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On the Complexity of Solving Stochastic Differential Equations Klaus Ritter TU Darmstadt 0

On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

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Page 1: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

On the Complexity of Solving

Stochastic Differential Equations

Klaus Ritter

TU Darmstadt

0

Page 2: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Introduction

Stochastic differential equation

��� ������ ���� ��� ������������� ����� ��� ����������� �������� ������ ���!

Computational problem, e.g.,

" Input:

,�,�#�

,�

. Output:�

.

Strong/pathwise approximation.

" Input:

,�,�#�

, $ . Output: % � $ �&� ��' �����.

(Related to) weak approximation.

Minimal errors

(min

�*)+�,� -/.10325476�6�896:�<; �>=�;algorithm with ? 8�@BAC�<; �ED )GF��

the intrinsic difficulty of the computational problem.

Requires definition of

" Class of all algorithms:

partial information about

,�,�

, or $ .

Exmp.: finitely many function values.

" Error and cost of an algorithm.

1

Page 3: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Minimal errors

(min

�*)+�,� -/.10325476�6�896:�<; �>=�;algorithm with ? 8�@BAC�<; �ED )GF�

Typical result (min

�&) � � )������consists of

" upper bound: existence (construction) of algorithms; �such that

? 8�@BAC�<; � �ED ) � 476�6�896:�<; � �,� � ) ���� �" lower bound: � � � �

algorithm; =

? 8�@BA ��; �ED ) � 4 6�6�896 ��; ��� �� )�����

Key: partial information.

Information-based Complexity

? 8�� � ��� �,� - .�032 ? 8�@ A:�<; �>=!;algorithm with

476�6�896:�<; �ED ��F�

Traub, Wasilkowski, Wozniakowski (1988), Novak (1988),

Werschulz (1991), Plaskota (1996), Traub, Werschulz

(1998), Ritter (2000), Muller-Gronbach (2002).

2

Page 4: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Strong Approximation

1. The Setting.

2. Minimal Error for Scalar Equations and ��� -Norm.

3. Further Results.

3

Page 5: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

The Setting

��� ������ ���� ��� ������������� ����� ��� ����������� �������� ������ ���!

Approximate� �����

for all��� � �������

.

Algorithms

" Partial information about�

: values at sequentially

(adaptively) chosen knots���

. Cost one per evaluation.

" Complete information about

,�, and

� �.

Average error of an algorithm �( � � �,� % � � � � � ��� ���� � �

where

� � ��� ��� � ��� � � � � �� � � ��� ��� ��� � ���3��� � � � ��� ��� � Average cost of an algorithm � (at least)

) � � � �expected number of evaluations of

4

Page 6: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Formal definition of an algorithm � via Borel-measurable

mappings (here: scalar case)

� � = � � ��������� � � � � ��� � ������� �� � = � � ��������� � � �

�2STOP

�GO

F��� � = � � ��������� � � �

� � �� � � �����*��

Sequential evaluation of�

" first step:��� �

� yields � ���� � � � ���

���� �

,

" -th step:

� � ��� �,� � � ����� � �

� � �3� 7�B� � � ���� ��� � � � �

��� ���yields � � ��� �,� � ��� � ��� ��� � �

.

Termination criterion and output

If � ����� �� � �

��� ��� � � ����� ��� � � � � ���,� STOP�

then � ��� �,� � ������� � �

��� ��� � � � � � �3� � � ��� ���3 Thus ),� � �,� % � � -/. 2 = � � �

STOPF ��

5

Page 7: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Scalar Equations and � � -Norm

Assumption For $ � and $ � �

��� � � � ���B� � � � ����� � �+� � � �� $ ���3��� ��� $ ���3� � � � D � � � � � � � �� $ ���!��� ��� $ ��� ��� � � D � � ��� � � � � � � � � � � � �

� $�� �� � ���3��� � � $������ ���3� � � � D � � � � � � �

6

Page 8: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Recall (min

�&) �,� -/.�032 ( � � �>=�),� � �ED )GF�

Theorem Hofmann, Muller-Gronbach, R. (2001)

(min

�&) ��� �� � � � � � �����C� � ) � ��� �

with

� � � ��� � �,� �� % � � � � ���3��� �������������3

Optimal algorithm: Milstein plus Euler with adaptive

step-size control.

