator): G t (x, y) = d(x,y) 2 ( ) e 4t (4πt) N/2 G (0) (x, y) + G (1) (x, y)t + G (2) (x, y)t n +..., d(x, y) geodesic distance (McKean-Singer, Greiner,..). Parametrix approximation (related to WKBJ) G L (t, x, y) ) t (j n)/2 p j (x, t 1/2 (x y) e (x y)t A(x) 1 (x y) 4t, j 0 p j (x, w) a polynomial of degree j in w (Melrose, Taylor,...). Hermite function expansion (Y. Ait-Sahalia).

13 Set up Green s Function Known results Heat Kernel short-time asymptotics Short-time asymptotic expansions well-known in literature: Geodesic flow (L = Laplace-Beltrami operator): G t (x, y) = d(x,y) 2 ( ) e 4t (4πt) N/2 G (0) (x, y) + G (1) (x, y)t + G (2) (x, y)t n +..., d(x, y) geodesic distance (McKean-Singer, Greiner,..). Parametrix approximation (related to WKBJ) G L (t, x, y) ) t (j n)/2 p j (x, t 1/2 (x y) e (x y)t A(x) 1 (x y) 4t, j 0 p j (x, w) a polynomial of degree j in w (Melrose, Taylor,...). Hermite function expansion (Y. Ait-Sahalia).

14 Set up Green s Function Known results Implementation issues Our method is also based on a parametrix, but more easily implementable. It combines well with numerical methods to solve over large time intervals. Geodesic flow approximation is very accurate, but difficult to implement in practice. Except in special cases, geodesics must be computed numerically. Also, extension to time dependent coefficients not straightforward. Challenging to solve the PDE directly (unbounded domain, high dimensionality in certain problems, degeneracy of L) Monte Carlo methods to obtain the p.d.f. of a Markov process are generally slow and not very accurate.

15 Set up Green s Function Known results Implementation issues Our method is also based on a parametrix, but more easily implementable. It combines well with numerical methods to solve over large time intervals. Geodesic flow approximation is very accurate, but difficult to implement in practice. Except in special cases, geodesics must be computed numerically. Also, extension to time dependent coefficients not straightforward. Challenging to solve the PDE directly (unbounded domain, high dimensionality in certain problems, degeneracy of L) Monte Carlo methods to obtain the p.d.f. of a Markov process are generally slow and not very accurate.

16 Set up Green s Function Known results Implementation issues Our method is also based on a parametrix, but more easily implementable. It combines well with numerical methods to solve over large time intervals. Geodesic flow approximation is very accurate, but difficult to implement in practice. Except in special cases, geodesics must be computed numerically. Also, extension to time dependent coefficients not straightforward. Challenging to solve the PDE directly (unbounded domain, high dimensionality in certain problems, degeneracy of L) Monte Carlo methods to obtain the p.d.f. of a Markov process are generally slow and not very accurate.

17 Preliminaries Main result Our expansion Based on Taylor expansion of parabolic rescaling L s,z of operator L: L s,z := N i,j=1 a s,z ij (x) i j + s N i=1 f s,z (t, x) := f (s 2 t, z + s(x z)), b s,z i (x) i + s 2 c s,z (x), z dilation center, s dilation parameter (eventually, s = t). Dilation center z allowed to be a function of x, y (may improve accuracy). Taylor expansion coupled with time-ordered perturbative expansion via Duhamel s principle.

18 Preliminaries Main result Our expansion Based on Taylor expansion of parabolic rescaling L s,z of operator L: L s,z := N i,j=1 a s,z ij (x) i j + s N i=1 f s,z (t, x) := f (s 2 t, z + s(x z)), b s,z i (x) i + s 2 c s,z (x), z dilation center, s dilation parameter (eventually, s = t). Dilation center z allowed to be a function of x, y (may improve accuracy). Taylor expansion coupled with time-ordered perturbative expansion via Duhamel s principle.

