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Strong Approximation of Stochastic Differential Equations under Non-Lipschitz Assumptions

Andreas Neuenkirch

U Mannheim & TU Kaiserslautern

14.02.2012

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 1/26

Outline

Part I: Introduction Part II: Euler Schemes under Non-Lipschitz Assumptions Part III: Strong Approximation of Square-root Diffusions

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 2/26

Stochastic Differential Equations

Continuous time random dynamics in Rd (SDE) dXt = a(Xt)dt +

m

• j=1

b(j)(Xt)dW (j)

t

, t ∈ [0, T] X0 = x0 ∈ Rd where

• a : Rd → Rd

drift coefficient

• b = (b(1), . . . , b(m)) with b(j) : Rd → Rd

diffusion coeff.

• W = (W (1), . . . , W (m))′

m-dimensional Brownian motion Assumption (SDE) has unique strong solution X = Φa,b,x0(W )

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 3/26

Computational SDEs

X = Φa,b,x0(W ) Problems (i) Approximate Itˆ

• map Φa,b,x0

strong / pathwise approximation

(ii) Approximate law PX

weak approximation

(iii) Compute expectation Ef (X) for f : C([0, T]; Rd) → R

quadrature

(iv) ... Maruyama (1955) ... Milstein (1974) ... Kloeden, Platen (1992) ... Classically: a, b globally Lipschitz, i.e. there exists L > 0 s.th. (Lip) |a(x) − a(y)| + |b(x) − b(y)| ≤ L · |x − y|, x, y ∈ Rd

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 4/26

Euler Scheme

Equidistant discretization ti = i∆ where ∆ = T/n

• X (∆)

= x0

• X (∆)

ti+1 =

X (∆)

ti

+ a( X (∆)

ti

)∆ + b( X (∆)

ti

) ∆iW , i = 0, . . . , n − 1 with ∆iW = Wti+1 − Wti Extension to [0, T] by piecewise linear interpolation, i.e.

• X (∆)

t

= ti+1 − t ∆

• X (∆)

ti

+ t − ti ∆

• X (∆)

ti+1 ,

t ∈ [ti, ti+1] Theorem (strong error)

Faure (1992); ...

Under (Lip)

• E max

t∈[0,T] |Xt −

Xt|21/2 ≤ c(a, b, x0) · (∆| log(∆)|)1/2

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 5/26

Two SDEs from Mathematical Finance

Heston model dSt = µSt dt +

• |Vt|St
• 1 − ρ2dW (1)

t

+ ρdW (2)

t

• ,

s0 > 0 dVt = κ(λ − Vt) dt + θ

• |Vt| dW (2)

t

, v0 > 0 where ρ ∈ (−1, 1), κ, λ, θ > 0, µ ∈ R (Vt)t∈[0,T]: Cox-Ingersoll-Ross process (CIR) 3/2-model dSt = µSt dt +

• |Vt|St
• 1 − ρ2dW (1)

t

+ ρdW (2)

t

• ,

s0 > 0 dVt = c1Vt(c2 − Vt) dt + c3|Vt|3/2 dW (2)

t

, v0 > 0 with c1, c2, c3 > 0

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 6/26

Properties (1) SDEs take values in subsets D ⊂ R2 only:

• Heston model D = (0, ∞) × [0, ∞)
• 3/2-model D = (0, ∞) × (0, ∞)

(2) Coefficients not globally Lipschitz on D (3) Coefficients smooth on interior of D Standard theory does not apply! Pioneering works on stochastic Euler schemes under non-standard assumptions:

• I. Gy¨
• ngy (1998); D. Higham, X. Mao, A. Stuart (2002)

Many contributions since then; several talks here at MCQMC 2012

• n numerics of SDEs under non-standard assumptions

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 7/26

Part II: Euler Schemes under Non-Lipschitz Assumptions

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 8/26

Euler Scheme for SDEs on Domains

dXt = a(Xt) dt + b(Xt) dWt, X0 = x0 SDE with values in a domain D, i.e. D ⊂ Rd open and (S) P(Xt ∈ D, t ≥ 0) = 1 Euler scheme