Remark Strong approximation related to ��� -approximation

of�

w.r.t. random weight function���� � � � ���3��� �������

.

7

Page 9: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Further Results

Scalar Equations

" Error at discrete points, in particular

( � � �,� % � � ���C��� � ���C� � � ���� � See Clark, Cameron (1980), Rumelin (1982), Cambanis,

Hu (1996), and Muller-Gronbach (2002).

� Integration of stoch. proc. w.r.t. random weight.

" ��� -norm,� D � D �

,

( � � �,� % � � � � ���� ��� �with

� D � � �. See Hofmann, Muller-Gronbach, R.

(2000), Muller-Gronbach (2002).

� Approximation of stoch. proc. w.r.t. random weight.

8

Page 10: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

" More general information: multiple stochastic integrals.

Used in Ito-Taylor schemes, due to Milstein, Wagner,

Platen, Kloeden.

Conjecture: For � � -norm

(min

�&) ���const

�info

� � �� � �C���#�C� � ) � ��� � �where

�� � � ���#� �,� �� % � � � � ��� ��� �������������

Known: const�point eval.

� � ��� � �,

const�lin. funct.

� � �����.

See Hofmann, Muller-Gronbach, R. (2002),

and Hofmann, Muller-Gronbach (2003).

" Key quantity: local mean-square smoothness of�

,

% �*� ��� � � ��� � ������� � � � �����,� � � � � ���3��� � � � � ��������

9

Page 11: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Systems of SDEs

" Key quantities: local smoothness and ‘amount of

non-commutativity’. See Muller-Gronbach (2002).

Stochastic Integration

See Wasilkowski, Wozniakowski (2001), Hertling (2001).

Equations Driven by Fractional Brownian Motion

See Neuenkirch (2005).

Stochastic Partial Differential Equations

See Davie, Gaines (2001), Muller-Gronbach, R. (2005a, b).

10

Page 12: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Weak Approximation

1. The Setting.

2. Minimal Errors for Holder Classes.

3. Further Results.

11

Page 13: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

The Setting

��� ������ ���� ��� ������������� ����� ��� ����������� �������� ������ ���!

Consider a class � of drift coefficients

= � � ��' ��� �������

and the particular case

� � � � � �������� For fixed $ = � �

��

,' � �

, and� �

� � �� � % � $ �&� �*' �����3 Approximate

� = � ��

. Note that

� � �� � ��� ����� ���where

��� � ��� �

� � ���� � ���� � ��' � � � � $

Algorithms: deterministic or randomized/Monte-Carlo

" Partial information about

: values at sequentially

(adaptively) chosen knots��� � ��� � � � � � ��' ��� � �

.

12

Page 14: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Approximate� = � �

�, where

� � �� � % � $ �&� �*' �����3

Worst case error and cost

" of a deterministic algorithm��

( � �� �� @ � ������ � � �� � � �� �� � � �

� �� �� @ � ������

��evaluations of

� �arithm. op.

!�

" of a randomized algorithm��

( � �� �,� @ � �����

% � � �� ��� �� � �� � � �� � � � � �� �,� @ � �

����% � evaluations of

� �arithm. op.

� �calls of random number gen.

Minimal errors

( detmin

�*)+�,� - .�032 ( � �� �>= �� det. algorithm with � �� �ED )GF��( ran

min

�&) �,� -/.�0 2 ( � �� �>= �� ran. algorithm with � �� �ED ) F

Question: ( ranmin

�&) �� ( detmin

�*)+�?