19 Preliminaries Main result Definition z : R 2N R N is admissible if z(x, x) = x, and all derivatives are bounded. Ex: z = x, z = x+y 2. For each µ Z + and z = z(x, y) admissible, let: G [µ,z] t (x, y) := t N/2 µ l=0 where P l (z, x, y) = a α,β smooth, bounded. t l/2 P l ( z, z + x z t 1/2, z + y z ) t1/2 1 4πt N det(a(z) e (x y)t A 1 (z)(x y) 4t ), α l,β 3l a α,β (z)(x z) α (x y) β,

20 Preliminaries Main result Definition z : R 2N R N is admissible if z(x, x) = x, and all derivatives are bounded. Ex: z = x, z = x+y 2. For each µ Z + and z = z(x, y) admissible, let: G [µ,z] t (x, y) := t N/2 µ l=0 where P l (z, x, y) = a α,β smooth, bounded. t l/2 P l ( z, z + x z t 1/2, z + y z ) t1/2 1 4πt N det(a(z) e (x y)t A 1 (z)(x y) 4t ), α l,β 3l a α,β (z)(x z) α (x y) β,

21 Preliminaries Main result Main Result Theorem Let µ Z +, z = z(x, y) an admissible function. Then, a α,β explicitly computable such that e t L f (x) = G [µ,z] t R N (x, y)f (y)dy + t (µ+1)/2 E [µ,z] f (x). where, for any a R, m R +, 1 < p <, k Z +, E [µ,z] t f W m+k,p Ct k/2 f W m,p, G [µ,z] t (x, y) is the µth-order approximate kernel for the solution operator e t L. E [µ,z] t is the error operator. t

22 Preliminaries Main result Main Result Theorem Let µ Z +, z = z(x, y) an admissible function. Then, a α,β explicitly computable such that e t L f (x) = G [µ,z] t R N (x, y)f (y)dy + t (µ+1)/2 E [µ,z] f (x). where, for any a R, m R +, 1 < p <, k Z +, E [µ,z] t f W m+k,p Ct k/2 f W m,p, G [µ,z] t (x, y) is the µth-order approximate kernel for the solution operator e t L. E [µ,z] t is the error operator. t

23 Preliminaries Main result Some remarks Expansion agrees with usual parametrix if z = x and L time independent. Can optimize choice of z(x, y) (work in progress). When z = x, our construction is equivalent to Taylor s expanding the Green s function. Error estimates are global on R N (generalize to non-compact, complete manifolds). In 1D, solution has closed form in term of Error Functions if initial data is piece-wise polynomial (e.g. pricing of contingent claims).

24 Preliminaries Main result Some remarks Expansion agrees with usual parametrix if z = x and L time independent. Can optimize choice of z(x, y) (work in progress). When z = x, our construction is equivalent to Taylor s expanding the Green s function. Error estimates are global on R N (generalize to non-compact, complete manifolds). In 1D, solution has closed form in term of Error Functions if initial data is piece-wise polynomial (e.g. pricing of contingent claims).

25 Preliminaries Main result Some remarks Expansion agrees with usual parametrix if z = x and L time independent. Can optimize choice of z(x, y) (work in progress). When z = x, our construction is equivalent to Taylor s expanding the Green s function. Error estimates are global on R N (generalize to non-compact, complete manifolds). In 1D, solution has closed form in term of Error Functions if initial data is piece-wise polynomial (e.g. pricing of contingent claims).

26 Preliminaries Main result Some remarks Expansion agrees with usual parametrix if z = x and L time independent. Can optimize choice of z(x, y) (work in progress). When z = x, our construction is equivalent to Taylor s expanding the Green s function. Error estimates are global on R N (generalize to non-compact, complete manifolds). In 1D, solution has closed form in term of Error Functions if initial data is piece-wise polynomial (e.g. pricing of contingent claims).

