Non-parametric methods for

Lévy-based financial models

José Enrique Figueroa-López1

1Department of Statistics

Purdue University

WORKSHOP ON INFINITELY DIVISIBLE PROCESSES

CIMAT, Guanajuato, Gto., México

March 16, 2009

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Introduction Some Lévy-based financial models

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Some Lévy-based financial models

Overview of Lévy-based financial models

1 Lévy process: {Xt}t≥0(i) X0 = 0, (ii) Independent and stationary increments,

(iii) càdlàg paths, (iv) no fixed jump times P(∆Xt 6= 0) = 0.2 Exponential Lévy models: St = S0eXt .

• Examples: Variance Gamma, CGMY, generalized Hyperbolic, etc.

• Advantages: Return distributions with heavy-tails, high kurtosis, asymmetry.

• Drawbacks: Lack of volatility clustering and leverage.

3 Time-changed Lévy models (Carr, et. al.): St = S0eXτ(t) , whereτ(t) =

∫ t0 rudu, drt = α(m − rt )dt + v

√rt dWt .

• Advantages: Introduce stochastic clustering.

• Drawbacks: Lack of leverage.

4 Stochastic volatility driven by OU (Barndorff-Nielsen and Shephard):

St := S0 exp{∫ t

0 budu +∫ t

0 σuWu}

, where σ2t = σ20 +

∫ t0 ασ

2s ds + Xαt .

Introduction Formulation of the problems

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Introduction Formulation of the problems

Formulation of the problems

1 Set-up:• {Xt}t≥0 is a Lévy process with Lévy measure ν.• ν(dx) = s(x)dx , where s is the Lévy density;

s(x) = intensity of jumps with size close to x .

• The process is discretely sampled at π : 0 = t0 < · · · < tn = T .

2 Objectives:

• Construct functional estimators ŝπ for s on a domain D = (a, b) (a > 0),

relying only on “qualitative" assumptions on s;

• Compared with the “best" estimator based on continuous sampling on [0,T ];

• Devise (asymptotic) point-wise confidence intervals:

Pˆs(x) ∈ (ŝπ(x)± σ̂(x)zα/2)

˜ T→∞,π̄→0−→ 1− α, for each x ∈ D;(π̄ := max{ti+1 − ti})

• Construct (asymptotic) confidence bands:

Pˆs(x) ∈ (ŝπ(x)± dα/2(x)),∀x ∈ D

˜ T→∞,π̄→0−→ 1− α.

Introduction Formulation of the problems

Formulation of the problems

1 Set-up:• {Xt}t≥0 is a Lévy process with Lévy measure ν.• ν(dx) = s(x)dx , where s is the Lévy density;

s(x) = intensity of jumps with size close to x .

• The process is discretely sampled at π : 0 = t0 < · · · < tn = T .

2 Objectives:

• Construct functional estimators ŝπ for s on a domain D = (a, b) (a > 0),

relying only on “qualitative" assumptions on s;

• Compared with the “best" estimator based on continuous sampling on [0,T ];

• Devise (asymptotic) point-wise confidence intervals:

Pˆs(x) ∈ (ŝπ(x)± σ̂(x)zα/2)

˜ T→∞,π̄→0−→ 1− α, for each x ∈ D;(π̄ := max{ti+1 − ti})

• Construct (asymptotic) confidence bands:

Pˆs(x) ∈ (ŝπ(x)± dα/2(x)),∀x ∈ D

˜ T→∞,π̄→0−→ 1− α.

Introduction Formulation of the problems

Formulation of the problems

1 Set-up:• {Xt}t≥0 is a Lévy process with Lévy measure ν.• ν(dx) = s(x)dx , where s is the Lévy density;

s(x) = intensity of jumps with size close to x .

• The process is discretely sampled at π : 0 = t0 < · · · < tn = T .

2 Objectives:

• Construct functional estimators ŝπ for s on a domain D = (a, b) (a > 0),

relying only on “qualitative" assumptions on s;

• Compared with the “best" estimator based on continuous sampling on [0,T ];

• Devise (asymptotic) point-wise confidence intervals:

Pˆs(x) ∈ (ŝπ(x)± σ̂(x)zα/2)

˜ T→∞,π̄→0−→ 1− α, for each x ∈ D;(π̄ := max{ti+1 − ti})

• Construct (asymptotic) confidence bands:

Pˆs(x) ∈ (ŝπ(x)± dα/2(x)),∀x ∈ D

˜ T→∞,π̄→0−→ 1− α.

