Accurate approximation ofstochastic differential equations

Simon J.A. Malham and Anke Wiese(Heriot–Watt University, Edinburgh)

Birmingham: 6th February 2009

Stochastic differential equations

dyt = V0(yt)dt + V1(yt)dW 1t + · · · + Vd(yt)dW d

t

Four approaches to approximation, Solve:

◮ related PDE

◮ for a weak approximation (Monte–Carlo)

◮ for a strong approximation

◮ pathwise

First three: expectation and higher moments of solution sought.

Basic setting

yt = y0 +

∫ t

0V0(yτ )dτ +

d∑

i=1

∫ t

0Vi (yτ )dW i

τ

◮ Stratonovich form (familiar, easier)

◮ Non-commuting vector fields: Vi =∑n

j=1 V ji (y)∂yj

◮ d-dimensional driving signal: (W 1, . . . ,W d)

◮ Convention: W 0t ≡ t

◮ Solution process: y : R+ → RN ,

Wiener process

0 h 2h . (N−1) h T=Nh−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

∆ t

Q ∆ t

◮ Wt − Ws ∼ N(0,√

t − s)

◮ Independent increments

◮ Continuous, potentially nowhere differentiable

Applications

◮ Finance: Heston model for pricing options. The stock price ismodelled as a stochastic process u with stochastic volatility v .⇒ in Ito form:

dut = µut dt +√

vt ut dW 1t ,

dvt = α(θ − vt)dt + βρ√

vt dW 1t + β

√

1 − ρ2√

vt dW 2t ,

◮ Neuronal dynamics (Coombes & Lord)

◮ Molecular DNA damage dynamics (Chickarmane et al. )

◮ Chemical reactions (K. Burrage)

◮ Ocean/weather modelling

◮ Linear-quadratic optimal stochastic control.

Weak approximation

◮ Replace Gaussian increments ∆W i (tn, tn+1) by simpler RVs∆W i (tn, tn+1) with appropriate moment properties, eg. bybranching process:

P(∆W i (tn, tn+1) = ±

√h)

= 12

◮ Expectation of approximation yT across all paths at the finaltime T is close to the expectation of the true solution:

‖E (yT ) − E (yT )‖ = O(hp)

◮ No pathwise comparison: these paths not “close” to Wienerpaths

Strong approximation (case hereafter)

◮ Generate approximate Wiener process paths

◮ Pick the increments ∆W i (tn, tn+1) independently fromN(0,

√h)

◮ Since we have followed Wiener path approximations, weexpect to be able to compare yT with yT ; they’re close in thesense:

E ‖yT − yT‖ = O(h

p2)

Stochastic chain rule (Stratonovich)

yt = y0 +∑

i

∫ t

0Vi ◦ yτ dW i

τ

Ito lemma (stochastic chain rule) ⇒

f ◦ yt = f ◦ y0 +∑

j

∫ t

0Vj ◦ f ◦ yτ dW j

τ

E.g. choose f = Vi ⇒

Vi ◦ yt = Vi ◦ y0 +∑

j

∫ t

0Vj ◦ Vi ◦ yτ dW j

τ

yt = y0 +∑

i

∫ t

0

Vi ◦ y0 +∑

j

∫ τ1

0Vj ◦ Vi ◦ yτ2 dW j

τ2

dW iτ1

Stochastic Taylor series

yt = y0 +∑

i

∫ t

0dW i

τ1Vi ◦ y0 +

∑

i ,j

∫ t

0

∫ τ1

0Vj ◦Vi ◦ yτ2 dW j

τ2dW i

τ1

Now choose f = Vj ◦ Vi ⇒

yt = y0+∑

i

∫ t

0dW i

τ1

︸ ︷︷ ︸

Ji (t)

Vi◦y0+∑

i ,j

∫ t

0

∫ τ1

0dW j

τ2dW i

τ1

︸ ︷︷ ︸

Jji (t)

Vj◦Vi◦y0+· · ·

◮ Feynman–Dyson path ordered exponential, Neumann series,Peano–Baker series, Chen-Fleiss series, stochastic B-series,. . .

◮ Euler-Maruyama and Milstein methods, RK methods (Kloeden& Platen)

◮ Need approximations for iterated integrals: quadrature later

Flow map

yt = ϕt ◦ y0

⇒ ϕt = id +∑

i

JiVi +∑

i ,j

JjiVj ◦ Vi +∑

i ,j ,k

JkjiVk ◦ Vj ◦ Vi + · · ·

⇒ ϕt = id +∑

w∈A+

JwVw

Here A+ =

{non-empty words overA = {0, 1, . . . , d}

}

Remainders and local error

Stochastic integral properties

Expectations:

E(Ji ) = 0

E(Jii ) = E(12J2

i ) = 12h

E(Jij) = 0, i 6= j

Shuffle relations: JuJv =∑

w∈ sh(u,v) Jw

Ja1Ja2 = Ja1a2 + Ja2a1

Ja1Ja2a3 = Ja1a2a3 + Ja2a1a3 + Ja2a3a1

Exponential Lie series

Set ϕt = expψt then

ψt = lnϕt

= (ϕt − id) − 12(ϕt − id)2 + 1

3(ϕt − id)3 + · · ·

=d∑

i=0

JiVi +∑

i>j

12(Jij − Jji )[Vi ,Vj ] + · · ·

Local error: R ls = expψt − exp ψt = ψt − ψt + o(ψt − ψt)

◮ Magnus 1954, Chen 1957, Kunita 1980, Strichartz 1987, BenArous 1989, Castell 1993, Castell–Gaines 1995, Burrage 1999,Misawa 2001, P-C. Moan 2004.

Castell–Gaines (ODE) method

Truncated exponential Lie series across [tn, tn+1]:

ψtn,tn+1 = J1V1 + J2V2 + J0V0 + 12(J12 − J21)[V1,V2] .

Approximate solution:

ytn+1 ≈ exp(ψtn,tn+1) ◦ ytn .

Castell–Gaines: solve ODE

u′(τ) = ψtn,tn+1 ◦ u(τ)

across τ ∈ [0, 1] with u(0) = ytn gives u(1) ≈ ytn+1 .

Quadrature

How do we strongly approximate J12(tn, tn+1)?

◮ By its conditional expection J12(tn, tn+1)

◮ Karhunen–Loeve (Fourier) expansion

◮ Wiktorsson’s method — looks at the joint probabilitydistribution function for J1, J2 and J12.

Error: ⇒∥∥∥J12(tn, tn+1) − J12(tn, tn+1)

∥∥∥

L2

= h/√

Q

◮ Rough paths (Lyons)

◮ Wiktorsson improves to h/Q (SDELab: Gilsing & Shardlow)

Geometric stochastic integration

◮ M is a smooth submanifold of Rn

◮ Lie group G with corresponding Lie algebra g

◮ Lie group action Λy0 : G → M; starting point y0 ∈ M fixed

◮ Λ transitive, effective

◮ Vector fields Vi , i = 0, 1, . . . , d , are each infinitesimal Liegroup actions generated by some ξi ∈ g via Λy0 , i.e.

Vi = λξi

(Fundamental vector fields.)

Homogeneous manifolds

y

logexp

Λ y

0

S

S^id

oσt

σt^

t

t

t

M

G

g

0

y

y0

yt

^

Λ−1

Example: Stiefel manifold

M = Vn,k ≡ {y ∈ Rn×k : yTy = I}

1. G = SO(n) ⇒ g = so(n)

2. Λy0 ◦ S ≡ S y0

Direct calculation ⇒

λξ ◦ y =((Λy0)∗Xξ

)◦ y = ξ(y) y

Note S2 ∼= V3,1.

Stochastic Munthe-Kaas methods

Given smooth map ξ : M → g.

vξ ◦ σ = dexp−1σ ◦ ξ

(Λy0 ◦ expσ

)

Xξ ◦ S = ξ(Λy0 ◦ S

)S

λξ =(Λy0

)

∗Xξ

vξi

exp∗−−−−→ Xξi

(Λy0 )∗−−−−→ λξi

Vi ≡ λξi⇒ equivalent to original SDE

Rigid body (satellite): simulation

Rigid body: S2 adherence

0 1 2 3 4 5 6 7 8 9 10−16

−14

−12

−10

−8

−6

−4

−2

0

time

log

(dis

tan

ce fro

m m

an

ifold

)

Stochastic TaylorCastell−GainesMunthe−Kaas

Basic idea

ϕ = id +∑

w∈A+

Jw Vw

Suppose:

ψ = F (ϕ) =∞∑

k=1

Ck (ϕ− id)k =∑

w∈A+

Kw Vw

where

Kw =

|w |∑

k=1

Ck

∑

u1,...,uk∈A+

u1u2···uk=w

Ju1Ju2 · · · Juk(t)

Goal: ϕ = F−1(ψ); choose best Ck .

Hopf algebraic structure

ϕ∗ = 1 ⊗ 1 +∑

w∈A+

w ⊗ w

⇒ ψ∗ =∑

k≥1

Ck

(ϕ∗ − 1 ⊗ 1

)k

=∑

k≥1

Ck

(∑

w∈A+

w ⊗ w

)k

=∑

k≥1

Ck

(∑

u1,...,uk∈A+

(u1 xxy . . . xxy uk) ⊗ (u1 . . . uk)

)

=∑

w∈A∗

(|w |∑

k=1

Ck

∑

u1,...,uk∈A+

w=u1...uk

u1 xxy . . . xxy uk

)

⊗ w

=∑

w∈A∗

(K ◦ w) ⊗ w .