27 Pertubative expansion Commutator Estimates Error Estimates Proof - Duhamel s Formula By Duhamel s principle, reduce to t u Lu = 0. e tl analytic semigroup on W m,p a u(t) W r,p a Ct (s r)/2 f s,p W, t (0, 1]. a r s, 1 < p <, C independent of t and a in bounded set. The map is smooth for any s, r. (0, ) t e tl B(Wa s,p, Wa r,p ) Set V = L L 0, L 0 given operator in the same class as L.

28 Pertubative expansion Commutator Estimates Error Estimates Proof - Duhamel s Formula By Duhamel s principle, reduce to t u Lu = 0. e tl analytic semigroup on W m,p a u(t) W r,p a Ct (s r)/2 f s,p W, t (0, 1]. a r s, 1 < p <, C independent of t and a in bounded set. The map is smooth for any s, r. (0, ) t e tl B(Wa s,p, Wa r,p ) Set V = L L 0, L 0 given operator in the same class as L.

29 Pertubative expansion Commutator Estimates Error Estimates Proof - Duhamel s Formula By Duhamel s principle, reduce to t u Lu = 0. e tl analytic semigroup on W m,p a u(t) W r,p a Ct (s r)/2 f s,p W, t (0, 1]. a r s, 1 < p <, C independent of t and a in bounded set. The map is smooth for any s, r. (0, ) t e tl B(Wa s,p, Wa r,p ) Set V = L L 0, L 0 given operator in the same class as L.

30 Pertubative expansion Commutator Estimates Error Estimates Iterating Duhamel s give time-ordered (Dyson) expansion: e tl = e tl 0 + t e tτ 0L 0 Ve tτ 1L 0 dτ Σ 1 + t Σ 2 e tτ 0L 0 Ve tτ 1L 0 Ve tτ 2L 0 dτ t d Σ p e tτ 0L 0 Ve tτ 1L 0... e tτ d 1L 0 Ve tτ d L 0 dτ + t d+1 Σ d+1 e tτ 0L 0 Ve tτ 1L 0... e tτ d L 0 Ve τ d+1l dτ, Σ k k-dim unit simplex, d iteration level. Integrals are Banach-valued Riemann integrals.

31 Pertubative expansion Commutator Estimates Error Estimates Proof - Dilation and Taylor expansion Given fixed point z (center), s > 0, recall L s,z := N i,j=1 a s,z ij (x) i j + s N i=1 b s,z i (x) i + s 2 c s,z (x) G L t (x, y) = s N G Ls,z 1 (z + s 1 (x z), z + s 1 (y z)), t = s 2. Taylor expand L s,z to order n = d in s at 0: L s,z = n m=0 s m L z m + s n+1 L s,z n+1.

32 Pertubative expansion Commutator Estimates Error Estimates Proof - Dilation and Taylor expansion Given fixed point z (center), s > 0, recall L s,z := N i,j=1 a s,z ij (x) i j + s N i=1 b s,z i (x) i + s 2 c s,z (x) G L t (x, y) = s N G Ls,z 1 (z + s 1 (x z), z + s 1 (y z)), t = s 2. Taylor expand L s,z to order n = d in s at 0: L s,z = n m=0 s m L z m + s n+1 L s,z n+1.

33 Pertubative expansion Commutator Estimates Error Estimates Proof - Dilation and Taylor expansion Given fixed point z (center), s > 0, recall L s,z := N i,j=1 a s,z ij (x) i j + s N i=1 b s,z i (x) i + s 2 c s,z (x) G L t (x, y) = s N G Ls,z 1 (z + s 1 (x z), z + s 1 (y z)), t = s 2. Taylor expand L s,z to order n = d in s at 0: L s,z = n m=0 s m L z m + s n+1 L s,z n+1.