• Xti+1 =

Xti + a( Xti) ∆ + b( Xti) ∆iW ,

• X0 = x0

Extension to [0, T] by piecewise linear interpolation Theorem (pathwise error)

Gy¨

• ngy (1998)

If (S), a ∈ C 1(D; Rd), b ∈ C 1(D; Rd,m), then for all ε > 0 max

t∈[0,T] |Xt(ω) −

Xt(ω)| ≤ Cε(ω) · ∆1/2−ε for almost all ω ∈ Ω, where Cε almost surely finite random variable

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 9/26

Theorem

Gy¨

• ngy (1998)

If (S), a ∈ C 1(D; Rd), b ∈ C 1(D; Rd,m), then for all ε > 0 max

t∈[0,T] |Xt(ω) −

Xt(ω)| ≤ Cε(ω) · ∆1/2−ε for almost all ω ∈ Ω Remarks

• Proof uses localization strategy
• Applies to Heston model if 2κλ ≥ θ2 and to 3/2-model
• For D = Rd: use suitable modification of the coefficients
• utside D for better numerical stability, e.g. x+ instead of |x|
• Above result can be extended to general Itˆ
• -Taylor schemes

Jentzen, Kloeden, N (2009)

Strong convergence of Euler scheme?

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 10/26

Strong Convergence

Theorem

Higham, Mao, Stuart (2002)

If (S), a ∈ C 1(D; Rd), b ∈ C 1(D; Rd,m) and (M1) E max

t∈[0,T] |Xt|p < ∞,

(M2) sup

∆>0

E max

t∈[0,T] |

X (∆)

t

|p < ∞ for some p > 2, then E max

t∈[0,T] |Xt −

X (∆)

t

|2 → 0 for ∆ → 0 Proof previous Theorem and integration to the limit using (M) Remarks

• Original proof did not use Gy¨
• ngy’s result
• Applies to CIR if 2κλ ≥ θ2 (strictly positive sample paths)
• Euler scheme strongly convergent for CIR also for 2κλ < θ2

Higham, Mao (2005)

Condition (M2) ’technical nuisance’?

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 11/26

Volatility process in 3/2-model dVt = 1.2Vt(0.8 − Vt) dt + |Vt|3/2 dWt, v0 = 0.5 Euler based Monte-Carlo estimator 1 N

N

• i=1

| V (i)

4 |

for E|V4| = 0.5662... where V (i)

4

iid copies of V4

repetitions / stepsize ∆ = 20 2−2 2−4 2−6 2−8 2−10 N = 103 6.3272 Inf Inf 0.5502 0.5535 0.5551 104 6.8947 Inf Inf Inf 0.5627 0.5634 105 7.4306 Inf Inf Inf 0.5662 0.5671 106 7.2274 Inf Inf Inf Inf 0.5658 107 7.2792 Inf Inf Inf Inf Inf

Empirical first moment explodes!

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 12/26

Moment Explosions

Here m = d = 1 Superlinearly growing coefficients: let c ≥ 1, β > α > 1 s.th. (G) max{|a(x)|, |b(x)|} ≥ 1 c · |x|β , min{|a(x)|, |b(x)|} ≤ c · |x|α for |x| ≥ c. Theorem

Hutzenthaler, Jentzen, Kloeden (2011)

Let p > 1 s.th. supt∈[0,T] E|Xt|p < ∞ and let b(x0) = 0. If (G), then lim

∆→0 E|

X (∆)

T

|p = ∞ Remarks

• Moment explosion caused by very large increments of

Brownian motion (rare events)

• 3/2-model: β = 2, α = 3/2. Can modification of coefficients
• utside (0, ∞) prevent moment explosion?

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 13/26

Drift-implicit Euler Scheme

Higham, Mao, Stuart (2002): Drift-implicitness provides numerical stability for SDEs with

• ne-sided Lipschitz drift coefficients, i.e.

