Introduction to the Numerical Solution of SDEs Selected topics

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0 / 23 Introduction to the Numerical Solution of SDEs Selected topics Andreas Rößler Summer School on Numerical Methods for Stochastic Differential Equations at Vienna University of Technology, Austria September 02-04, 203

0 / 23 Contents I Brief introduction to SDEs Stochastic integration Itô formula Stochastic differential equations II Introduction to numerical methods for SDEs Strong approximation Stochastic Taylor-approximation SRK methods

0 / 23 Contents I Brief introduction to SDEs Stochastic integration Itô formula Stochastic differential equations II Introduction to numerical methods for SDEs Strong approximation Stochastic Taylor-approximation SRK methods III Selected topics Weak approximation Multilevel Monte Carlo simulation

Time discrete approximation for SDEs / 23

/ 23 Time discrete approximation for SDEs Let (Ω,F, P) be a P-space, (F t) t 0 filtration (fulfills usual conditions), I = [t 0, T]. B,...,B m independent Brownian motions w.r.t. (F t) t 0. Consider the Itô SDE system m dx t = a(t, X t) dt + b j (t, X t) db j t, Xt = x. 0 j=

Time discrete approximation for SDEs Let (Ω,F, P) be a P-space, (F t) t 0 B,...,B m independent Brownian motions w.r.t. (F t) t 0. Consider the Itô SDE system dx t = a(t, X t) dt + filtration (fulfills usual conditions), I = [t 0, T]. m b j (t, X t) db j t, Xt = x. 0 j= Discretization: I h = {t 0, t, t 2,...,t N } [t 0, T] with t 0 < t < t 2 <... < t N = T. Y n = Y(t n) = Y tn approximation at time t n. h n = t n+ t n time increment (step size), h = max 0 n<n h n. B j n = B j t n+ B j t n N(0, h n) Brownian increment Euler-Maruyama approximation: Y n+ = Y n + a(t n, Y n) h n + m b j (t n, Y n) B j n j= for n = 0,,...,N with initial value Y 0 = X t0. / 23

2 / 23 Weak approximation Definition [Weak convergence (bias)] A time discrete approximation Y h converges with order p in the weak sense to X at time T as h 0, if there exist C f > 0 and δ 0 > 0 with E P (f(x T )) E P(f(Y T )) C f h p for all f C 2(p+) P (R d,r) and all h ]0,δ 0 [. 0 9 solutions. moment 90 80 solutions 2. moment 8 70 7 60 6 50 5 40 4 3 30 2 20 0 0 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3 / 23 Stochastic B-series analysis Now: Trees in TS with possibly more than one ramification at the root τ γ. j 2 j 3 j 4 t I = j t II = j t III = j j 2

3 / 23 Stochastic B-series analysis Now: Trees in TS with possibly more than one ramification at the root τ γ. j 2 j 3 j 4 t I = j t II = j t III = j j 2 Theorem [R. (2009)] Let p N and a, b j C p+,2p+2 P (I R d,r d ) for j =,...,m. If the coefficients fulfill for all t TS with ρ(t) p + the moment conditions 2 E P (I t;t,t+h ) = E P(Φ S (t; t, t + h)) for t, t + h I, then the SRK approximations are of weak order p, i.e. E P (f(x T )) E P(f(Y T )) C f h p for all f C 2(p+) P (R d,r), h ]0,δ 0 ] and some C f > 0.

4 / 23 Weak approximation method SRK method of weak order 2.0 for multi-dim. Itô SDEs with Y 0 = x 0 and s s m ( ) Y n+ = Y n + α i a(h (0) i ) h n+ β () i Î (k) +β (2) Î (k,k) i hn b k (H (k) i ) i= for n = 0,,...,N with stages H (0) i = Y n+ H (k) i = Y n+ Ĥ (k) i = Y n+ s j= s j= s j= + i= k= s m ( i= k= A (0) ij a(h (0) j ) h n+ A () ij a(h (0) j ) h n + A (2) ij a(h (0) j ) h n + β (3) i s j= s j= s j= ) Î (k) +β (4) hn m l= B () ij m l= l k B (0) ij i b l (H (l) j ) Î(r) b k (H (k) j ) h n B (2) ij b l (H (l) j ) Î(k,l) hn b k (Ĥ(k) i )

