[PPT] - Outline 1 The General Framework A Central Limit Theorem for PowerPoint Presentation

SLIDE 1

The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms

A Central Limit Theorem for Truncating Stochastic Algorithms

J´ erˆ

me Lelong

http://cermics.enpc.fr/∼lelong

Tuesday September 5, 2006

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me Lelong (CERMICS)

Tuesday September 5, 2006 1 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms

Outline

1 The General Framework

A standard stochastic algorithm Truncating Algorithm: Chen’s technique

2 A motivating example in finance

An adaptive Importance Sampling Technique

3 Convergence Rate

Already known results New Results

4 Scheme of the proof

J´ erˆ

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Tuesday September 5, 2006 2 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework

1 The General Framework

A standard stochastic algorithm Truncating Algorithm: Chen’s technique

2 A motivating example in finance

An adaptive Importance Sampling Technique

3 Convergence Rate

Already known results New Results

4 Scheme of the proof

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Tuesday September 5, 2006 3 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework

General Framework

Let u: θ ∈ Rd − → u(θ) ∈ Rd, be a continuous function defined as an expectation on a probability space (Ω, A, P). u : Rd − → Rd θ − → E[U(θ, Z)]. Z is a r.v. in Rm and U a measurable function defined on Rd × Rm.

Hypothesis 1 (convexity)

∃! θ⋆ ∈ Rd, u(θ⋆) = 0 and ∀θ ∈ Rd, θ = θ⋆, (θ − θ⋆|u(θ)) > 0. Remark: if u is the gradient of a strictly convex function, then u satisfies Hypothesis 1. Problem: How to find the root of u ?

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Tuesday September 5, 2006 4 / 32

SLIDE 2

The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework A standard stochastic algorithm

For “sub-linear” functions

Assume that for all θ ∈ Rd E[U(θ, Z)2] ≤ K(1 + θ2). (1) We define for θ0 ∈ Rd θn+1 = θn − γn+1U(θn, Zn+1). (2) (Zn)n is i.i.d. following the law of Z. γn > 0, γn ց 0, γi = ∞ and γ2

i < ∞.

Theorem 1 (Robbins Monro)

Assume Hypothesis 1 and that Equation (1) is true then, the sequence (2) converges a.s. to θ⋆. Condition (1) is barely satisfied in practice.

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Tuesday September 5, 2006 5 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework Truncating Algorithm: Chen’s technique

For “fast” growing functions: an intuitive approach

Consider an increasing sequence of compact sets (Kj)j such that ∞

j=0 Kj = Rd.

Consider (Zn)n and (γn)n as defined previously. Prevent the algorithm from blowing up: At each step, θn should remain in a given compact set. If such is not the case, reset the algorithm and consider a larger compact set. This is due to Chen (see [Chen and Zhu, 1986]).

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Tuesday September 5, 2006 6 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework Truncating Algorithm: Chen’s technique

For “fast” growing functions

+

θ⋆ Kσn Kσn+1

+

θn

+

θ0

+

θn+1 =θn+ 1

2

= Kσn+1

+

θn+ 1

2

θn+1 = = Kσn+1

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Tuesday September 5, 2006 7 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework Truncating Algorithm: Chen’s technique

For “fast” growing functions: mathematical approach

For θ0 ∈ K0 and σ0 = 0, we define (θn)n and (σn)n      θn+ 1

2 = θn − γn+1U(θn, Zn+1),

if θn+ 1

2 ∈ Kσn

θn+1 = θn+ 1

2

and σn+1 = σn, if θn+ 1

2 /

∈ Kσn θn+1 = θ0 and σn+1 = σn + 1. (3) σn counts the number of truncations up to time n. Fn = σ(Zk; k ≤ n). θn is Fn−measurable and Zn+1 independent

f Fn.

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Tuesday September 5, 2006 8 / 32

SLIDE 3

The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework Truncating Algorithm: Chen’s technique

It is often more convenient to rewrite (3) as follows θn+1 = θn − γn+1u(θn)

Newton algorithm

− γn+1δMn+1

noise term
standard Robbins Monro algorithm

+ γn+1pn+1

truncation term

(4) where δMn+1 = U(θn, Zn+1) − u(θn), and pn+1 =

u(θn) + δMn+1 +

1 γn+1 (θ0 − θn)

if θn+ 1

2 /

∈ Kσn,

therwise.

δMn is a martingale increment, pn is the truncation term.

