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
X
(j)
b (j) (Xt )dWt ,
t ∈ [0, T ]
j=1
X0 = x0 ∈ R
d
where
• a : Rd → Rd
• b=
drift coefficient
(b (1) , . . . , b (m) )
• W =
with b (j) : Rd → Rd
(W (1) , . . . , W (m) )0
Assumption
diffusion coeff.
m-dimensional Brownian motion
(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 Itô 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
b (∆) = x0
X
0
bt(∆) = X
bt(∆) + a(X
bt(∆) )∆ + b(X
bt(∆) ) ∆i W ,
X
i+1
i
i
i
i = 0, . . . , n − 1
with
∆i W = Wti+1 − Wti
Extension to [0, T ] by piecewise linear interpolation, i.e.
bt(∆) = ti+1 − t X
bt(∆) + t − ti X
bt(∆) ,
X
i
i+1
∆
∆
Theorem (strong error)
Under (Lip)
bt |2
E max |Xt − X
t∈[0,T ]
Andreas Neuenkirch
t ∈ [ti , ti+1 ]
Faure (1992); ...
1/2
≤ c(a, b, x0 ) · (∆| log(∆)|)1/2
Strong Approximation of SDEs under Non-Lipschitz Assumptions
5/26
Two SDEs from Mathematical Finance
Heston model
p
p
(1)
(2) |Vt |St
1 − ρ2 dWt + ρdWt ,
p
(2)
dVt = κ(λ − Vt ) dt + θ |Vt | dWt ,
dSt = µSt dt +
s0 > 0
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 +
p
p
(1)
(2) |Vt |St
1 − ρ2 dWt + ρdWt ,
3/2
dVt = c1 Vt (c2 − Vt ) dt + c3 |Vt |
(2)
dWt ,
s0 > 0
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. Gyöngy (1998); D. Higham, X. Mao, A. Stuart (2002)
Many contributions since then; several talks here at MCQMC 2012
on 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
bt = X
bt + a(X
bt ) ∆ + b(X
bt ) ∆i W ,
X
i+1
i
i
i
b0 = x0
X
Extension to [0, T ] by piecewise linear interpolation
Theorem (pathwise error) Gyöngy (1998)
If (S), a ∈ C 1 (D; Rd ), b ∈ C 1 (D; Rd,m ), then for all ε > 0
bt (ω)| ≤ Cε (ω) · ∆1/2−ε
max |Xt (ω) − X
t∈[0,T ]
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
Gyöngy (1998)
If (S), a ∈ C 1 (D; Rd ), b ∈ C 1 (D; Rd,m ), then for all ε > 0
bt (ω)| ≤ Cε (ω) · ∆1/2−ε
max |Xt (ω) − X
t∈[0,T ]
for almost all ω ∈ Ω
Remarks
• Proof uses localization strategy
• Applies to Heston model if 2κλ ≥ θ 2 and to 3/2-model
• For D 6= Rd : use suitable modification of the coefficients
outside D for better numerical stability, e.g. x + instead of |x|
• Above result can be extended to general Itô-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 |Xt |p < ∞,
(M2)
t∈[0,T ]
bt(∆) |p < ∞
sup E max |X
∆>0
t∈[0,T ]
for some p > 2, then
bt(∆) |2 → 0
E max |Xt − X
for
∆→0
t∈[0,T ]
Proof previous Theorem and integration to the limit using (M)
Remarks
• Original proof did not use Gyöngy’s result
• Applies to CIR if 2κλ ≥ θ 2 (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
N
1 X b (i)
|V4 |
N
i=1
b (i) iid copies of V
b4
for E|V4 | = 0.5662... where V
4
repetitions / stepsize
N = 103
104
105
106
107
∆ = 20
6.3272
6.8947
7.4306
7.2274
7.2792
2−2
Inf
Inf
Inf
Inf
Inf
2−4
Inf
Inf
Inf
Inf
Inf
2−6
0.5502
Inf
Inf
Inf
Inf
2−8
0.5535
0.5627
0.5662
Inf
Inf
2−10
0.5551
0.5634
0.5671
0.5658
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.
1
(G) max{|a(x)|, |b(x)|} ≥ · |x|β , min{|a(x)|, |b(x)|} ≤ c · |x|α
c
for |x| ≥ c.
Theorem Hutzenthaler, Jentzen, Kloeden (2011)
Let p > 1 s.th. supt∈[0,T ] E|Xt |p < ∞ and let b(x0 ) 6= 0.
If (G), then
b (∆) |p = ∞
lim E|X
T
∆→0
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
outside (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
one-sided Lipschitz drift coefficients, i.e.