4 / 23 Weak approximation method SRK method of weak order 2.0 for multi-dim. Itô SDEs with Y 0 = x 0 and s s m ( ) Y n+ = Y n + α i a(h (0) i ) h n+ β () i Î (k) +β (2) Î (k,k) i hn b k (H (k) i ) i= Extended Butcher tableau: + i= k= s m ( i= k= β (3) i ) Î (k) +β (4) hn i b k (Ĥ(k) i ) A (0) B (0) A () B () A (2) B (2) α β () β (2) β (3) β (4)

5 / 23 Weak approximation In the case of weak approximation, we can use random variables that are defined by 2 (Î(k)Î(l) h n Ĩ (k) ) if k < l Î (k,l) = 2 (Î(k)Î(l) + h n Ĩ (l) ) if l < k hn) if k = l 2 (Î2 (k) for k, l m with independent random variables Î(k), k m, and random variables Ĩ(k), k m, possessing the moments 0 for q {, 3, 5} E(Îq ) = (k) (q )h q/2 n for q {2, 4}, O(h q/2 n ) for q 6 and 0 for q {, 3} E(Ĩq ) = (k) h n for q = 2. O(h q/2 n ) for q 4 Thus, only 2m independent random variables are needed for each step n = 0,,...,N.

6 / 23 Weak approximation Proposition [R. (2009)] The SRK method attains weak order of convergence, if. α T e = 2. β (4)T e = 0 3. β (3)T e = 0 4. (β ()T e) 2 = 5. β (2)T e = 0 6. β ()T B () e = 0 7. β (4)T A (2) e = 0 8. β (3)T B (2) e = 0 9. β (4)T (B (2) e) 2 = 0 If, in addition, some further equations 0 59 are fulfilled, the SRK method attains the weak order of convergence 2. Well-known schemes of Runge-Kutta type: i= k,l= (e.g. Platen (992); Tocino, Vigo-Aguiar (2002); R. (2003); Komori (2007)) s m β (k,l) i b k (H (l) i )

6 / 23 Weak approximation Proposition [R. (2009)] The SRK method attains weak order of convergence, if. α T e = 2. β (4)T e = 0 3. β (3)T e = 0 4. (β ()T e) 2 = 5. β (2)T e = 0 6. β ()T B () e = 0 7. β (4)T A (2) e = 0 8. β (3)T B (2) e = 0 9. β (4)T (B (2) e) 2 = 0 If, in addition, some further equations 0 59 are fulfilled, the SRK method attains the weak order of convergence 2. Costs of the order 2 scheme proposed by Platen: Drift: 2 d Diffusion: 2 d m 2 + d m Random numbers: m(m + )/2 Costs of the SRK scheme RI6: Drift: 2 d Diffusion: 5 d m Random numbers: 2m

7 / 23 Coefficients for an explicit SRK scheme RI6 with s = 3 stages: (2, 2) A (0) B (0) 0 0 0 0 A () B () 0 0 A (2) B (2) 0 0 0 0 2 2 0 2 2 4 4 4 0 4 0 2 2 2 2 α T β ()T /β (3)T β (2)T /β (4)T

8 / 23 Numerical example Nonlinear SDE system with non-commutative noise: X 243 t d Xt 2 54 X t 27 77 X t 2 + 23 54 X t 3 65 54 X t 4 27 X 3 = 77 X t 243 54 X t 2 + 65 54 X t 3 23 54 X t 4 5 dt t Xt 4 + 9 54 X t 6 54 X t 2 + 62 77 X t 3 36 77 X t 4 6 54 X t 5 54 X t 2 + 36 77 X t 3 62 77 X t 4 3 4 db t + 3 8 5 6 5 db3 t + c 5 4 6 5 db5 t + c 2 3 (X 2 t )2 +(X 3 t )2 + 2 23 + c (Xt 2 )2 +(Xt 2)2 + 9 + c 2 (Xt 0 )2 +(Xt 3)2 + 3 (X 4 t )2 +(X t )2 + (X 3 t )2 +(X 4 t )2 + 3 29 (X 2 t )2 +(X 4 t )2 + 2 25 4 6 db2 t 6 2 8 9 db4 t 8 9 2 3 db6 t 6 3