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Tuesday September 5, 2006 9 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms The General Framework Truncating Algorithm: Chen’s technique

a.s convergence of Chen’s procedure

Hypothesis 2 (integrability)

For all p > 0, the series

n γn+1δMn+11θn≤p converges a.s.

Hypothesis 2 is satisfied as soon as u and θ − → E[U(θ, Z)2] are bounded on any compact sets (or continuous).

Hint

Theorem 2

Under Hypotheses 1 and 2, the sequence (θn)n defined by (3) converges a.s. to θ⋆ and the sequence (σn)n is a.s. finite. A proof of this theorem can be found in [Delyon, 1996].

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Tuesday September 5, 2006 10 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms A motivating example in finance

1 The General Framework

A standard stochastic algorithm Truncating Algorithm: Chen’s technique

2 A motivating example in finance

An adaptive Importance Sampling Technique

3 Convergence Rate

Already known results New Results

4 Scheme of the proof

J´ erˆ

me Lelong (CERMICS)

Tuesday September 5, 2006 11 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms A motivating example in finance An adaptive Importance Sampling Technique

General Problem

Option pricing problem in a Brownian driven model (no jump): compute E[ψ(G)] by a MC method with G ∼ N(0, Id). One way of reducing the variance is to perform importance sampling techniques. For all θ ∈ Rd, E[ψ(G)] = E

ψ(G + θ)e−θ·G− |θ|2

2

.

(5) Minimise v(θ) = E

ψ(G + θ)2e−2θ·G−|θ|2

= E

ψ(G)2e−θ·G+ |θ|2

2

.

∇v(θ) = E

(θ − G)ψ(G)2e−θ·G+ |θ|2

2

.

(6)

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Tuesday September 5, 2006 12 / 32

SLIDE 4

The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms A motivating example in finance An adaptive Importance Sampling Technique

Analysis of the problem

To use the previous algorithms, we set u = ∇v and Z = G. v is strictly convex, hence Hypothesis 1 is satisfied. ∇v is not ”sub-linear”. ⇒ Cannot use a standard S.A.

Theorem

Hypothesis 2 holds even if ψ is of an exponential type. Theorem 2 holds and provides a way to compute θ⋆.

Theorem

More details in [Arouna, 2004].

Example J´ erˆ

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Tuesday September 5, 2006 13 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate

1 The General Framework

A standard stochastic algorithm Truncating Algorithm: Chen’s technique

2 A motivating example in finance

An adaptive Importance Sampling Technique

3 Convergence Rate

Already known results New Results

4 Scheme of the proof

J´ erˆ

me Lelong (CERMICS)

Tuesday September 5, 2006 14 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate Already known results

A CLT for Standard S.A. I

First, let us consider the standard algorithm θn+1 = θn − γn+1u(θn) − γn+1δMn+1 with γn =

γ (n+1)α , 1/2 < α ≤ 1.

For n ≥ 0, we define the renormalised error ∆n = θn − θ⋆ √γn .

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Tuesday September 5, 2006 15 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate Already known results

A CLT for Standard S.A. II

Hypothesis 3

There exists a function y : Rd → Rd×d satisfying limx→0 y(x) = 0 and a symmetric definite positive matrix A such that u(θ) = A(θ − θ⋆) + y(θ − θ⋆)(θ − θ⋆). There exists a real number ρ > 0 such that κ = sup

n E

δMn2+ρ

< ∞. There exists a symmetric definite positive matrix Σ such that E (δMnδM ′

n|Fn−1) P

− − − − →

n→∞ Σ.

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Tuesday September 5, 2006 16 / 32

SLIDE 5

The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate Already known results

A CLT for Standard S.A. III

Under Hypotheses 1 and 3 and if the algorithm converges a.s., then ∆n

L

− → N(0, V ). A proof of this result can be found in [Kushner and Yin, 2003], [Benveniste et al., 1990], [Bouton, 1985], [Duflo, 1997] for instance.

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me Lelong (CERMICS)

Tuesday September 5, 2006 17 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate New Results

Why we cannot deduce a CLT for truncating S.A. from the CLT for Standard S.A.

The number of projections is a.s. finite. ∀ω ∈ A, P(A) = 1, ∃N(ω), ∀n > N(ω), pn(ω) = 0. N is r.v. a.s. finite but not bounded. Hence, one cannot use a time shifting argument. Random time shifting does not preserve convergence in

distribution. If Xn

L

− → X and τ < ∞ a.s., one can show trivial examples where Xn+τ does not converge. Choose for instance τ and τ ′ 2 independent r.v. on {0, 1} with parameter 1/2 and set Xn := (−1)n(τ − τ ′).

