Numerical approximation of two-scale SDEs

Numerical approximation of two-scale SDEs

Camilo Andr´ es Garc´ıa Trillos

Camilo-Andres.GARCIA@unice.fr

Colloque de Jeunes Probabilistes et Statisticiens

CIRM Marseille - Avril 18 2012

Stochastic Volatility

Stochastic volatility

dS _t = b(S _t )dt + σ(S _t , v _t )dW _t where v _t is also given as the solution of an SDE.

Empirical studies on high frequency data (Ex: S&P [Fouqu´ e et al. (98)] ; IBOVESPA [Souza et al. (06)] ) support financial asset modeling using Stochastic Volatility with

• Mean reverting volatility

• Fast reverting process: mean reverting time in the order of

days ( maturity of instruments/derivatives).

Stochastic Volatility

Stochastic volatility

dS _t = b(S _t )dt + σ(S _t , v _t )dW _t where v _t is also given as the solution of an SDE.

Empirical studies on high frequency data (Ex: S&P [Fouqu´ e et al.

(98)] ; IBOVESPA [Souza et al. (06)] ) support financial asset modeling using Stochastic Volatility with

• Mean reverting volatility

• Fast reverting process: mean reverting time in the order of days ( maturity of instruments/derivatives).

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 2 / 16

A general two-scale SDE

We consider the following two-scale SDE X _t

= x ₀ +

Z t 0

f (X _s

, Y _s

) ds + Z t

0

g (X _s

, Y _s

) dW _s Y _t

= y 0 +

Z t 0

b(X _s

, Y _s

)ds +

Z t

0

σ(X _s

, Y _s

)d W ˜ s ,

We will assume

• 1

• W and ˜ W are independent

• f (x, y) and g (x, y) C ¹ with linear growth in x, C _b

in y

• b(x, y), C _b

in both x, y

• Non-degenerate: λ _M > σσ

(x, y) > λ _m > 0. And mean reverting

|y|→∞

lim b(x, y)y = −∞

A general two-scale SDE

We consider the following two-scale SDE X _t

= x ₀ +

Z t 0

f (X _s

, Y _s

) ds + Z t

0

g (X _s

, Y _s

) dW _s Y _t

= y 0 +

Z t 0

b(X _s

, Y _s

)ds +

Z t

0

σ(X _s

, Y _s

)d W ˜ s , We will assume

• 1

• W and ˜ W are independent

• f (x, y ) and g (x, y ) C ¹ with linear growth in x, C _b

in y

• b(x, y), C _b

in both x, y

• Non-degenerate: λ _M > σσ

(x, y) > λ _m > 0. And mean reverting

|y|→∞

lim b(x, y)y = −∞

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 3 / 16

Homogenization result

Theorem [Pardoux & Veretennikov (01, 03)]

Under the hypothesis the rescaled, frozen parameter diffusion Y _t ^x = y 0 +

Z t 0

b(x, Y _s ^x )ds + Z t

0

σ(x, Y _s ^x )d W ˜ s ,

is ergodic with invariant measure µ ^x . X

−→

X where X solves X t = x 0 +

Z t 0

F (X s )ds + Z t

0

G (X s )dW s (Effective equation), with G = √

A, and F and A are averages of f and a = gg

with respect to µ ^x .

F (x) = Z

f (x, y )µ ^x (dy ) A(x) = Z

g · g

(x, y)µ ^x (dy )

Homogenization result

Theorem [Pardoux & Veretennikov (01, 03)]

Under the hypothesis the rescaled, frozen parameter diffusion Y _t ^x = y 0 +

Z t 0

b(x, Y _s ^x )ds + Z t

0

σ(x, Y _s ^x )d W ˜ s ,

is ergodic with invariant measure µ ^x . X

−→

X where X solves X t = x 0 +

Z t 0

F (X s )ds + Z t

0

G (X s )dW s (Effective equation), with G = √

A, and F and A are averages of f and a = gg

with respect to µ ^x .

• Problem: In general we do not know µ ^x explicitly

• Our goal: Propose a numerical method to approximate the Effective equation

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 4 / 16

Effective equation approximation algorithm

• Suppose we have good (possibly random) estimates ˜ F ⁽ⁿ⁾ ≈ F , G ˜ ⁽ⁿ⁾ ≈ G

• Discretize the approximated equation. Euler scheme: For t _k = k/n

X ˜ _t ⁿ

= 1 n F ˜ ⁽ⁿ⁾

X ˜ _t ⁿ

+ 1

√ n G ˜ ⁽ⁿ⁾

X ˜ _t ⁿ

U _k+1 where U k ∼ N (0, 1)

• Similar approach used for deterministic [Fatkullin,

Vanden-Eijnden (04)] and stochastic setup [E et al. (04)] .

• Our approach: Choose estimators for the averages ˜ F ⁽ⁿ⁾ , G ˜ ⁽ⁿ⁾

allowing us to develop a C.L.T like result for the strong error.