9 / 23 Numerical example Initial value: X 0 = ( 8, 8,, 8 )T Solution: E(X i T) = X i 0 exp(2t) for i =, 2, 3, 4 0 2 4 ld(error) 6 8 0 2 4 6 Euler ExEu PLWM RI6 RI5 2 3 4 5 6 7 8 9 0 ld(computational effort) c = 0, c 2 = 0, m = 2

9 / 23 Numerical example Initial value: X 0 = ( 8, 8,, 8 )T Solution: E(X i T) = X i 0 exp(2t) for i =, 2, 3, 4 0 2 4 ld(error) 6 8 0 2 4 6 Euler ExEu PLWM RI6 RI5 3 4 5 6 7 8 9 0 ld(computational effort) c =, c 2 = 0, m = 4

9 / 23 Numerical example Initial value: X 0 = ( 8, 8,, 8 )T Solution: E(X i T) = X i 0 exp(2t) for i =, 2, 3, 4 0 2 4 ld(error) 6 8 0 2 4 6 Euler ExEu PLWM RI6 RI5 4 5 6 7 8 9 0 2 ld(computational effort) c =, c 2 =, m = 6

0 / 23 Further topics of interest: SDAEs (circuit simulation) Step size control algorithms Stability analysis A-stable drift-implicit schemes Schemes for continuous approximation Coefficients for optimal schemes with minimized error constant

/ 23 Monte Carlo simulation Approximation of E P (f(x T )) by classical Monte Carlo simulation: E P (f(y(t))) ŶMC = M M f(y i (T)) based on M independent realizations Y (T),...,Y M (T). i= 0.452 Monte Carlo Simulation 0.45 0.448 0.446 0.444 0.442 0.44 0.438 0 0000 20000 30000 40000 50000 60000 70000 80000 90000 00000

/ 23 Monte Carlo simulation Approximation of E P (f(x T )) by classical Monte Carlo simulation: E P (f(y(t))) ŶMC = M M f(y i (T)) based on M independent realizations Y (T),...,Y M (T). i= L 2 -error (RMSE): ( e(ŷmc) = E P ((ŶMC E P (f(x T ))) 2)) /2 = (Var P(ŶMC)+ E P (f(x T )) E P(f(Y(T))) 2) /2 D. Duffie, P. Glynn (995): e(ŷmc) = O(C(ŶMC) p 2p+) Thus: Order for classical Monte Carlo simulation is bounded by O(C(ŶMC) 2 )

2 / 23 Multi-level Monte Carlo simulation Multi-level Monte Carlo method (M. Giles (2008)): Let t 0 = 0 and T =. Choose a number of levels L N. Sequence of step sizes: h l = M l, l = 0,,...,L In the following: M = 2 Apply on level l the discretization I hl w.r.t. h l for l = 0,,...,L. Consider approximation process on level l: Y l = (Y l (t)) t Ihl, l = 0,,...,L Main idea: L E P(f(Y L (T))) = E P(f(Y 0 (T)))+ E P(f(Y l (T)) f(y l (T))) l=

3 / 23 As an estimator for E P(f(Y 0 (T))) apply Ŷ 0 = N 0 N 0 i= f(y 0 (T)) and for E P(f(Y l (T)) f(y l (T)) apply Ŷ l = N l N l i= (f(y l (T)) f(y l (T))) for l =,...,L with Y l and Y l based on the same realizations of the underlying driving stochastic process (Wiener process).

3 / 23 As an estimator for E P(f(Y 0 (T))) apply Ŷ 0 = N 0 N 0 i= f(y 0 (T)) and for E P(f(Y l (T)) f(y l (T)) apply Ŷ l = N l N l i= (f(y l (T)) f(y l (T))) for l =,...,L with Y l and Y l based on the same realizations of the underlying driving stochastic process (Wiener process). Multi-level Monte Carlo estimator: Ŷ ML = L l=0 Ŷ l is unbiased, i.e. E P(ŶML) = E P(f(Y L (T))).

4 / 23 If the Euler-Maruyama scheme is applied with B j n N(0, h n): Then it holds for some Lipschitz function f : R d R (strong convergence): E P ( f(x(t)) f(y h (T)) 2 ) c h. Therefore, we obtain for the variance Var P (f(y l (T)) f(y l (T))) ( ( VarP (f(y l (T)) f(x(t))) ) /2 + ( Var P (f(y l (T)) f(x(t))) ) ) /2 2 ( ( EP ( f(y l (T)) f(x(t)) 2 ) ) /2 + ( E P ( f(y l (T)) f(x(t)) 2 ) ) ) /2 2 c 2 h l. Further, for the bias holds E P (f(x T )) E P(f(Y l (T))) c h l for all l = 0,,...,L.