Sinh-log expansion

K ◦ w =

|w |∑

k=1

Ck

∑

u1,...,uk∈A+

w=u1...uk

u1 xxy u2 xxy . . . xxy uk

u1 xxy u2 xxy . . . xxy uk −→ c |u1|−1sc |u2|−1s . . . sc |uk |−1

Sinh-log series coefficients ⇒

K = 12

(cn + (c − s)n

).

Future directions

◮ Numerical stability (Buckwar et al. )

◮ Positivity preservation (Andersen, . . . )

◮ Implicit methods? (Kahl, Alfonsi, . . . )

◮ Symplectic methods (Tretyakov, Bou–Rabee)

◮ Driving fractional Brownian motions (Baudoin, . . . )

◮ Driving processes with jumps, eg. Levy processes

◮ PSDEs (Brown report)

Introductory references

◮ Theory:

L.C. Evans: An introduction to stochastic differentialequations

http://math.berkeley.edu/∼evans◮ Numerics:

D.J. Higham: An algorithmic introduction to numericalsimulation of stochastic differential equations

SIAM Review 43 (2001), pp. 525–546

Brown report

Applied mathematics at the US Department of Energy: fromSections 1 & 2:

Develop new approaches for efficient modeling of largestochastic systems.

Significantly advance the theory and tools for quantifyingthe effects of uncertainty and numerical simulation erroron predictions using complex models and when fittingcomplex models to observations.

Ito’s lemma

dyt = V0 ◦ yt dt +d∑

i=1

Vi ◦ yt dW it

⇒ df (yt) = V0◦f ◦yt dt+d∑

i=1

Vi ◦f ◦yt dW it +

d∑

i=1

V 2i ◦f ◦yt “dt”

⇒ df (yt) =

(

V0 +d∑

i=1

V 2i

)

◦ f ◦ yt dt +d∑

i=1

Vi ◦ f ◦ yt dW it

Formally use Ito product rule:d(W iW j) = W idW j + W jdW i + δijdt(prove using Ito integral definition)

Related PDE

◮ Consider Ito SDE for f ◦ yt(x) with initial data y0(x) ≡ x :

⇒ f ◦ yt(x) = f ◦ x +

∫ t

0

(

V0 +d∑

i=1

V 2i

)

◦ f ◦ yτ (x)dτ

+d∑

i=1

∫ t

0Vi ◦ f ◦ yτ (x)dW i

τ

◮ Feynman–Kac ⇒ solution u(t, x) of PDE

∂tu +

(

V0 +d∑

i=1

V 2i

)

◦ u = 0 with u(0, x) = f (x)

is u(t, x) = E(f (yt(x))

)

◮ (Roughly, take the expectation of Ito SDE for f )

Black–Scholes–Merton PDE

◮ Good explanation Evans notes, p. 114

◮ Constant volatility v , stock/index value ut evolves:

dut = µut dt +√

v ut dWt

◮ Current price of option at time t is C (t) = f (t, ut)

◮ Ito formula and financial argument to duplicate C by aportfolio consisting of investment of u and a bond (risk-freewith interest rate r) ⇒

∂t f + ru ∂uf + 12vu2 ∂uuf − r f = 0

Stratonovich form (hereafter)

◮ Ito: ∫ T

0W i

τ dW iτ = 1

2

(W i

T

)2 − 12T

◮ Stratonovich:

∫ T

0W i

τ ◦ dW iτ = 1

2

(W i

T

)2

◮ Ito to Stratonovich:

V0 = V0 − 12

d∑

i=1

V 2i

◮ Stratonovich calculus familiar, easier, swapping to/back to Itoform trivial

Quadrature error I

With τq = tn + q∆t, q = 0, . . . ,Q − 1, Q∆t = h

J12(tn, tn+1) =

∫ tn+1

tn

∫ τ

tn

dW 1τ1

dW 2τ

=

Q−1∑

q=0

∫ τq+1

τq

W 1τ − W 1

tndW 2

τ

=

Q−1∑

q=0

∫ τq+1

τq

(W 1τ − W 1

τq) + (W 1

τq− W 1

tn)dW 2

τ

=

Q−1∑

q=0

J12(τq, τq+1) +

Q−1∑

q=0

(W 1

τq− W 1

tn

)∆W 2(τq)

Quadrature error II

With τq = tn + q∆t, q = 0, . . . ,Q − 1, Q∆t = h

∥∥∥J12(tn, tn+1) − J12(tn, tn+1)

∥∥∥

2

L2

=

Q−1∑

q=0

∥∥J12(τq, τq+1)

∥∥2

L2

=

Q−1∑

q=0

(∆t)2

= Q(∆t)2

= h2/Q

⇒∥∥∥J12(tn, tn+1) − J12(tn, tn+1)

∥∥∥

L2

= h/√

Q

◮ Wiktorsson improves to h/Q (SDELab: Gilsing & Shardlow)