Effective equation approximation algorithm

• Suppose we have good (possibly random) estimates ˜ F ⁽ⁿ⁾ ≈ F , G ˜ ⁽ⁿ⁾ ≈ G

• Discretize the approximated equation. Euler scheme: For t _k = k/n

X ˜ _t ⁿ

k+1

= 1

n F ˜ ⁽ⁿ⁾ X ˜ _t ⁿ

k

+ 1

√ n G ˜ ⁽ⁿ⁾

X ˜ _t ⁿ

k

U _k+1 where U k ∼ N (0, 1)

• Similar approach used for deterministic [Fatkullin,

Vanden-Eijnden (04)] and stochastic setup [E et al. (04)] .

• Our approach: Choose estimators for the averages ˜ F ⁽ⁿ⁾ , G ˜ ⁽ⁿ⁾ allowing us to develop a C.L.T like result for the strong error.

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 5 / 16

Effective equation approximation algorithm

• Suppose we have good (possibly random) estimates ˜ F ⁽ⁿ⁾ ≈ F , G ˜ ⁽ⁿ⁾ ≈ G

• Discretize the approximated equation. Euler scheme: For t _k = k/n

X ˜ _t ⁿ

k+1

= 1

n F ˜ ⁽ⁿ⁾ X ˜ _t ⁿ

k

+ 1

√ n G ˜ ⁽ⁿ⁾

X ˜ _t ⁿ

k

U _k+1 where U k ∼ N (0, 1)

• Similar approach used for deterministic [Fatkullin,

Vanden-Eijnden (04)] and stochastic setup [E et al. (04)] .

• Our approach: Choose estimators for the averages ˜ F ⁽ⁿ⁾ , G ˜ ⁽ⁿ⁾

allowing us to develop a C.L.T like result for the strong error.

Invariant approximation algorithm: Decreasing Euler step

Decreasing Euler step

Let {γ _k } be a sequence of decreasing positive reals tending to zero, and Γ _M := P M

k=0 γ _k , and ¯ U _k ∼ N (0, 1)

• Decreasing step Euler scheme:

Y ¯ _k+1 ^x = ¯ Y _k ^x + γ _{k +1} b x, Y ¯ _k ^x + √

γ _k+1 σ x, Y ¯ _k ^x U ¯ _k+1 ,

• Average estimator:

ν(f , x; M ) := 1 Γ _M

M

X

k=1

γ k f x , Y ¯ _k−1 ^x

≈ 1 Γ

Z

0

f (x, Y

) ds

≈ lim

M→∞

1 Γ

Z

0

f (x, Y

) ds = Z

f (x , y)µ

(dy ) = F (x)

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 6 / 16

Invariant approximation algorithm: Decreasing Euler step

Decreasing Euler step

Let {γ _k } be a sequence of decreasing positive reals tending to zero, and Γ _M := P M

k=0 γ _k , and ¯ U _k ∼ N (0, 1)

• Decreasing step Euler scheme:

Y ¯ _k+1 ^x = ¯ Y _k ^x + γ _{k +1} b x, Y ¯ _k ^x + √

γ _k+1 σ x, Y ¯ _k ^x U ¯ _k+1 ,

• Average estimator:

ν(f , x; M ) := 1 Γ _M

M

X

k=1

γ k f x , Y ¯ _k−1 ^x

≈ 1 Γ

Z

0

f (x, Y

) ds

≈ lim

M→∞

1 Γ

Z

0

f (x, Y

) ds = Z

f (x, y)µ

(dy ) = F (x)

Invariant approximation algorithm

Properties [Lamberton, Pag` es (02)]

Under some hypothesis on the step sizes γ _k ,

1

Almost sure convergence: For x fixed, ν(f , x; M ) −→ ^a.s. F (x)

2

C.L.T.: For x fixed,

p Γ M (ν(f , x ; M) − F (x)) −→ N

(0, Ξ(x))

Ξ depends on µ ^x and the coefficients of the ergodic diffusion, but not on the choice of γ k .

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 7 / 16

Effective equation approximation

X ¯ _t ⁽ⁿ⁾

= 1

n ν f , X ¯ t

; M(n) + 1

√ n q

ν gg

, X ¯ t

; M (n) U k+1

i.e. it is an Euler scheme for which we use independent realizations of the decreasing Euler estimator at each discretization step.

1

We fix γ k = k

for 1/3 < θ < 1.

2

We relate the two parameters M and n. The most efficient way to do so is by fixing M (n) such that

Γ M ∝ n

Result 1 : Strong convergence

n→∞ lim E sup

0≤t≤T

X t − X ¯ _t ⁽ⁿ⁾

2 !