34 Pertubative expansion Commutator Estimates Error Estimates Choose L 0 = L z 0 = ij aij (z) i j, constant-coefficient operator: e Lz 0 (x, y) = 1 4π det(a(z)) e (x y)t A 1 (z)(x y) 4. For each µ n, Dyson expansion becomes: e Ls,z = e Lz 0 + µ l=1 max(l,n+1) s l Λ l z + s l Λ l z = l=µ+1 E s,z µ error operator, Λ l z = Λ l z,s, if l > n. µ l=0 s l Λ l z + s µ+1 E s,z µ,

35 Pertubative expansion Commutator Estimates Error Estimates Choose L 0 = L z 0 = ij aij (z) i j, constant-coefficient operator: e Lz 0 (x, y) = 1 4π det(a(z)) e (x y)t A 1 (z)(x y) 4. For each µ n, Dyson expansion becomes: e Ls,z = e Lz 0 + µ l=1 max(l,n+1) s l Λ l z + s l Λ l z = l=µ+1 E s,z µ error operator, Λ l z = Λ l z,s, if l > n. µ l=0 s l Λ l z + s µ+1 E s,z µ,

36 Pertubative expansion Commutator Estimates Error Estimates Where, with Λ 0 z = e Lz 0 set: Λ l z := α A l Λ α,z, A k,l := {α = (α 1, α 2,..., α k ) N k : α = l}, A l := l k=1 A k,l. And if 1 k n, Λ α,z := e τ0lz 0L z α1 e τ1lz 0L z α2 L z α k e τ k Lz 0dτ, Σ k while if k = n + 1, Λ α,z := e τ0lz 0L z α1 e τ1lz 0L z α2 L z α d+1 e τd+1ls,z dτ. Σ d+1

37 Pertubative expansion Commutator Estimates Error Estimates Where, with Λ 0 z = e Lz 0 set: Λ l z := α A l Λ α,z, A k,l := {α = (α 1, α 2,..., α k ) N k : α = l}, A l := l k=1 A k,l. And if 1 k n, Λ α,z := e τ0lz 0L z α1 e τ1lz 0L z α2 L z α k e τ k Lz 0dτ, Σ k while if k = n + 1, Λ α,z := e τ0lz 0L z α1 e τ1lz 0L z α2 L z α d+1 e τd+1ls,z dτ. Σ d+1

38 Pertubative expansion Commutator Estimates Error Estimates Where, with Λ 0 z = e Lz 0 set: Λ l z := α A l Λ α,z, A k,l := {α = (α 1, α 2,..., α k ) N k : α = l}, A l := l k=1 A k,l. And if 1 k n, Λ α,z := e τ0lz 0L z α1 e τ1lz 0L z α2 L z α k e τ k Lz 0dτ, Σ k while if k = n + 1, Λ α,z := e τ0lz 0L z α1 e τ1lz 0L z α2 L z α d+1 e τd+1ls,z dτ. Σ d+1

39 Pertubative expansion Commutator Estimates Error Estimates Proof - Commutators If α A l, L z α differential operator of order l + 2 and degree l polynomial coefficients (say L z α D(l, l + 2)). Campbell-Baker-Hasdorff formula then gives: e θl 0 L z α = P α (θ)e θl 0, where P α (θ) = P α (L 0, L z α; θ, x, ) D(l, l + 2) given by P l (θ) := l k=0 θ k k! adk L 0 (L z α)

40 Pertubative expansion Commutator Estimates Error Estimates Proof - Commutators If α A l, L z α differential operator of order l + 2 and degree l polynomial coefficients (say L z α D(l, l + 2)). Campbell-Baker-Hasdorff formula then gives: e θl 0 L z α = P α (θ)e θl 0, where P α (θ) = P α (L 0, L z α; θ, x, ) D(l, l + 2) given by P l (θ) := l k=0 θ k k! adk L 0 (L z α)

41 Pertubative expansion Commutator Estimates Error Estimates Proof - Expansion revisited For l n, then have Λ z α = with Σ k i=1 k P αi (L z 0, Lz α i ; 1 σ i, x, )dσe Lz 0 := Pα (x, z, )e Lz 0, P α (x, z, ) = β l γ l+2k a β,γ smooth, bounded function. a β,γ (z)(x z) β γ x, Using explicit formula for e Lz 0(x, y), dilating back, substituting z = z(x, y) gives the final formula for the approximation kernel.