13

Page 15: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Minimal Errors for Holder Classes

Let � � � �and � � � �������

, put

� �� � �

unit ball in Holder class � �� � �

� � � �� �� � 2: � � �

� � =�@ � � �+������ � � ��' � � � � � �� � � F� � � � �� � �

Theorem Petras, R. (2004)

� � � � ) � � =( det

min

�&) ��� �� )���� �( ran

min

�&) ��� �� )�� � �� � � � A simple consequence

Euler scheme��

, step-size' ���

, and � � �simulations:

� -���@ � �� �,� � ��� � � � if � � � � � ,

cost � �� �,� � ��� � � � � ��� � �� =�),

worst case error ( � �� �,� � �*) � ��� � � .Thus Euler is better than every deterministic method if

� � � � � � � � � � � � � ��� �� 14

Page 16: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Proof of the lower bound for ( detmin (here

� � �)

Series representation

� � �� � � �� � �

� �� � � � � � � � �

� ��� �� -times

with linear functionals

� � � = � � � � ��' ��� � � � ��

being weighted integrals.

If � � � is ‘small’ then

� �� �� � �� � � � � ����3

In � �� �

‘unfavorable’ functions for weighted integration are

‘small’.

Thus computation of� � ��

is not easier than computation of� � � ���� .Use the lower bounds for weighted integration due to

Bakhvalov (1959) and Novak (1988).

In this way: a general theorem.

15

Page 17: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Upper bounds

As previously, � � � � � � ����� � � �and

� � � �� �� � 2 � � �

� � =�@ � � �+�� � � � ��� ' � � � � � � � � ��F Theorem Petras, R. (2004)

For every� � �

( detmin

�&) �,� � )�� � � ��� �( ran

min

�&) �,� � )�� � � ��� � ���

Remark

" Upper and lower bounds arbitrarily close in the power

scale.

" Analogous result for unbounded coefficients under

suitable growths conditions.

16

Page 18: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Proof of the upper bound for ( detmin and construction of an

algorithm (here� � �

)

Series representation

� ���� � � �� � �

� �� � � � � � � � �

� � � �� -times

��

Algorithm

�� �� �,� � �� � �

� �� � � �� � � � � � � � � �� �

where�� � � is a Smolyak algorithm (sparse grid) for

approximation of functions

� � ��' ��� � � � �� � ���

(Tensor product algorithms (full grids) need too many arith.

operations.) Requires precomputing.

Based on

" Wasilkowski, Wozniakowski (1995), Li (2002),

" Wasilkowski, Wozniakowski (2001),

" Deck, Kruse (2002).

17

Page 19: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Further Results

Feynman-Kac Path Integration

See Plaskota, Wasilkowski, Wozniakowski (2000), Kwas, Li

(2003), Kwas (2004) for the Gaussian case.

Variable and path-dependent functionals

� � $ �� % � $ �&� ���

with $ = � � � � ��' �����

and fixed drift and diffusion

coefficients

and�, resp.

See Dereich, Muller-Gronbach, R. (2005).

18

Page 20: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Summary

Partial information, lower bounds, minimal errors:

concepts from Information-based Complexity.

Apply results and ideas for linear problems to computational

problems for SDEs

" strong approximation: integration and approximation of

stochastic processes,

" weak approximation: integration of multivariate functions.

Yields

" sharp bounds for the minimal error,

" and almost optimal algorithms.

19

Page 21: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

References

20

Page 22: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Bakhvalov, N. S. (1959), On approximate computation of integrals (in Russian),

Vestnik MGV, Ser. Math. Mech. Astron. Phys. Chem. 4, 3–18.

Cambanis, S., Hu, Y. (1996). Exact convergence rate of the Euler-Maruyamascheme, with application to sampling design. Stochastics Stochastics Rep. 59,

211–240.

Clark, J. M. C., Cameron, R. J. (1980). The maximum rate of convergence ofdiscrete approximations. In Stochastic Differential Systems (B. Grigelionis, ed.)

162–171. Lect. Notes Control Inf. Sci. 25. Springer, Berlin.

Davie, A. M., Gaines, J. G. (2001), Convergence of numerical schemes for thesolution of parabolic stochastic partial differential equations, Math. Comp. 70,

121–134.

Deck, T., and Kruse, S. (2002), Parabolic differential equations with unboundedcoefficients - a generalization of the parametrix method, Acta Appl. Math. 74,

71–91.