5 / 23 Approximation bias: E P (f(x(t))) E P(ŶML(T)) c h L Variances: thus total variance Var P(Ŷ l ) = N l Var P ( ) f(y l (T)) f(y l (T)) = c 2 h l N l Var P(ŶML) = c 2 L l=0 N l h l Computational cost for the multi-level Monte Carlo method: C(ŶML) = L l=0 N l C(Y) T t 0 h l

6 / 23 Let some prescribed error bound e(ŷml) ε be given. What is the best choice for L and for N l, l = 0,,...,L, to fulfill the error bound?

6 / 23 Let some prescribed error bound e(ŷml) ε be given. What is the best choice for L and for N l, l = 0,,...,L, to fulfill the error bound? Minimization problem: ( L min N l,l c 2 l=0 N l h l + ch 2 L 2 ) under constraint L l=0 N l C(Y) T t 0 h l = s Solving the optimization problem yields: Number of levels: L = log M ( 2 c (T t 0 )ε ) Number of independent samples on level l: N l = 2ε 2 (L+) c 2 h l

7 / 23 Assumptions: Cost model: C(Y(T)) denotes the number of drift and diffusion evaluations that are necessary to compute Y(T). ) E(f(X(T))) E(f(Y L (T))) c h α L 2) Var(f(Y l (T)) f(y l (T))) c 2 h β l 3) C(Y l (T), Y l (T)) c 3 h γ l for l =,...,L, some α,β, c, c 2, c 3 > 0 and γ. Theorem [M. Giles (2008)] For every ε > 0 there exist L and N l for l = 0,,...,L, such that e(ŷml) < ε with computational costs bounded by O(ε 2 ) if β > γ, α γ, 2 C(ŶML) = O(ε 2 (log(ε)) 2 ) if β = γ, α γ 2, O(ε 2 γ β α ) if β < γ, α β, 2

Modified multi-level Monte Carlo simulation Goal: Improve the MLMC method by reducing the computational effort. Let a prescribed error bound e(ŷml) ε be given. Let t 0 = 0 and T = for simplicity. We claim that for some q ]0, [ Var P(ŶML) = L Var P(Ŷ l ) ( q)ε 2 and E P (f(x T )) E P(f(Y L (T))) 2 qε 2 l=0 i.e. e(ŷml) ε is fulfilled. 8 / 23

Modified multi-level Monte Carlo simulation Goal: Improve the MLMC method by reducing the computational effort. Let a prescribed error bound e(ŷml) ε be given. Let t 0 = 0 and T = for simplicity. We claim that for some q ]0, [ L Var P(ŶML) = Var P(Ŷ l ) ( q)ε 2 and E P (f(x T )) E P(f(Y L (T))) 2 qε 2 l=0 i.e. e(ŷml) ε is fulfilled. New idea: Apply on levels l = 0,,...,L a cheap weak order α approximation Y l such that E P (f(x T )) E P(f(Y l (T))) c hl α Var P(f(Y l (T)) f(y l (T))) c 2 h β l and apply on level L a higher oder (i.e. p > α) weak approximation Ỹ L such that E P (f(x T )) E P(f(Ỹ L (T))) c,l h p L Var P(f(Ỹ L (T)) f(y L (T))) c 2,L h β L L. 8 / 23

9 / 23 Assumption on costs: C(Y l (T), Y l (T)) = ĉ 3 h γ l, for l =, 2,...,L, C(Ỹ L (T), Y L (T)) = ĉ 3,L h γ L L. In case of β = γ and β L = γ L we get a reduction of the number of levels L p = log p M (q /2 cε ) as well as a reduction of Monte Carlo simulations due to N l = q ε 2 c h l (L p + ).