42 Pertubative expansion Commutator Estimates Error Estimates Proof - Expansion revisited For l n, then have Λ z α = with Σ k i=1 k P αi (L z 0, Lz α i ; 1 σ i, x, )dσe Lz 0 := Pα (x, z, )e Lz 0, P α (x, z, ) = β l γ l+2k a β,γ smooth, bounded function. a β,γ (z)(x z) β γ x, Using explicit formula for e Lz 0(x, y), dilating back, substituting z = z(x, y) gives the final formula for the approximation kernel.

43 Pertubative expansion Commutator Estimates Error Estimates Proof - Expansion revisited For l n, then have Λ z α = with Σ k i=1 k P αi (L z 0, Lz α i ; 1 σ i, x, )dσe Lz 0 := Pα (x, z, )e Lz 0, P α (x, z, ) = β l γ l+2k a β,γ smooth, bounded function. a β,γ (z)(x z) β γ x, Using explicit formula for e Lz 0(x, y), dilating back, substituting z = z(x, y) gives the final formula for the approximation kernel.

44 Pertubative expansion Commutator Estimates Error Estimates Error Estimates Let z = z(x, y) admissible, s = t 1/2. Two types of error terms in E [µ,z] t, operator with kernel: s N E s,z µ (z + s 1 (x z), z + s 1 (y z)). 1. For µ < l n, operators L s,l with kernel s N Λ l z(z + s 1 (x z), z + s 1 (y z)). 2. For l n + 1, operators L s,l with kernel s N Λ s,l z (z + s 1 (x z), z + s 1 (y z)).

45 Pertubative expansion Commutator Estimates Error Estimates Error Estimates Let z = z(x, y) admissible, s = t 1/2. Two types of error terms in E [µ,z] t, operator with kernel: s N E s,z µ (z + s 1 (x z), z + s 1 (y z)). 1. For µ < l n, operators L s,l with kernel s N Λ l z(z + s 1 (x z), z + s 1 (y z)). 2. For l n + 1, operators L s,l with kernel s N Λ s,l z (z + s 1 (x z), z + s 1 (y z)).

46 Pertubative expansion Commutator Estimates Error Estimates 1. For µ < l n, obtain error bounds uniformly in s in W r,p a, for all r R by showing: L s,l = b s (x, ), b s (x, ξ) = a s (x, sξ), for some family of symbols a s bounded in S 0 1,0. 2. For l n + 1, use Riesz Lemma along with: β x β z y β Λ s,l z (x, y) = β δ x, β z Λ s,l z β δ y L s,l W r,p W r,p C T t r/2, t (0, T ]. This bound not optimal, but sufficient to prove sharp estimate for E [µ,z] t by choosing n > µ + N 1.

47 Pertubative expansion Commutator Estimates Error Estimates 1. For µ < l n, obtain error bounds uniformly in s in W r,p a, for all r R by showing: L s,l = b s (x, ), b s (x, ξ) = a s (x, sξ), for some family of symbols a s bounded in S 0 1,0. 2. For l n + 1, use Riesz Lemma along with: β x β z y β Λ s,l z (x, y) = β δ x, β z Λ s,l z β δ y L s,l W r,p W r,p C T t r/2, t (0, T ]. This bound not optimal, but sufficient to prove sharp estimate for E [µ,z] t by choosing n > µ + N 1.

48 1D Models Numerics - 1D Formulas Set z = x and denote G µ,z t =: G µ t. If L(x) = 1 2 Σ(x)2 x 2 + µ(x) x ρ(x), then first-order approx kernel given by: G [1] t (x, y; z) = 1 2π(t)Σ(z) 2 x y 2 [( e 2tΣ(z) Σ(z)Σ (z) 2µ(z) 2Σ(z) 2 Σ (z) 2tΣ(z) 3 (x y)3 + (x z) ( (x y) 2 tσ(z) 2 tσ(z) 3 ) (x y) )].