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Tuesday September 5, 2006 18 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate New Results

A CLT for Randomly Truncating S.A.

Consider Chen’s Algorithm

Definition .

Theorem 3

Assume Hypotheses 1 and 3. If there exists η > 0 such that ∀n ≥ 0 d(θ⋆, ∂Kn) > η, then ∆n

L

− → N(0, V ). if 1/2 < α < 1 with V = ∞ exp (−At)Σ exp (−At)dt. if α = 1 and γA − I

2 > 0 with

V = γ ∞ exp I 2 − γA

t
Σ exp

I 2 − γA

t
dt.

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Tuesday September 5, 2006 19 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate New Results

A functional CLT for Chen’s algorithm I

We introduce a sequence of interpolating times {tn(u); u ≥ 0, n ≥ 0} tn(u) = sup

k ≥ 0

;

n+k

i=n

γi ≤ u

.

(7) with the convention sup ∅ = 0.

+ + + + + + +

γn γn + γn+1 Ptn(T ) i=0 γn+i T

∆n(0) = ∆n and ∆n(t) = ∆n+tn(t)+1 for t ≥ 0. For t ∈ n+p

i=n γi , n+p+1 i=n

γi

, tn(t) = p and ∆n(t) = ∆n+p+1.

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Tuesday September 5, 2006 20 / 32

SLIDE 6

The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate New Results

A functional CLT for Chen’s algorithm II

We introduce Wn(.) Wn(0) = 0 and Wn(t) =

n+tn(t)+1

i=n+1

√γiδMi for t > 0. (8)

Theorem 4

Assume the Hypotheses of Theorem 3. (∆n(·), Wn(·))

D×D

= = = ⇒ (∆, W)

n any finite time interval

where ∆ is a stationary Ornstein Uhlenbeck process of initial law N(0, V ) and W a Wiener process F∆,W −measurable with covariance matrix Σ.

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Tuesday September 5, 2006 21 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate New Results

Averaging Algorithms I

We restrict to γn =

γ (n+1)α with 1/2 < α < 1. For any t > 0, we

introduce a moving window average of the iterates ˆ θn(t) = γn t

n+

t γn

i=n

θi. (9) We study the renormalised error ˆ ∆n(t) = ˆ θn(t) − θ⋆ √γn = √γn t

n+

t γn

i=n

(θi − θ⋆). (10) We need to characterise the limit law of (∆n, ∆n+1, . . . , ∆n+p) when (n, p) → ∞.

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Tuesday September 5, 2006 22 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Convergence Rate New Results

A CLT for Averaging Algorithms

Theorem 5

Under the Hypotheses of Theorem 3, ˆ ∆n(t)

L

− →

n N(0, ˆ

V ) where ˆ V = 1 t A−1ΣA−1 + A−2(e−At − I)V + V A−2(e−At − I) t2

J´ erˆ

me Lelong (CERMICS)

Tuesday September 5, 2006 23 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Scheme of the proof

1 The General Framework

A standard stochastic algorithm Truncating Algorithm: Chen’s technique

2 A motivating example in finance

An adaptive Importance Sampling Technique

3 Convergence Rate

Already known results New Results

4 Scheme of the proof

J´ erˆ

me Lelong (CERMICS)

Tuesday September 5, 2006 24 / 32

SLIDE 7

The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Scheme of the proof

Major steps

∆n is tight in R.

Tightness

Use a localisation technique to prove that (supt∈[0,T ] ∆n(t))n is tight. Prove a Donsker Theorem for martingales increment.

Theorem

∆n(·) satisfies Aldous’ criteria.

Tightness in D

(Wn(·), ∆n(·))n is tight in D × D and converges in law to (W, ∆) where W is a Wiener process with respect to the smallest σ−algebra that measures (W(·), ∆(·)) with covariance matrix Σ and ∆ is the stationary solution of d∆(t) = −Q∆(t)dt − dW(t).

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Tuesday September 5, 2006 25 / 32 The General Framework A motivating example in finance Convergence Rate Scheme of the proof TCL for truncating Stochastic Algorithms Scheme of the proof

Donsker’s Theorem for martingales increments

Theorem 6

Let (Mn(t))n be a sequence of martingales. Assume that (Mn(·))n is tight in D and satisfies a C−tightness criteria. (Mn(t))n is a uniformly square integrable family for each t. Mnt

P

− →

n t

Then, Mn(·)

D[0,T ]

= = = = ⇒

n

B.M. Wn(t) satisfies Theorem 6.

