Interpolation and function representation with grids in stochastic control

Transcription

1 with in classical Sparse for Sparse for with in FIME EDF R&D & FiME, Laboratoire de Finance des Marchés de l Energie ( September 2015

2 Schedule with in classical Sparse for Sparse for 1 2 classical 3 Sparse for Sparse for

3 Where is it possible to get software with effective in? with in classical Sparse for Sparse for Should be open for the end of this year, Most of what is presented below is included 1 C++ library, multi OS 2 MPI threaded version 3 Python binding for python users 4 Extensive documentation...

4 Including the following resolution with in classical Sparse for Sparse for 1 Regression (adaptive local, sparse, global polynomials) 2 Dynamic Programming with Longstaff Schwartz dealing with stocks (for storage etc...) 3 for HJB equations (with linear interpolator, high order interpolators for full sparse ) 4 Stochastic Dual Dynamic Programming for multi stock optimization 5 Framework for optimization simulation of the optimal.

5 Should be considered in future versions with in classical Sparse for Sparse for Adaptation for sparse (local adaptation dimension adaptation) Finite Difference for singular problems, When open, don t hesitate to test send feed back..

6 Interest of the /interpolation of a function in with in classical Sparse for Sparse for Calculation of conditional expectation (s...) Evaluation of a function depending on stocks (interpolation due to dynamic programming) (characteristics...) needs interpolation on a grid...

7 Linear interpolation with in classical Sparse for Sparse for The most common interpolation : given some regular meshes, interpolation linearly between the values on the mesh. Loved the theorist (permit to keep monotone, stable etc...) easy to tensorize in dimension d Slow convergence of the interpolator I 1, x : mesh x = ( x 1,..., x d ), f C k+1 (R d ) with k 1 f I 1, x f c If f is only Lispchitz d i=1 x k+1 i f I 1, x f K sup x i i sup k+1 f x [ 1,1] d x k+1 i

8 Slow convergence while refining with in classical Sparse for Sparse for Figure: Linear interpolator error for sin(2πx)

9 Spectral finite elements interpolator with in classical Sparse for Sparse for On a grid [ 1, 1], N + 1 points X = (x 0,.., x N ), interpolate by the unique polynomial of degree N such that I X N (f )(x i) = f (x i ), 0 i N. Introduce the Lagrange polynomials li X, 0 i N (satisfying li X (x j ) = δ i=j ) I X N (f )(x) = N i=0 f (x i )li X (x)

10 Stability convergence of the interpolator with in Stard results with λ N (X ) = max N x [ 1,1] i=0 l i X (x) the Lebesgue constant : IN X (f )(x) λ N (X ) f Difficult to classical Sparse for Sparse for I X N (f )(x) f Cλ N (X )w(f, 1 N ) where w is the modulus of continuity w(f, δ) = sup x 1, x 2 [ 1, 1] x 1 x 2 < δ f (x 1 ) f (x 2 ) Try the find the with the best λ N. Erdös theorem : λ N (X ) > 2 log(n + 1) C Π

11 Runge effect :uniform grid X u is not optimal with in λ N (X u ) 2N+1 enlnn classical Sparse for Sparse for

12 with optimal Lebesgue constant : Gauss Legendre Lobatto with in classical Sparse for Sparse for The in [ 1, 1] : η 1 = 1, η N+1 = 1, η i (i = 2,..., N) zeros of L N. Legendre polynomials satisfies : (N + 1)L N+1 (x) = (2N + 1)xL N (x) NL N 1 (x) Lebesgue constant λ N (X GLL ) 2 Πln(N + 1). formula on [ 1, 1] I N (f ) = N f k L k (x), k=0 f k = 1 N ρ i f (η i )L k (η i ), γ k γ k = i=0 N L k (η i ) 2 ρ i, i=0 ρ i = 2. (M+1)ML 2 M (η i ), 1 i N + 1.