49 1D Models Numerics - 1D Formulas Set z = x and denote G µ,z t =: G µ t. If L(x) = 1 2 Σ(x)2 x 2 + µ(x) x ρ(x), then first-order approx kernel given by: G [1] t (x, y; z) = 1 2π(t)Σ(z) 2 x y 2 [( e 2tΣ(z) Σ(z)Σ (z) 2µ(z) 2Σ(z) 2 Σ (z) 2tΣ(z) 3 (x y)3 + (x z) ( (x y) 2 tσ(z) 2 tσ(z) 3 ) (x y) )].

50 1D Models Numerics In pricing of contingent claims, initial data u(x, 0) = (x K ) +. h is the claim pay-off and parameter K is the strike price. Then, first-order approx solution given by u [1] (t, x) = t 2 2π e + 1 ( 2 erf (x K ) 2 ( 2Σ(x) 2 t 2Σ(x) Σ (x)(x K ) ) ( ) ) x K + 1 (µ(x)t + x K ) 2tΣ(x) Can easily compute 2nd-order approximation as well.

51 1D Models Numerics In pricing of contingent claims, initial data u(x, 0) = (x K ) +. h is the claim pay-off and parameter K is the strike price. Then, first-order approx solution given by u [1] (t, x) = t 2 2π e + 1 ( 2 erf (x K ) 2 ( 2Σ(x) 2 t 2Σ(x) Σ (x)(x K ) ) ( ) ) x K + 1 (µ(x)t + x K ) 2tΣ(x) Can easily compute 2nd-order approximation as well.

52 1D Models Numerics An example Consider the CEV model (J.C. Cox, S. A. Ross) for pricing contingent claims: L(x) = 1 2 σ2 x 2β 2 x + rx x r, β > 0. Exact formula for β = 1, 2/3, otherwise series solution (Cox-Ross, D.Emanuel-J. MacBeth) in terms of Bessel s functions slow to compute. Varadhan metric is complete if β 1, our theorem applies. For β = 1, metric is Poincaré metric. Varadhan metric is incomplete if β < 1, but our formula gives a very accurate approximation nevertheless.

53 1D Models Numerics An example Consider the CEV model (J.C. Cox, S. A. Ross) for pricing contingent claims: L(x) = 1 2 σ2 x 2β 2 x + rx x r, β > 0. Exact formula for β = 1, 2/3, otherwise series solution (Cox-Ross, D.Emanuel-J. MacBeth) in terms of Bessel s functions slow to compute. Varadhan metric is complete if β 1, our theorem applies. For β = 1, metric is Poincaré metric. Varadhan metric is incomplete if β < 1, but our formula gives a very accurate approximation nevertheless.

54 1D Models Numerics An example Consider the CEV model (J.C. Cox, S. A. Ross) for pricing contingent claims: L(x) = 1 2 σ2 x 2β 2 x + rx x r, β > 0. Exact formula for β = 1, 2/3, otherwise series solution (Cox-Ross, D.Emanuel-J. MacBeth) in terms of Bessel s functions slow to compute. Varadhan metric is complete if β 1, our theorem applies. For β = 1, metric is Poincaré metric. Varadhan metric is incomplete if β < 1, but our formula gives a very accurate approximation nevertheless.

55 1D Models Numerics Assume again initial data u(x, 0) = h(x) = (x K ) +. Our 1st-order approximate solution for z = x: u [1] CEV (t, x) = σx β 1 t (x K ) 2 2 2π e 2σ 2 x 2β t ((2 β)x βk ) ( erf ( x K 2tσx β ) ) + 1 ((1 + rt)x K ) Compare exact formula with 1st-order approximation, when β = 2/3. Closed-form solution avoid issues with numerical integration against Green s function. In practice, important to approximate solution well near x = K.