Hertling, P. (2000). Nonlinear Lebesgue and Ito integration problems of highcomplexity. J. Complexity 17, 366–387.

Hofmann, N., M uller-Gronbach, T. (2004), On the global error of Ito-Taylor sche-

mes for strong approximation of scalar stochastic differential equations, J. Com-

plexity 20, 732–752.

Hofmann, N., M uller-Gronbach, T., Ritter, K. (2000), Optimal approximation of

stochastic differential equations by adaptive step-size control, Math. Comp. 69,

1017–1034.

Hofmann, N., M uller-Gronbach, T., Ritter, K. (2000), Step-size control for the

uniform approximation of systems of stochastic differential equations with ad-

ditive noise, Ann. Appl. Prob. 10, 616–633.

Hofmann, N., M uller-Gronbach, T., Ritter, K. (2001), The optimal discretizationof stochastic differential equations, J. Complexity 17, 117–153.

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Page 23: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Hofmann, N., M uller-Gronbach, T., Ritter, K. (2002), Linear vs. standard infor-

mation for scalar stochastic differential equations, J. Complexity 18, 394–414

Kwas, M. (2004), Complexity of multivariate Feynman-Kac path integration in

randomized and quantum settings, Preprint, Department of Computer Science,Columbia Univ., New York.

Kwas, M., and Li, Youming (2003), Worst case complexity of multivariate Feynman-Kac path integration, J. Complexity 19, 730–743.

Li, Youming (2002), Applicability of Smolyak’s algorithm to certain Banach spaces

of multivariate functions, J. Complexity 18, 792–814.

Novak, E. (1988), Deterministic and Stochastic Error Bounds in Numerical Ana-

lysis, Lect. Notes in Math. 1349, Springer-Verlag, Berlin.

M uller-Gronbach, T. (2002), The optimal uniform approximation of systems of

stochastic differential equations, Ann. Appl. Probab. 12, 664-690.

M uller-Gronbach, T. (2004), Optimal pointwise approximation of SDEs basedon Brownian motion at discrete points, Ann. Appl. Probab. 14, 1605–1642.

M uller-Gronbach, T., Ritter, K. (2006), Lower bounds and non-uniform time dis-cretization for approximation of stochastic heat equations, Found. Comp. Math.,

to appear.

Neuenkirch, A. (2006), Approximation of stochastic differential equations driven

by fractional Brownian noise, Ph.D. Dissertation, TU Darmstadt.

Petras, K., Ritter, K. (2006), On the complexity of parabolic initial value pro-

blems with variable drift, J. Complexity to appear.

Plaskota, L. (1996) Noisy Information and Computational Complexity, Cambrig-de Univ. Press, Cambridge.

Plaskota, L., Wasilkowski, G. W., and Wozniakowski, H. (2000), A new algo-rithm and worst case complexity for Feynman-Kac path integration, J. Comput.

Phys. 164, 335–353.

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Page 24: On the Complexity of Solving Stochastic Differential Equations · Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes in Math. 1733, Springer-Verlag, Berlin

Ritter, K. (2000), Average-Case Analysis of Numerical Problems, Lect. Notes

in Math. 1733, Springer-Verlag, Berlin.

Ruemelin, W. (1982), Numerical treatment of stochastic differential equations,SIAM J. Numer. Anal. 19, 604–613.

Traub, J. F., Wasilkowski, G. W., Wozniakowski, H. (1988), Information-Based

Complexity, Academic Press, New York.

Traub, J. F., Werschulz, A. G. (1998), Complexity and Information, CambrigdeUniv. Press, Cambridge.

Wasilkowski, G. W., and Wozniakowski, H. (1995), Explicit cost bounds of al-

gorithms for multivariate tensor product problems, J. Complexity 11, 1–56.

Wasilkowski. G. W., Wozniakowski, H. (2001). On the complexity of stochasticintegration. Math. Comp. 70, 685–698.

Werschulz, A. G. (1991), The Computational Complexity of differential and in-

tegral equations, Oxford Univ. Press, Oxford.

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