13 Runge effect on Gauss Legendre Lobatto with in classical Sparse for Sparse for

14 Back to the problem of the meshes with GLL interpolator with in Keep 4 meshes increase the polynomial approximation on each mesh. classical Sparse for Sparse for

15 Mesh construction dimension d with in Define approximation by tensorization : classical Sparse for Sparse for Figure: Gauss Legendre Lobatto points on 2 2 meshes

16 results in dimension d with meshes of size x = ( x 1, x 2.., x d ) with in classical Sparse for Sparse for If f C k+1 ([ 1, 1] d ), k N f I X N, x f c (1 + λ N(X )) d N k d i=1 x k+1 i sup k+1 f x [ 1,1] d x k+1 Of course, accuracy limited by the regularity of the solution. Is there a more effective way than tensorization to deal with functions in dimension d? i

17 Linear hierarchical in 1D with in classical Sparse for Sparse for Hat function : φ (L) (x) = max(1 x, 0), By dilatation ( level l) translation (given by i) φ (L) l,i (x) = φ (L) (2 l x i) { } W (L) l := span φ (L) l,i (x) : 1 i 2 l 1, i odd space V n = W (L) l l n Nodal equivalent : { } V n = span φ (L) n,i (x) : 1 i 2l 1

18 Example in 1 D with in classical Sparse for Sparse for Figure: One dimensional W (L) spaces : W (L) 1, W (L) 2, W (L) 3, W (L) 4 the nodal W (L,N) 4

19 in 1 D with in I (L) (f )(x) = l n,1 i 2 l 1, α (L) l,i φ (L) l,i (x) i odd where for m = x l,i classical Sparse for Sparse for α (L) (m) := α (L) l,i = f (m) 0.5(f (e(m)) + f (w(m))) where e(m) is the east neighbor of m w(m) the west one.

20 Nodal versus hierarchical approach with in classical Sparse for Sparse for Figure: Example of hierarchical coefficients values : estimation of the discrete second derivative of the function : adaptation possible by refining at node with highest hierarchical values, Same solution as linear interpolation in 1 D, But basis function supports intersect a lot : full matrix appearing in numerical.

21 Extension of the Sparse grid to dimension d with in classical Sparse for Sparse for Basis functions : φ (L) l,i (x) = d j=1 φ(l) l j,i j (x j ) for x = (x 1,...x d ), a multi-level l := (l 1,.., l d ) a multi-index i := (i 1,.., i d ). with { } B l := i : 1 i j 2 l j 1, i j odd, 1 j d Sparse grid { } W (L) l := span φ (L) l,i (x) : i B l V n = W (L) l l 1 n+d 1 Full grid V F n = l n W (L) l Possible V n in term of nodal basis.

22 2 Full Sparse basis with in Representation of the W subspace for l 3 in dimension 2. classical Sparse for Sparse for Figure: The two dimensional subspace W (L) l up to l = 3. Additional hierarchical functions corresponding to the full grid in dashed lines.

23 Error associated to linear space grid with in classical Sparse for Sparse for With roughly a n,d = 2 n nd 1 (d 1)! [3, 5, 6] points, if 2d u x x2 d <, f I 1 (f ) = O(2 2n log(2 n ) d 1 ) (1) to compare to the full grid interpolator f I 1 (f ) = O(2 2n ) (2) with 2 dn points. The number of points increases slowly with the dimension with sparse with an error slowly above the full grid.

24 Incorporation boundary points with in Either add boundary points either modify the basis functions near the boundary. classical Sparse for Sparse for Figure: One dimensional W (L) spaces with linear functions with exact boundary (left) modified boundary (right) : W (L) 1, W (L) 2, W (L) 3, W (L) 4

25 Grids with boundary points with in classical Sparse for Sparse for Figure: Sparse grid in dimension 2 3 with boundary points

26 Grids with extrapolation with in classical Sparse for Sparse for Figure: Sparse grid in dimension 2 3 without boundary points

27 Incorporation boundary points with in classical Sparse for Sparse for first method : explosion of the number of points with the dimension in dimension 5, 2.8 millions points for 6401 inside the domain, can t deal with some very high dimension problems second method : only effective if boundary points non important Depending on the problem, stocks problems need accurate boundary treatment (dimension limited to 5) : first method needed PDE resolution in infinite domain, for conditional expectation will use the second method (dimension 7 to 10 )

28 Quadratic sparse with in classical Sparse for Sparse for How to upgrade the convergence properties of sparse without adding points... On [2 l (i 1), 2 l (i + 1)] by with φ (Q) (x) = 1 x 2. I (Q) (f )(x) = φ (Q) l,i (x) = φ (Q) (2 l x i) l n,1 i 2 l 1,i odd α (Q) l,i φ (Q) l,i (x)

29 quadratic (cont) with in classical Sparse for Sparse for where for m = x l,i α(m) (Q) = f (m) ( 3 8 f (w(m)) f (e(m)) 1 8 f (ee(m))) = α(m) (L) (m) 1 4 α(m)(l) (e(m)) = α(m) (L) (m) 1 4 α(m)(l) (df (m)) where df (m) is the direct father of the node m in the tree.