Introduction Formulation of the problems

Formulation of the problems

1 Set-up:• {Xt}t≥0 is a Lévy process with Lévy measure ν.• ν(dx) = s(x)dx , where s is the Lévy density;

s(x) = intensity of jumps with size close to x .

• The process is discretely sampled at π : 0 = t0 < · · · < tn = T .

2 Objectives:

• Construct functional estimators ŝπ for s on a domain D = (a, b) (a > 0),

relying only on “qualitative" assumptions on s;

• Compared with the “best" estimator based on continuous sampling on [0,T ];

• Devise (asymptotic) point-wise confidence intervals:

Pˆs(x) ∈ (ŝπ(x)± σ̂(x)zα/2)

˜ T→∞,π̄→0−→ 1− α, for each x ∈ D;(π̄ := max{ti+1 − ti})

• Construct (asymptotic) confidence bands:

Pˆs(x) ∈ (ŝπ(x)± dα/2(x)),∀x ∈ D

˜ T→∞,π̄→0−→ 1− α.

Introduction Formulation of the problems

Formulation of the problems

1 Set-up:• {Xt}t≥0 is a Lévy process with Lévy measure ν.• ν(dx) = s(x)dx , where s is the Lévy density;

s(x) = intensity of jumps with size close to x .

• The process is discretely sampled at π : 0 = t0 < · · · < tn = T .

2 Objectives:

• Construct functional estimators ŝπ for s on a domain D = (a, b) (a > 0),

relying only on “qualitative" assumptions on s;

• Compared with the “best" estimator based on continuous sampling on [0,T ];

• Devise (asymptotic) point-wise confidence intervals:

Pˆs(x) ∈ (ŝπ(x)± σ̂(x)zα/2)

˜ T→∞,π̄→0−→ 1− α, for each x ∈ D;(π̄ := max{ti+1 − ti})

• Construct (asymptotic) confidence bands:

Pˆs(x) ∈ (ŝπ(x)± dα/2(x)),∀x ∈ D

˜ T→∞,π̄→0−→ 1− α.

Introduction Formulation of the problems

Formulation of the problems

1 Set-up:• {Xt}t≥0 is a Lévy process with Lévy measure ν.• ν(dx) = s(x)dx , where s is the Lévy density;

s(x) = intensity of jumps with size close to x .

• The process is discretely sampled at π : 0 = t0 < · · · < tn = T .

2 Objectives:

• Construct functional estimators ŝπ for s on a domain D = (a, b) (a > 0),

relying only on “qualitative" assumptions on s;

• Compared with the “best" estimator based on continuous sampling on [0,T ];

• Devise (asymptotic) point-wise confidence intervals:

Pˆs(x) ∈ (ŝπ(x)± σ̂(x)zα/2)

˜ T→∞,π̄→0−→ 1− α, for each x ∈ D;(π̄ := max{ti+1 − ti})

• Construct (asymptotic) confidence bands:

Pˆs(x) ∈ (ŝπ(x)± dα/2(x)),∀x ∈ D

˜ T→∞,π̄→0−→ 1− α.

Introduction Formulation of the problems

Why is the estimation problem interesting?

1 The Lévy measure ν determines the jump behavior of the process:

ν(A) =1t· E{

∑s≤t

1{∆Xs∈A}}.

2 The sizes of the jumps {∆Xs : s ≤ t ,∆Xs ∈ D} are unobservable basedon discrete data observations

Xti+1 − Xti = Pure-jump Lévy (ν)︸ ︷︷ ︸Jumps on (ti ,ti+1]

+σ(Wti+1 −Wti )︸ ︷︷ ︸White noise

+ b∆t︸︷︷︸Drift

3 Intuition from the finite-jump activity case:

• {Xti − Xti−1}i will recover the jumps of X in [0,T ] if maxi{ti − ti−1} → 0;• Need T →∞ for consistency of the estimation.

Introduction Formulation of the problems

Why is the estimation problem interesting?