(one-sided Lip) x −y, a(x)−a(y) ≤ L|x −y|2, x, y ∈ Rd Example: a(x) = x − x3 Drift-implicit Euler scheme X ti+1 = X ti + a(X ti+1) ∆ + b(X ti) ∆iW , X 0 = x0 Extension to [0, T] by piecewise linear interpolation Note: implicit equations of the form y − a(y)∆ = c with c ∈ Rd have to be solved

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 14/26

Theorem

Szpruch, Mao (2012)

If a ∈ C 1

pol(Rd; Rd), b ∈ C 1 pol(Rd; Rd,m), a one-sided Lipschitz and

(monotone) x, a(x) + 1 2|b(x)|2 ≤ α + β|x|2, x ∈ Rd for some α, β > 0, then lim

∆→0 E max t∈[0,T] |Xt − X (∆) t

|p = 0 for all p ∈ [1, 2) Remarks

• Similar for Ait-Sahalia interest model

Szpruch et al. (2011)

• Standard L2-convergence rate (∆| log(∆)|)1/2 recovered

if b additionally globally Lipschitz

Higham, Mao, Stuart (2002)

• Solving implicit equations avoided by tamed Euler scheme

Hutzenthaler, Jentzen, Kloeden (2012)

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 15/26

Summary of Part II

dXt = a(Xt) dt + b(Xt) dWt, X0 = x0 with C 1-coefficients Euler scheme

• Xti+1 =

Xti + a( Xti) ∆ + b( Xti) ∆iW ,

• X0 = x0
• pathwise convergence
• strong convergence under a moment condition
• moment explosions possible

Drift-implicit Euler scheme X ti+1 = X ti + a(X ti+1) ∆ + b(X ti) ∆iW , X 0 = x0

• monotone condition + drift one-sided Lipschitz + ... :

strong convergence

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 16/26

Part III: Strong Approximation of Square-root Diffusions

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 17/26

Two Square-root SDEs

log-Heston model d log(St) =

• µ − 1

2Vt

• dt +
• |Vt|
• 1 − ρ2d

Wt + ρdWt

• ,

s0 > 0 dVt = κ(λ − Vt) dt + θ

• |Vt| dWt,

v0 > 0 where ρ ∈ (−1, 1), κ, λ, θ > 0, µ ∈ R and W one-dim. Brownian motion, independent of W Strong app. of CIR process V : Numerous schemes proposed, convergence rates analyzed typically by numerical tests only Wright-Fisher SDE (WF) dXt = (α − βXt) dt + γ

• |Xt(1 − Xt)| dWt,

x0 ∈ (0, 1) with α, β, γ > 0

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 18/26

Strong Approximation of CIR

A method from Alfonsi (2005): U0 = √v0 Utk+1 = Utk + θ/2 ∆kW 2 + κ∆ +

• (Utk + θ/2 ∆kW )2

(2 + κ∆)2 + κλ − θ2/4 2 + κ∆ ∆ and V

2 tk = Utk

Extension to [0, T] by piecewise linear interpolation Properties

• Scheme well defined for 4κλ ≥ θ2
• Non-negativity of CIR preserved for 4κλ ≥ θ2
• Method arises by discretizing SDE for √Vt with drift-implicit

Euler scheme

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 19/26

CIR: Strong Convergence Rates

Theorem

Dereich, N, Szpruch (2012)

If 1 ≤ p < 2κλ θ2 , then there exists Kp > 0 such that

• E max

t∈[0,T] |Vt − V t|p1/p ≤ Kp · (∆| log(∆)|)1/2

Remarks

• Matches standard convergence rates for SDEs with Lipschitz

coefficients

• Integration of log-Heston SDE using ’reasonable’ schemes

based on V :

(1) convergence rate for CIR carries over (2) exponential integrability?

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 20/26

Idea of the Scheme

If 2κλ ≥ θ2 , by Itˆ

• ’s lemma,

d√Vt = a(√Vt) dt + θ

2 dWt

where a(x) = 4κλ − θ2 8 · 1 x − κ 2 · x

• ne-sided Lipschitz on D = (0, ∞), i.e.