30 Quadratic basis functions incorporating boundary points with in classical Sparse for Sparse for Figure: One dimensional W (Q) spaces with quadratic with exact boundary (left) modified boundary (right) : W (Q) 1, W (Q) 2, W (Q) 3, W (Q) 4

31 High order error with in classical Sparse for Sparse for The methodology can be used for Cubic, Quartic... interpolators, { } if sup αi {2,..,p+1} α α d u < then for x α x α d d I 2 := I (Q), I 3 := I (C) by : f I p (f ) = O(2 n(p+1) log(2 n ) d 1 ), p = 2, 3

32 Adaptation for sparse with in classical Sparse for Sparse for Two strategies: Local adaptation : Refine nodes with highest surplus adding sons (left right) in all directions, Derefine (if use in temporal problem) if surplus calculated to small, Not sure to refine/derefine at the good points, Necessity to be sure that all added nodes have fathers in all directions (if not, add points) Dimension adaptation: select a multilevel according to an error estimation refine all the points at this level in all directions. The adaptation to choose should depend on the problem...

33 Benchmarks for bermudean options with in classical Sparse for Sparse for d assets with same characteristics S i 0 = 1, σi = 0.2, no correlation Maturity T = 1, interest rate r = 0.05, strike K = 1., t = 1 10, Exercise dates j t, j = 1, T / t, pay off (K 1 d ST i d )+ basket american put. Reference calculated in [7]. i=1 Use sparse with approximated boundary treatment.

34 First results dimension 1, 3 with in classical Sparse for Sparse for Type LINEAR QUADRATIC Nb particles 1e5 1e5 1e5 1e5 Sparse grid level Results Error 1.8e-05 1.e-5 7.9e e-05 Time (seconds) Table: Results in dimension 1 Type LINEAR QUADRATIC Nb particles 5e5 5e5 5e5 5e5 5e5 5e5 Sparse grid level Results Error 4.4e e-4 3e-5 4.4e e-4 4e-5 Time (seconds) Table: Results in dimension 3 No interest in high order approximation : solution not enough regular?

35 First results dimension 5, 6 (Linear sparse ) with in classical Sparse for Sparse for Nb particles 2.5e6 2.5e6 2.5e6 Sparse grid level Results Error 5e-4 3.e-4 1.4e-4 Time (seconds) Table: Results in dimension 5 Nb particles 12.5e6 12.5e6 12.5e6 Sparse grid level Results Error Time (seconds) Table: Results in dimension 6

36 First conclusion for the use of sparse for s with in classical Sparse for Sparse for Till dimension 6, provide good results for american style basket options, No reference in dimension above 6, Comparing to [7], less particles seem to be necessary. Not sure sparse for s are competitive (till dimension 6), Cost of the method is in the matrix construction (nearly a full one).

37 HJB problem with in classical Sparse for Sparse for ( v 1 (t, x) inf t a A 2 tr(σ a(t, x)σ a (t, x) T D 2 v(t, x)) + b a (t, x)dv(t, x) ) +c a (t, x)v(t, x) + f a (t, x) = 0 in Q v(0, x) = g(x) in R d (3) where Q := (0, T ] R d, A complete metric space, σ a (t, x) is a d q matrix, b a f a coefficients functions defined on Q in R d R. HJB associated to a led process Xs t,x Brownian motion: dx t,x s with W s a q dimensional = b a (s, Xs t,x )dt + σ a (t, Xs t,x )dw s (4)

38 1D example of scheme with in classical Sparse for Sparse for Using development of φ φ(x + b ah + σ a h) = φ(x) + hbadφ(x) + hσ adφ + σ2 a h 2 D2 φ + σ 2 a h3/2 D 3 φ + O(h 2 ) 6 φ(x + b ah σ a h) = φ(x) + hbadφ(x) hσ adφ + σ2 a h 2 D2 φ σ 2 a h3/2 D 3 φ + O(h 2 ) 6 So (φ(x + b ah + σ a h) + φ(x + bah σ a h) 2φ(x)) 2hbaDφ(x) + hσ 2 a D2 φ + O(h 2 ) And use explicit scheme 1 v(t + h, x) = v(t, x) + inf a A 2 [(v(t, φ+ a,h (t, x)) + v(t, φ a,h (t, x)) 2v(t, x)) + hc a(t, x)v(t, x) + hf a(t, x)] φ + a,h (t, x) = x + ba(t, x)h + σa(t, x) h φ a,h (t, x) = x + ba(t, x)h σa(t, x) h