56 1D Models Numerics Assume again initial data u(x, 0) = h(x) = (x K ) +. Our 1st-order approximate solution for z = x: u [1] CEV (t, x) = σx β 1 t (x K ) 2 2 2π e 2σ 2 x 2β t ((2 β)x βk ) ( erf ( x K 2tσx β ) ) + 1 ((1 + rt)x K ) Compare exact formula with 1st-order approximation, when β = 2/3. Closed-form solution avoid issues with numerical integration against Green s function. In practice, important to approximate solution well near x = K.

57 1D Models Numerics Numerical Test - CEV Model β = 2 3, K = 15, σ = 0.3, r = 0.1 Error is O(10 3 ) U u exact formula 1st order approx exact formula 1st order approx =K X t = =K x t = 0.5

58 Bootstrap Scheme Numerics Bootstrap to large time Discuss only second-order approximation. e tl = ( e t/nl) n. Observe Error in the second-order approx over time t is O(t) 3/2 ). Time step t/n small if n large, so error over one time step is small error O( ( t n) 3/2). Denote by G [2] t/n step t/n. the approximate solution operator over the time ( n. G t/n) [2] Compare e tl with After n steps, total error is O ( (t/n) 3/2) n = O ( t 3/2 / n ) 0, n.

59 Bootstrap Scheme Numerics Bootstrap to large time Discuss only second-order approximation. e tl = ( e t/nl) n. Observe Error in the second-order approx over time t is O(t) 3/2 ). Time step t/n small if n large, so error over one time step is small error O( ( t n) 3/2). Denote by G [2] t/n step t/n. the approximate solution operator over the time ( n. G t/n) [2] Compare e tl with After n steps, total error is O ( (t/n) 3/2) n = O ( t 3/2 / n ) 0, n.

60 Bootstrap Scheme Numerics Bootstrap to large time Discuss only second-order approximation. e tl = ( e t/nl) n. Observe Error in the second-order approx over time t is O(t) 3/2 ). Time step t/n small if n large, so error over one time step is small error O( ( t n) 3/2). Denote by G [2] t/n step t/n. the approximate solution operator over the time ( n. G t/n) [2] Compare e tl with After n steps, total error is O ( (t/n) 3/2) n = O ( t 3/2 / n ) 0, n.

61 Bootstrap Scheme Numerics Error is not improved by bootstrap for first-order approximation (error is O(t 1/2 ). Prove rigorous error bound by using equivalent norm for which e tl is contractive. Neglect error from numerical quadrature at each time step. After first time step, no closed-form solutions available. Quadrature error negligible if space discretization fine enough. For t > 0 integrand is smooth, can use high-order methods away from y = x (e.g. high-order Gaussian quadrature). Neglect error from truncation of domain (small if use compactly supported data).

62 Bootstrap Scheme Numerics Error is not improved by bootstrap for first-order approximation (error is O(t 1/2 ). Prove rigorous error bound by using equivalent norm for which e tl is contractive. Neglect error from numerical quadrature at each time step. After first time step, no closed-form solutions available. Quadrature error negligible if space discretization fine enough. For t > 0 integrand is smooth, can use high-order methods away from y = x (e.g. high-order Gaussian quadrature). Neglect error from truncation of domain (small if use compactly supported data).

63 Bootstrap Scheme Numerics Error is not improved by bootstrap for first-order approximation (error is O(t 1/2 ). Prove rigorous error bound by using equivalent norm for which e tl is contractive. Neglect error from numerical quadrature at each time step. After first time step, no closed-form solutions available. Quadrature error negligible if space discretization fine enough. For t > 0 integrand is smooth, can use high-order methods away from y = x (e.g. high-order Gaussian quadrature). Neglect error from truncation of domain (small if use compactly supported data).

64 Bootstrap Scheme Numerics Numerical test - CEV Model β = 1, K = 20, σ = 0.3, r = 0.1, t = 1, n = 10 Error is O(10 5 ) with bootstrap nd order approx. with bootstrap 1st order approx. 2nd order approx. 1st order approx. with bootstrap nd order approx. with bootstrap 1st order aprox. 2nd order approx. 1st order approx. with bootstrap u 1 u =K x Truncate at x = =K x Truncate at x = 400