1 The Lévy measure ν determines the jump behavior of the process:

ν(A) =1t· E{

∑s≤t

1{∆Xs∈A}}.

2 The sizes of the jumps {∆Xs : s ≤ t ,∆Xs ∈ D} are unobservable basedon discrete data observations

Xti+1 − Xti = Pure-jump Lévy (ν)︸ ︷︷ ︸Jumps on (ti ,ti+1]

+σ(Wti+1 −Wti )︸ ︷︷ ︸White noise

+ b∆t︸︷︷︸Drift

3 Intuition from the finite-jump activity case:

• {Xti − Xti−1}i will recover the jumps of X in [0,T ] if maxi{ti − ti−1} → 0;• Need T →∞ for consistency of the estimation.

Introduction Formulation of the problems

Why is the estimation problem interesting?

1 The Lévy measure ν determines the jump behavior of the process:

ν(A) =1t· E{

∑s≤t

1{∆Xs∈A}}.

2 The sizes of the jumps {∆Xs : s ≤ t ,∆Xs ∈ D} are unobservable basedon discrete data observations

Xti+1 − Xti = Pure-jump Lévy (ν)︸ ︷︷ ︸Jumps on (ti ,ti+1]

+σ(Wti+1 −Wti )︸ ︷︷ ︸White noise

+ b∆t︸︷︷︸Drift

3 Intuition from the finite-jump activity case:

• {Xti − Xti−1}i will recover the jumps of X in [0,T ] if maxi{ti − ti−1} → 0;• Need T →∞ for consistency of the estimation.

Introduction Formulation of the problems

Why is the estimation problem interesting?

1 The Lévy measure ν determines the jump behavior of the process:

ν(A) =1t· E{

∑s≤t

1{∆Xs∈A}}.

2 The sizes of the jumps {∆Xs : s ≤ t ,∆Xs ∈ D} are unobservable basedon discrete data observations

Xti+1 − Xti = Pure-jump Lévy (ν)︸ ︷︷ ︸Jumps on (ti ,ti+1]

+σ(Wti+1 −Wti )︸ ︷︷ ︸White noise

+ b∆t︸︷︷︸Drift

3 Intuition from the finite-jump activity case:

• {Xti − Xti−1}i will recover the jumps of X in [0,T ] if maxi{ti − ti−1} → 0;• Need T →∞ for consistency of the estimation.

Introduction Formulation of the problems

Why is the estimation problem interesting?

1 The Lévy measure ν determines the jump behavior of the process:

ν(A) =1t· E{

∑s≤t

1{∆Xs∈A}}.

2 The sizes of the jumps {∆Xs : s ≤ t ,∆Xs ∈ D} are unobservable basedon discrete data observations

Xti+1 − Xti = Pure-jump Lévy (ν)︸ ︷︷ ︸Jumps on (ti ,ti+1]

+σ(Wti+1 −Wti )︸ ︷︷ ︸White noise

+ b∆t︸︷︷︸Drift

3 Intuition from the finite-jump activity case:

• {Xti − Xti−1}i will recover the jumps of X in [0,T ] if maxi{ti − ti−1} → 0;• Need T →∞ for consistency of the estimation.

Estimation of Lévy densities Minimax rate of convergence

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Estimation of Lévy densities Minimax rate of convergence

Minimax rate of convergence

1 What is the best possible rate of convergence when T →∞?• Suppose that s is "smooth" on a domain D = [a, b] (away from the origin):

Θα = {s : |s(k)(x)− s(k)(y)| ≤ L|x − y |β , ∀x , y ∈ D} (k ∈ N, 0 < β ≤ 1)

α = k + β is called the "Degree of Smoothness" of s.

• ST : All estimators ŝT based on {Xt}t≤T .• [F. 2008]: Optimal minimax rate is O

“T−2α/(2α+1)

”;

lim infT→∞

T2α

2α+1 infŝT ∈ST

sups∈Θα

EsZ

D(ŝT (x)− s(x))

2 dx > 0.

2 Problem: Devise discrete-based estimator s̃T that attains the optimal rate

O(T−2α/(2α+1)

)?