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Tuesday September 5, 2006 26 / 32 Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Integrability

Comments on Hypothesis 2

Mn is a martingale Mn =

n

i=1

γiδMi1θi−1≤p, E (Mn|Fn−1) = Mn−1 + γn1θn−1≤pE (δMn|Fn−1) . if supnMn < ∞ then Mn converges a.s. Mn =

n

i=1

γ2

i E(δM 2 i |Fi−1)1θi−1≤p.

E(δM 2

i |Fi−1)

= E

U(θ, Z)2

|θ=θi−1 − u(θi−1)2

≤ E

U(θ, Z)2

|θ=θi−1 .

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Tuesday September 5, 2006 27 / 32 Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Random Time shifting

Random Time shifting

Let τ and τ ′ be 2 independent r.v. on {0, 1} with parameter 1/2. We set Xn := (−1)n(τ − τ ′). τ − τ ′ is symmetric. Hence, Xn is constant in law. E(eiuXn+τ ) = E(eiu(−1)n+τ (τ−τ ′)), = 1 2

E(eiu(−1)n+τ τ) + E(eiu(−1)n+τ(τ−1))
,

= 1 4

1 + eiu(−1)n+11 + eiu(−1)n(−1) + 1
,

= 1 2

1 + eiu(−1)n+1

. Hence, (Xn+τ)n does not converge in distribution.

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Tuesday September 5, 2006 28 / 32

SLIDE 8

Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Numerical implementation

Basket Option

1

N

i=0 Si T − K

+

basket size : 6 initial value : 110 110 120 80 90 100 maturity time : 1 strike : 120 interest rate : 0.02 volatility : 0.2 step size : 6 Sample Number : 5000 Figure: Evolution of the drift vector

1e3 2e3 3e3 4e3 5e3 −0.3 0.1 0.5 0.9 1.3

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Tuesday September 5, 2006 29 / 32 Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Numerical implementation

Basket Option

Figure: Importance Sampling

1e3 2e3 3e3 4e3 5e3 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

variance = 0.879592

Figure: Standard MC

1e3 2e3 3e3 4e3 5e3 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

variance = 11.8096

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Tuesday September 5, 2006 30 / 32 Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Numerical implementation

Atlas Option

Consider a basket of 16 stocks. At maturity time, remove the stocks with the three best and the three worst performances. It pays off 105% of the average of the remaining stocks.

basket size : 16 maturity time : 10 interest rate : 0.02 volatility : 0.2 step size : 0.1 Sample Number : 2000 Figure: Evolution of the drift vector

200 400 600 800 1000 1200 1400 1600 1800 2000 0.02 0.04 0.06 0.08

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Tuesday September 5, 2006 31 / 32 Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Numerical implementation

Atlas Option

Figure: Importance Sampling

200 400 600 800 1000 1200 1400 1600 1800 2000 0.2 0.4 0.6 0.8 1.0

variance = 0.0275906

Figure: Standard MC

200 400 600 800 1000 1200 1400 1600 1800 2000 0.2 0.4 0.6 0.8 1.0

variance = 0.206152

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Tuesday September 5, 2006 32 / 32

SLIDE 9

Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Numerical implementation

Arouna, B. (Winter 2003/2004). Robbins-monro algorithms and variance reduction in finance. The Journal of Computational Finance, 7(2). Benveniste, A., M´ etivier, M., and Priouret, P. (1990). Adaptive algorithms and stochastic approximations, volume 22 of Applications of Mathematics (New York). Springer-Verlag, Berlin. Translated from the French by Stephen S. Wilson. Bouton, C. (1985). Approximation Gaussienne d’algorithmes stochastiques ` a dynamique Markovienne. PhD thesis, Universit´ e Pierre et Marie Curie - Paris 6. Chen, H. and Zhu, Y. (1986). Stochastic Approximation Procedure with randomly varying truncations. Scientia Sinica Series.

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Tuesday September 5, 2006 32 / 32 Integrability Random Time shifting Numerical implementation TCL for truncating Stochastic Algorithms Numerical implementation

Delyon, B. (1996). General results on the convergence of stochastic algorithms. IEEE Transactions on Automatic Control, 41(9):1245–1255. Duflo, M. (1997). Random Iterative Models. Springer-Verlag Berlin and New York. Kushner, H. and Yin, G. (2003). Stochastic Approximation and Recursive Algorthims and Applications. Springer-Verlag New York, second edition.

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Tuesday September 5, 2006 32 / 32