Assumption on costs: C(Y l (T), Y l (T)) = ĉ 3 h γ l, for l =, 2,...,L, C(Ỹ L (T), Y L (T)) = ĉ 3,L h γ L L. In case of β = γ and β L = γ L we get a reduction of the number of levels L p = log p M (q /2 cε ) as well as a reduction of Monte Carlo simulations due to N l = q ε 2 c h l (L p + ). Apply the modified multi-level Monte Carlo estimator with Ŷ l L p Ŷ ML(α,p) = l=0 based on order α approximations for l = 0,,...,L p, however applying Ŷ l for level L p. Ŷ Lp = N Lp N Lp ( f( Ỹ Lp ) f(y Lp ) ) i= 9 / 23

Proposition [Debrabant, R. (203)] a) In case of β > γ, β γ < β L γ L and for sufficiently small ε > 0 holds C(ŶML(α,α))(ε) C(ŶML(α,p))(ε) > provided that α γ 2, p 2 max{γ,γ L} and p > 4 max{β +γ,β γ + 2γ L}. Further C(ŶMML(α,p))(ε) = O ( ε 2). b) In case of β = γ, β L γ L, p max{γ,γ 2 L} and α γ 2 C(ŶMML(α,α)) ( p ) 2 lim ε 0 C(ŶML(α,p)) α with C(ŶMML(α,p))(ε) = O ( ε 2 (log(ε)) 2). we get c) In case of β < γ, γ β = γ L β L and p 2 (max{γ,γ L} γ +β) holds C(ŶML(p,p))(ε) lim ε 0 C(ŶML(α,p))(ε) > ) if c 2 c 3 < c 2,L c 3,L. Further C(ŶMML(α,p))(ε) = O (ε 2 max{γ β,γ L β L } p. Especially, if q = γ β γ β + 2p then the computational costs are asymptotically minimal. 20 / 23

2 / 23 Numerical example Linear SDE (d = m = ): dx t = 3 Xt dt + Xt dbt 2 Initial condition: X 0 =, Solution: E(Xt) = exp( 3 t) 0 0 2 log 0 (error) 0.5.5 2 2.5 3 MLMC EM MLMC SRK 3.5 0 5 0 5 log 0 (computational effort)

22 / 23 Numerical example SDE with non-commutative noise: X 243 t d Xt 2 54 X t 27 77 X t 2 + 23 54 X t 3 65 54 X t 4 27 Xt 3 = 77 X t 243 54 X t 2 + 65 54 X t 3 23 54 X 4 t 5 Xt 4 54 X t 6 54 X t 2 + 62 77 X t 3 36 77 X t 4 dt 6 54 X t 5 54 X t 2 + 36 77 X t 3 62 77 X t 4 + (Xt 2 9 )2 +(Xt 3)2 + 2 3 4 23 db t + (Xt 4 3 8 )2 +(Xt )2 + 4 6 db2 t 6 5 2 + (Xt 2 )2 +(Xt 2)2 + 6 5 9 db3 t + (Xt 3 5 4 )2 +(Xt 4)2 + 3 8 9 29 db4 t 8 6 9 + (Xt 0 )2 +(Xt 3)2 + 5 3 db5 t + (Xt 2 3 )2 +(Xt 4)2 + 2 2 3 25 db6 t 6 3

23 / 23 Numerical example Initial condition: X 0 = ( 8, 8,, 8 )T Solution: E(X i T) = X i 0 exp(2t) for i =, 2, 3, 4, d = 4, m = 6 log 0 (error) 0 2 MLMC EM MLMC SRK 3 0 5 0 5 20 log 0 (computational effort)

23 / 23 Literature:.) Debrabant, Rößler: On the acceleration of the multi-level Monte Carlo method. Preprint arxiv:302.7650v (203). 2.) Giles, Debrabant, Rößler: Numerical analysis of multilevel Monte Carlo path simulation using the Milstein discretization. Preprint arxiv:302.4676v (203). 3.) Günther, Jüngel: Finanzderivate mit Matlab. Vieweg & Sohn Verlag, Wiesbaden (2003). 4.) Higham: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., Vol. 43, No. 3, 525-546 (200). 5.) Kloeden, Platen: Numerical solution of stochastic differential equations. Springer-Verlag, Berlin (999). 6.) Milstein: Numerical integration of stochastic differential equations. Kluwer Academic Publishers (995). 7.) Rößler: Second order Runge-Kutta methods for Itô stochastic differential equations, SIAM J. Numer. Anal., Vol. 47, No. 3, 73-738 (2009). 8.) Rößler: Strong and weak approximation methods for stochastic differential equations-some recent developments. Recent Developments in Applied Probability and Statistics, p. 27-53, Physica-Verlag/Springer (200).

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