Estimation of Lévy densities Minimax rate of convergence

Minimax rate of convergence

1 What is the best possible rate of convergence when T →∞?• Suppose that s is "smooth" on a domain D = [a, b] (away from the origin):

Θα = {s : |s(k)(x)− s(k)(y)| ≤ L|x − y |β , ∀x , y ∈ D} (k ∈ N, 0 < β ≤ 1)

α = k + β is called the "Degree of Smoothness" of s.

• ST : All estimators ŝT based on {Xt}t≤T .• [F. 2008]: Optimal minimax rate is O

“T−2α/(2α+1)

”;

lim infT→∞

T2α

2α+1 infŝT ∈ST

sups∈Θα

EsZ

D(ŝT (x)− s(x))

2 dx > 0.

2 Problem: Devise discrete-based estimator s̃T that attains the optimal rate

O(T−2α/(2α+1)

)?

Estimation of Lévy densities Minimax rate of convergence

Minimax rate of convergence

1 What is the best possible rate of convergence when T →∞?• Suppose that s is "smooth" on a domain D = [a, b] (away from the origin):

Θα = {s : |s(k)(x)− s(k)(y)| ≤ L|x − y |β , ∀x , y ∈ D} (k ∈ N, 0 < β ≤ 1)

α = k + β is called the "Degree of Smoothness" of s.

• ST : All estimators ŝT based on {Xt}t≤T .• [F. 2008]: Optimal minimax rate is O

“T−2α/(2α+1)

”;

lim infT→∞

T2α

2α+1 infŝT ∈ST

sups∈Θα

EsZ

D(ŝT (x)− s(x))

2 dx > 0.

2 Problem: Devise discrete-based estimator s̃T that attains the optimal rate

O(T−2α/(2α+1)

)?

Estimation of Lévy densities Minimax rate of convergence

Minimax rate of convergence

1 What is the best possible rate of convergence when T →∞?• Suppose that s is "smooth" on a domain D = [a, b] (away from the origin):

Θα = {s : |s(k)(x)− s(k)(y)| ≤ L|x − y |β , ∀x , y ∈ D} (k ∈ N, 0 < β ≤ 1)

α = k + β is called the "Degree of Smoothness" of s.

• ST : All estimators ŝT based on {Xt}t≤T .• [F. 2008]: Optimal minimax rate is O

“T−2α/(2α+1)

”;

lim infT→∞

T2α

2α+1 infŝT ∈ST

sups∈Θα

EsZ

D(ŝT (x)− s(x))

2 dx > 0.

2 Problem: Devise discrete-based estimator s̃T that attains the optimal rate

O(T−2α/(2α+1)

)?

Estimation of Lévy densities Minimax rate of convergence

Minimax rate of convergence

1 What is the best possible rate of convergence when T →∞?• Suppose that s is "smooth" on a domain D = [a, b] (away from the origin):

Θα = {s : |s(k)(x)− s(k)(y)| ≤ L|x − y |β , ∀x , y ∈ D} (k ∈ N, 0 < β ≤ 1)

α = k + β is called the "Degree of Smoothness" of s.

• ST : All estimators ŝT based on {Xt}t≤T .• [F. 2008]: Optimal minimax rate is O

“T−2α/(2α+1)

”;

lim infT→∞

T2α

2α+1 infŝT ∈ST

sups∈Θα

EsZ

D(ŝT (x)− s(x))

2 dx > 0.

2 Problem: Devise discrete-based estimator s̃T that attains the optimal rate

O(T−2α/(2α+1)

)?

Estimation of Lévy densities Minimax rate of convergence

Minimax rate of convergence

1 What is the best possible rate of convergence when T →∞?• Suppose that s is "smooth" on a domain D = [a, b] (away from the origin):

Θα = {s : |s(k)(x)− s(k)(y)| ≤ L|x − y |β , ∀x , y ∈ D} (k ∈ N, 0 < β ≤ 1)

α = k + β is called the "Degree of Smoothness" of s.

• ST : All estimators ŝT based on {Xt}t≤T .• [F. 2008]: Optimal minimax rate is O

“T−2α/(2α+1)

”;

lim infT→∞

T2α

2α+1 infŝT ∈ST

sups∈Θα

EsZ

D(ŝT (x)− s(x))

2 dx > 0.

2 Problem: Devise discrete-based estimator s̃T that attains the optimal rate

O(T−2α/(2α+1)

)?