(one-sided Lip)
hx − y , a(x) − a(y )i ≤ L|x − y |2 ,
x, y ∈ Rd
Example: a(x) = x − x 3
Drift-implicit Euler scheme
X ti+1 = X ti + a(X ti+1 ) ∆ + b(X ti ) ∆i W ,
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
If a ∈
Szpruch, Mao (2012)
1 (Rd ; Rd ),
Cpol
(monotone)
1 (Rd ; Rd,m ), a one-sided Lipschitz and
b ∈ Cpol
1
hx, a(x)i + |b(x)|2 ≤ α + β|x|2 ,
2
x ∈ Rd
for some α, β > 0, then
(∆) p
lim E max |Xt − X t
∆→0
| =0
t∈[0,T ]
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
bt = X
bt + a(X
bt ) ∆ + b(X
bt ) ∆i W ,
X
i+1
i
i
i
b0 = x0
X
• 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 ) ∆i W ,
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
p
p
1 ft + ρdWt ,
d log(St ) = µ − Vt dt + |Vt | 1 − ρ2 d W
2
p
dVt = κ(λ − Vt ) dt + θ |Vt | dWt ,
s0 > 0
v0 > 0
f one-dim. Brownian
where ρ ∈ (−1, 1), κ, λ, θ > 0, µ ∈ R and W
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
p
|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 =
U tk+1 =
√
v0
θ/2 ∆k W
U tk +
2 + κ∆
s
+
(U tk + θ/2 ∆k W )2 κλ − θ2/4
∆
+
(2 + κ∆)2
2 + κ∆
and
2
V tk = U tk
Extension to [0, T ] by piecewise linear interpolation
Properties
• Scheme well defined for 4κλ ≥ θ 2
• Non-negativity of CIR preserved for 4κλ ≥ θ 2
√
• 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 |Vt − V t |p
t∈[0,T ]
1/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 Itô’s lemma,
√
√
d Vt = a( Vt ) dt + 2θ dWt
where
4κλ − θ2 1 κ
· − ·x
8
x
2
one-sided Lipschitz on D = (0, ∞), i.e.
a(x) =
(x − y )(a(x) − a(y )) ≤ − κ2 (x − y )2 ,
x, y > 0
and −a coercive on D = (0, ∞), i.e.
limx→0 a(x) = ∞,
Drift-implicit Euler scheme for
limx→∞ a(x) = −∞
√
Vt :
θ
U tk+1 = U tk + a(U tk+1 )∆ + ∆k W
2
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 + 2θ dWt
with
a(x) =
4κλ − θ2 1 κ
· − ·x
8
x
2
Sketch of Proof
p
(1) Error analysis for Vtk − U tk :
(i) Drift one-sided Lipschitz: control of error propagation
(ii) Analysis of one step error, difficulty E V1q = ∞ if q ≥ 2κλ
θ2
t
(2) Error analysis for Vt − V t :
(i) Control transformation x 7→ x 2
(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
2−3
2−4
2−5
2−6
2−7
2−8
cost
32 · 26
42 · 28
52 · 210
62 · 212
72 · 214
82 · 216
mean
2.92775
2.96910
2.94909
2.95133
2.94731
2.95110
max
5.31776
4.20493
3.78347
3.31206
3.12879
3.04030
min
0.05180
1.74487
2.29099
2.62872
2.74758
2.86977
std
0.77934
0.40637
0.21749
0.11197
0.06161
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
p
|Xt (1 − Xt )| dWt ,
x0 ∈ (0, 1)
with α, β, γ > 0
Note: Lamperti-transformation
Z
Λ(x) =
For CIR: Λ(x) =
2√
θ x
For WF: Λ(x) =
2
γ
x
1
dξ
b(ξ)
√
√
arcsin( x) and Yt = 2 arcsin( X t ) satisfies
dYt = a(Yt ) dt + γdWt
where
x x γ2 γ2 a(x) = α −
cot
− β −α−
tan
,
x ∈ (0, π)
4
2
4
2
one-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
U tk+1 = U tk + a(U tk+1 )∆ + γ∆k W ,
√
U 0 = 2 arcsin( x0 )
and
X tk = sin2
U tk
2
Extension to [0, T ] by piecewise linear interpolation
Theorem
If
Dereich, N, Szpruch (2012)
2α 2(β − α)
1 ≤ p < min
,
,
γ2
γ2
then there exists Kp > 0 such that
1/p
E max |Xt − X t |p
≤ Kp · (∆| log(∆)|)1/2
t∈[0,T ]
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
1 -coeff., one-sided Lipschitz drift and monotone condition
Cpol
• ’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 < θ 2 : Convergence rates?
• General theory for approximation of square-root diffusions?
• ...
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Strong Approximation of SDEs under Non-Lipschitz Assumptions
26/26