(x − y)(a(x) − a(y)) ≤ − κ

2(x − y)2,

x, y > 0 and −a coercive on D = (0, ∞), i.e. limx→0 a(x) = ∞, limx→∞ a(x) = −∞ Drift-implicit Euler scheme for √Vt: Utk+1 = Utk + a(Utk+1)∆ + θ 2∆kW Quadratic equation, can be solved explicitly

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 21/26

Idea of the Proof

If 2κλ ≥ θ2 d√Vt = a(√Vt) dt + θ

2 dWt

with a(x) = 4κλ − θ2 8 · 1 x − κ 2 · x Sketch of Proof (1) Error analysis for

• Vtk − Utk:

(i) Drift one-sided Lipschitz: control of error propagation

(ii) Analysis of one step error, difficulty E 1

V q

t = ∞ if q ≥ 2κλ

θ2

(2) Error analysis for Vt − V t:

(i) Control transformation x → x2

(ii) Analysis of error of piecewise linear interpolation

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 22/26

Numerical Test: Option Pricing via Multi-level Monte Carlo

Heston Lookback Option, f (S, V ) = e−rT(K − maxt∈[0,T] St)+

s0 = 85, r = 0.0319, v0 = 0.01021, κ = 6.21, λ = 0.082, θ = 0.61, ρ = −0.7, K = 100, T = 1, M = 1000

eps cost mean max min std 2−3 32 · 26 2.92775 5.31776 0.05180 0.77934 2−4 42 · 28 2.96910 4.20493 1.74487 0.40637 2−5 52 · 210 2.94909 3.78347 2.29099 0.21749 2−6 62 · 212 2.95133 3.31206 2.62872 0.11197 2−7 72 · 214 2.94731 3.12879 2.74758 0.06161 2−8 82 · 216 2.95110 3.04030 2.86977 0.02899 Ef (S, V ) ≈ 2.95...

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 23/26

Strong Approximation of WF

dXt = (α − βXt) dt + γ

• |Xt(1 − Xt)| dWt,

x0 ∈ (0, 1) with α, β, γ > 0 Note: Lamperti-transformation Λ(x) = x 1 b(ξ) dξ For CIR: Λ(x) = 2

θ

√x For WF: Λ(x) = 2

γ arcsin(√x) and Yt = 2 arcsin(

√ X t) satisfies dYt = a(Yt) dt + γdWt where a(x) =

• α − γ2

4

• cot

x 2

• −
• β − α − γ2

4

• tan

x 2

• ,

x ∈ (0, π)

• ne-sided Lipschitz on (0, π) and −a coercive on (0, π)

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 24/26

WF: Strong Convergence Rates

Drift-implicit Euler method Utk+1 = Utk + a(Utk+1)∆ + γ∆kW , U0 = 2 arcsin(√x0) and X tk = sin2 Utk 2

• Extension to [0, T] by piecewise linear interpolation

Theorem

Dereich, N, Szpruch (2012)

If 1 ≤ p < min 2α γ2 , 2(β − α) γ2

• ,

then there exists Kp > 0 such that

• E max

t∈[0,T] |Xt − X t|p1/p ≤ Kp · (∆| log(∆)|)1/2

Proof analogous to CIR, also control of inverse moments required

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 25/26

Summary & Work in Progress

Summary

• Euler scheme converges pathwise for C 1-coefficients
• Moment explosions possible without additional conditions
• Drift-implicit Euler scheme converges strongly for SDEs with

C 1

pol-coeff., one-sided Lipschitz drift and monotone condition

• ’Drift-implicit-Lamperti’-Euler for CIR and WF: standard

convergence rate (∆| log(∆)|)1/2 is recovered Work in Progress

• Modified explicit Euler scheme to prevent moment explosions?
• CIR: Exponential integrability of scheme?
• CIR for 2κλ < θ2: Convergence rates?
• General theory for approximation of square-root diffusions?
• ...

Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 26/26