Estimation of Lévy densities Minimax rate of convergence

Some relevant results

1 [F. & Houdré 2006]: Proposes estimator s̃cT

such that

lim supT→∞

T2α

2α+1 sups∈Θα

Es∫

D

(s̃c

T(x)− s(x)

)2 dx

Estimation of Lévy densities Minimax rate of convergence

Some relevant results

1 [F. & Houdré 2006]: Proposes estimator s̃cT

such that

lim supT→∞

T2α

2α+1 sups∈Θα

Es∫

D

(s̃c

T(x)− s(x)

)2 dx

Estimation of Lévy densities Minimax rate of convergence

Some relevant results

1 [F. & Houdré 2006]: Proposes estimator s̃cT

such that

lim supT→∞

T2α

2α+1 sups∈Θα

Es∫

D

(s̃c

T(x)− s(x)

)2 dx

Estimation of Lévy densities Minimax rate of convergence

Some relevant results

1 [F. & Houdré 2006]: Proposes estimator s̃cT

such that

lim supT→∞

T2α

2α+1 sups∈Θα

Es∫

D

(s̃c

T(x)− s(x)

)2 dx

Estimation of Lévy densities Nonparametric sieve estimators

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Estimation of Lévy densities Nonparametric sieve estimators

The method of sievesGrenander (81), Birgé & Massart (97).

• Idea: Approximate s(·) on D by a linear combinationβ1ϕ1(·) + · · ·+ βdϕd (·) of orthonormal functions ϕ1, . . . , ϕd ∈ L2(D,dx)

• S = span{ϕ1, . . . , ϕd} is called the Sieve.

• Typical sieves: Splines, wavelets, trigonometric polynomials, etc.

• Under the condition s ∈ L2(D,dx), the best approximation of s in S is theorthogonal projection:

s∗(·) =(∫

ϕ1(x)s(x)dx)ϕ1(·) + · · ·+

(∫ϕd (x)s(x)dx

)ϕd (·)

• Goal: Find estimators β̂(ϕ) for β(ϕ) :=∫

D ϕ(x)s(x)dx :

ŝ := β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd

Estimation of Lévy densities Nonparametric sieve estimators

The method of sievesGrenander (81), Birgé & Massart (97).

• Idea: Approximate s(·) on D by a linear combinationβ1ϕ1(·) + · · ·+ βdϕd (·) of orthonormal functions ϕ1, . . . , ϕd ∈ L2(D,dx)

• S = span{ϕ1, . . . , ϕd} is called the Sieve.

• Typical sieves: Splines, wavelets, trigonometric polynomials, etc.

• Under the condition s ∈ L2(D,dx), the best approximation of s in S is theorthogonal projection:

s∗(·) =(∫

ϕ1(x)s(x)dx)ϕ1(·) + · · ·+

(∫ϕd (x)s(x)dx

)ϕd (·)

• Goal: Find estimators β̂(ϕ) for β(ϕ) :=∫

D ϕ(x)s(x)dx :

ŝ := β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd

Estimation of Lévy densities Nonparametric sieve estimators

The method of sievesGrenander (81), Birgé & Massart (97).

• Idea: Approximate s(·) on D by a linear combinationβ1ϕ1(·) + · · ·+ βdϕd (·) of orthonormal functions ϕ1, . . . , ϕd ∈ L2(D,dx)

• S = span{ϕ1, . . . , ϕd} is called the Sieve.

• Typical sieves: Splines, wavelets, trigonometric polynomials, etc.

• Under the condition s ∈ L2(D,dx), the best approximation of s in S is theorthogonal projection:

s∗(·) =(∫

ϕ1(x)s(x)dx)ϕ1(·) + · · ·+

(∫ϕd (x)s(x)dx

)ϕd (·)

• Goal: Find estimators β̂(ϕ) for β(ϕ) :=∫

D ϕ(x)s(x)dx :

ŝ := β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd

Estimation of Lévy densities Nonparametric sieve estimators

The method of sievesGrenander (81), Birgé & Massart (97).

• Idea: Approximate s(·) on D by a linear combinationβ1ϕ1(·) + · · ·+ βdϕd (·) of orthonormal functions ϕ1, . . . , ϕd ∈ L2(D,dx)

• S = span{ϕ1, . . . , ϕd} is called the Sieve.