→ 0

Sketch of the proof Stability technique

• A priori bounds

• Obtain a global L 2 error control from step-wise L 2 error control (Burkholder maximal inequality + Gronwall lemma)

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 9 / 16

Result 2 : Limit error distribution

Limit distribution

If the effective diffusion is non-degenarte, fixing Γ _M = n n ^1/2

X _t − X ¯ ⁽ⁿ⁾

⇒ ζ with ζ defined as the solution of

ζ t = Z t

0

∂ x F (X s )ζ(s)ds + Z t

0

∂ x G (X s )ζ (s )dW s

+ 1

√ 2

Z t 0

∂ x G (X s )G(X s )dB _s ¹ + Z t

0

p Ξ(X s )dB _s ²

with B ¹ and B ² are two independent standard Brownian motions.

Result 2 : Limit error distribution

Limit distribution

If the effective diffusion is non-degenarte, fixing Γ _M = n n ^1/2

X _t − X ¯ ⁽ⁿ⁾

⇒ ζ with ζ defined as the solution of

ζ t = Z

0

∂

F (X

)ζ(s )ds + Z

0

∂

G (X

)ζ(s)dW

+ 1

√ 2

Z

0

∂

G (X

)G(X

)dB

| {z }

Euler discretization

+ Z

0

p Ξ(X

)dB

| {z }

average approx.

with B ¹ and B ² are two independent standard Brownian motions.

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 10 / 16

Result 2 : Limit error distribution - outline of the proof

n ^1/2

X t − X ¯ ⁽ⁿ⁾

=: ζ ⁿ ⇒ ζ

Sketch of the proof

• Tightness

• Obtain an SDE for the error ζ

= ˜ ζ

+ R

• Prove R

−→

0

• Classical results (Kurtz and Protter) to deduce tightness of ˜ ζ

from convergence in law of the tuple of coefficients of related SDE.

• Convergence of the tuple: Use C.L.T of decreasing Euler, independence and convergence of the quadratic variations.

• Identification is straightforward thanks to strong convergence.

Result 3 : Romberg extrapolation

Let l ∈ N , l ≥ 2. Let c ₁ , . . . , c _l ∈ R satisfy the linear system

l

X

i=1

c i = 1

l

X

i =1



 c i

Γ _iM

iM

X

j=1

γ _j ^r



 = 0 for r = 2, . . . , l.

Define the approximation function

ˆ

ν(x , F ; M, l) =

l

X

i=1

c _i ν(x, F ; iM )

Convergence and limit error results using this approximation are unchanged (up to a constant), but in this case we may take

1

2l + 1 < θ < 1

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 12 / 16

Result 3 : Romberg extrapolation

Let l ∈ N , l ≥ 2. Let c ₁ , . . . , c _l ∈ R satisfy the linear system

l

X

i=1

c i = 1

l

X

i =1



 c i

Γ _iM

iM

X

j=1

γ _j ^r



 = 0 for r = 2, . . . , l.

Define the approximation function

ˆ

ν(x , F ; M, l) =

l

X

i=1

c _i ν(x, F ; iM )

Convergence and limit error results using this approximation are unchanged (up to a constant), but in this case we may take

1

2l + 1 < θ < 1

Efficiency analysis

Define

τ := # of operations Our algorithm with n steps requires

τ (n) = Kn

For a fixed strong error tolerance ∆ ( recall ∆(n) := Kn

) :

• Simple SDE (θ > 1/3) :

τ (∆) = K ∆

≥ K ∆

• Interpolated SDE (θ > _2l+1 ¹ ):

τ (∆) = K _l ∆

≥ K _l ∆

.

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 13 / 16

Efficiency analysis

Define

τ := # of operations Our algorithm with n steps requires

τ (n) = Kn

For a fixed strong error tolerance ∆ ( recall ∆(n) := Kn

) :

• Simple SDE (θ > 1/3) :

τ (∆) = K ∆

≥ K ∆

• Interpolated SDE (θ > _2l+1 ¹ ):

τ (∆) = K _l ∆

≥ K _l ∆

.

Numerical test

Test problem:

dX _t

= X _t

dt + X _t

Y _t

dW t

dY _t

=

s

1

2(1 + (X _t

) ² ) − Y _t

!

dt +

s

2(X _t

) ² + 1 (X _t

) ² + 1 dW t

We test the algorithm with

• γ _k = k

for the simple version (θ ≈ 1/3)

• γ _k = k

for the extrapolated version (θ ≈ 1/5)

Numerical approximation of two-scale SDEs CJPS - Avril 18 2012 14 / 16

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−10 −5 0 5 10

−10−50510

QQplot − SDE − Decreasing step

Normalized observed error (n^{1 2}⋅(X−X~))

Limit error (ζ)

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−10 −5 0 5 10

−10−50510

QQplot − SDE − Extrapolated

Normalized observed error (n^{1 2}⋅(X−X~))

Limit error (ζ)

5e−02 5e−01 5e+00 5e+01

0.10.20.51.02.0

SDE − L

error vs. Time

Time (s) Error

|

|

Simple (r=−0.18) Extrap. (r=−0.22)