39 Need for some interpolation with in classical Sparse for Sparse for v(t,.) represented on a grid, for v(t, φ ± a,h (t, x)) interpolation on a grid is required, Regular full with high order interpolators are possible [8] Results with sparse [9]

40 Portfolio optimization problem with in classical Sparse for Sparse for {S t, t [0, T ]} be an Ito process modeling the price evolution of n financial securities, {θ t, t [0, T ]} the investor strategy, Portfolio value dx θ t = θ t ds t S t +(X θ t θ t 1) ds0 t S 0 t Maximize the portfolio value : v 0 := sup E θ A [ exp = θ t ds t S t +(X θ t θ t 1)r t dt, ( )] ηxt θ.

41 The two dimensional case (r = 0) with in classical Sparse for Sparse for Asset dynamic (Heston model) HJB equation : ds t = µs t dt + Y t S t dw (1) t dy t = k(m Y t )dt + c ( ) Y t ρdw (1) t + 1 ρ 2 dw (2) t, Quasi explicit solution (Zariphopoulou) where the process Ỹ is defined by v(t, x, y) = e ηx 0 = v t k(m y)v y 1 2 c2 yv yy ( 1 sup θ R 2 θ2 yv xx + θ(µv x + ρcyv xy )) ( v(t, x, y) = e ηx exp 1 T µ 2 ) ds 2 t Ỹ s L 1 ρ 2 Ỹ t = y dỹ t = (k(m Ỹ t ) µcρ)dt + c Ỹ t dw t.

42 Two dimensional case : no adaptation, exact boundary treatment with in classical Sparse for Sparse for Parameters : η = 1, µ = 0.15, c = 0.2, k = 0.1, m = 0.3, Y 0 = m, ρ = 0, T = 1, X 0 = 1, reference Table: Portfolio optimization in dimension 2, no adaptation LINEAR QUADRATIC CUBIC Level Solution Time Solution Time Solution Time

43 Two dimensional case : extrapolated boundary with in classical Sparse for Sparse for Nearly same results but less expensive in computing time. Table: Portfolio optimization in dimension 2, no adaptation, extrapolated boundary treatment LINEAR QUADRATIC CUBIC Level Solution Time Solution Time Solution Time

44 two dimensional case : extrapolated boundary adaptation with in classical Sparse for Sparse for Table: Adaptation, extrapolated boundary, initial level 5 LINEAR QUADRATIC CUBIC Precision Solution Time Solution Time Solution Time e e Figure: Example of adapted meshes in dimension 2

45 A 5 dimensional problem with in classical Sparse for Sparse for OU process for the rate dr t = κ(b r t )dt + ζdw (0) t. Heston model for the second security, CEV-SV model for the first security ds (i) t = µ i S (i) t dt + σ i Y (i) t S (i) β i t dw (i,1) t, β 2 = 1, dy (i) t = k i ( ) m i Y (i) dt + c i t Y (i) t dw (i,2) t State (t, X t, r t, S (1) t, Y (1) t, Y (2) t ), value function v(t, x, r, s 1, y 1, y 2 ) solution of an HJB equation

46 Results in 5D with in classical Sparse for Sparse for Space discretization for the comms (2 dimensional) Table: No adaptation, extrapolated boundary treatment LINEAR QUADRATIC CUBIC Level Solution Time Solution Time Solution Time Table: adaptation, extrapolated boundary treatment, initial level equal to 7 LINEAR QUADRATIC CUBIC Precision Solution Time Solution Time Solution Time e e

47 References with in classical Sparse for Sparse for H.J. Bunged, M. Griebel, Sparse Grids, Acta Numerica, volume 13, (2004), pp D Pflüger, Spatially Adaptive Sparse Grids for High-Dimension problems, Dissertation, für Informatik, Technische Universität München, München (2010). H.-J. Bungartz.,Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation, Fakultät für Informatik, Technische Universität München, November T. Gerstner, M. Griebel,Dimension-Adaptive Tensor-Product Quadrature, Computing 71, (2003) H.-J. Bungartz.,Concepts for higher order finite elements on sparse, Proceedings of the 3.Int. Conf. on Spectral High Order Methods, pp , (1996)