• Typical sieves: Splines, wavelets, trigonometric polynomials, etc.

• Under the condition s ∈ L2(D,dx), the best approximation of s in S is theorthogonal projection:

s∗(·) =(∫

ϕ1(x)s(x)dx)ϕ1(·) + · · ·+

(∫ϕd (x)s(x)dx

)ϕd (·)

• Goal: Find estimators β̂(ϕ) for β(ϕ) :=∫

D ϕ(x)s(x)dx :

ŝ := β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd

Estimation of Lévy densities Nonparametric sieve estimators

The method of sievesGrenander (81), Birgé & Massart (97).

• Idea: Approximate s(·) on D by a linear combinationβ1ϕ1(·) + · · ·+ βdϕd (·) of orthonormal functions ϕ1, . . . , ϕd ∈ L2(D,dx)

• S = span{ϕ1, . . . , ϕd} is called the Sieve.

• Typical sieves: Splines, wavelets, trigonometric polynomials, etc.

• Under the condition s ∈ L2(D,dx), the best approximation of s in S is theorthogonal projection:

s∗(·) =(∫

ϕ1(x)s(x)dx)ϕ1(·) + · · ·+

(∫ϕd (x)s(x)dx

)ϕd (·)

• Goal: Find estimators β̂(ϕ) for β(ϕ) :=∫

D ϕ(x)s(x)dx :

ŝ := β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd

Estimation of Lévy densities Nonparametric sieve estimators

The method of sievesGrenander (81), Birgé & Massart (97).

• Idea: Approximate s(·) on D by a linear combinationβ1ϕ1(·) + · · ·+ βdϕd (·) of orthonormal functions ϕ1, . . . , ϕd ∈ L2(D,dx)

• S = span{ϕ1, . . . , ϕd} is called the Sieve.

• Typical sieves: Splines, wavelets, trigonometric polynomials, etc.

• Under the condition s ∈ L2(D,dx), the best approximation of s in S is theorthogonal projection:

s∗(·) =(∫

ϕ1(x)s(x)dx)ϕ1(·) + · · ·+

(∫ϕd (x)s(x)dx

)ϕd (·)

• Goal: Find estimators β̂(ϕ) for β(ϕ) :=∫

D ϕ(x)s(x)dx :

ŝ := β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd

Estimation of Lévy densities Nonparametric sieve estimators

The method of sievesGrenander (81), Birgé & Massart (97).

• Idea: Approximate s(·) on D by a linear combinationβ1ϕ1(·) + · · ·+ βdϕd (·) of orthonormal functions ϕ1, . . . , ϕd ∈ L2(D,dx)

• S = span{ϕ1, . . . , ϕd} is called the Sieve.

• Typical sieves: Splines, wavelets, trigonometric polynomials, etc.

• Under the condition s ∈ L2(D,dx), the best approximation of s in S is theorthogonal projection:

s∗(·) =(∫

ϕ1(x)s(x)dx)ϕ1(·) + · · ·+

(∫ϕd (x)s(x)dx

)ϕd (·)

• Goal: Find estimators β̂(ϕ) for β(ϕ) :=∫

D ϕ(x)s(x)dx :

ŝ := β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd (Projection Estimator)

Estimation of Lévy densities Nonparametric sieve estimators

Estimators for β(ϕ) := RD ϕ(x)s(x)dx .Realized ϕ−variation of X in [0,T ] per unit time: [Woerner 03], [F. 04]

β̂(ϕ) :=1tn

n∑k=1

ϕ(Xtk − Xtk−1

),

Properties: [Woerner 03, F. & Houdré 06; F. 07]

If ϕ satisfies some regularity conditions (?), then

as tn →∞ and π̄n := max{tk − tk−1} → 0,

1 Eβ̂(ϕ) −→ β(ϕ)

2 β̂(ϕ)P−→β(ϕ)

3 E{β̂(ϕ)− β(ϕ)

}2−→ 0, provided that ϕ2 meets (?).

4√

tn(β̂(ϕ)− β(ϕ)

)D→ β(ϕ2) 12 N (0,1), provided that π̄n

√tn → 0.

Estimation of Lévy densities The rate of convergence of the risk

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Estimation of Lévy densities The rate of convergence of the risk

Rate of convergence for spline estimators

Set-up:

1 Let Skm := span{ϕ1, . . . , ϕd} be all regular piece-wise polynomials onD = [a,b] of degree at most k with m classes (d = (k + 1)m).

2 s is bounded and smooth on the domain D with “smoothness" α.

3 Let ŝT ,m = β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd be the projection estimator on Skmbased on Xt0 , . . . ,Xtn (T = tn)

Theorem: [F. 2008]

If (i) maxk{tk − tk−1} = O(T−1), (ii) mT :=[T 1/(2α+1)

], and (iii) k + 1 > α, then

lim supT→∞

T2α

2α+1 sups∈Θα

E ‖s − ŝT ,mT ‖2

Estimation of Lévy densities The rate of convergence of the risk

Rate of convergence for spline estimators

Set-up:

1 Let Skm := span{ϕ1, . . . , ϕd} be all regular piece-wise polynomials onD = [a,b] of degree at most k with m classes (d = (k + 1)m).

2 s is bounded and smooth on the domain D with “smoothness" α.

3 Let ŝT ,m = β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd be the projection estimator on Skmbased on Xt0 , . . . ,Xtn (T = tn)

Theorem: [F. 2008]

If (i) maxk{tk − tk−1} = O(T−1), (ii) mT :=[T 1/(2α+1)

], and (iii) k + 1 > α, then

lim supT→∞

T2α

2α+1 sups∈Θα

E ‖s − ŝT ,mT ‖2

Estimation of Lévy densities The rate of convergence of the risk

Rate of convergence for spline estimators

Set-up:

1 Let Skm := span{ϕ1, . . . , ϕd} be all regular piece-wise polynomials onD = [a,b] of degree at most k with m classes (d = (k + 1)m).

2 s is bounded and smooth on the domain D with “smoothness" α.

3 Let ŝT ,m = β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd be the projection estimator on Skmbased on Xt0 , . . . ,Xtn (T = tn)

Theorem: [F. 2008]

If (i) maxk{tk − tk−1} = O(T−1), (ii) mT :=[T 1/(2α+1)

], and (iii) k + 1 > α, then

lim supT→∞

T2α

2α+1 sups∈Θα

E ‖s − ŝT ,mT ‖2

Estimation of Lévy densities The rate of convergence of the risk

Rate of convergence for spline estimators

Set-up:

1 Let Skm := span{ϕ1, . . . , ϕd} be all regular piece-wise polynomials onD = [a,b] of degree at most k with m classes (d = (k + 1)m).

2 s is bounded and smooth on the domain D with “smoothness" α.

3 Let ŝT ,m = β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd be the projection estimator on Skmbased on Xt0 , . . . ,Xtn (T = tn)

Theorem: [F. 2008]

If (i) maxk{tk − tk−1} = O(T−1), (ii) mT :=[T 1/(2α+1)

], and (iii) k + 1 > α, then

lim supT→∞

T2α

2α+1 sups∈Θα

E ‖s − ŝT ,mT ‖2

Estimation of Lévy densities The rate of convergence of the risk

Rate of convergence for spline estimators

Set-up:

1 Let Skm := span{ϕ1, . . . , ϕd} be all regular piece-wise polynomials onD = [a,b] of degree at most k with m classes (d = (k + 1)m).

2 s is bounded and smooth on the domain D with “smoothness" α.

3 Let ŝT ,m = β̂(ϕ1)ϕ1 + · · ·+ β̂(ϕd )ϕd be the projection estimator on Skmbased on Xt0 , . . . ,Xtn (T = tn)

Theorem: [F. 2008]

If (i) maxk{tk − tk−1} = O(T−1), (ii) mT :=[T 1/(2α+1)

], and (iii) k + 1 > α, then

lim supT→∞

T2α

2α+1 sups∈Θα

E ‖s − ŝT ,mT ‖2

Confidence intervals for the Lévy density A pointwise CLT

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Pointwise confidence intervals

1 Question: For given x ∈ D and s ∈ Θα, is it possible to construct a sieveestimator ŝT such that

Tα

2α+1 (ŝT (x)− s(x))D→ σ̄(x)N (0,1)?

2 Typical strategy for a CLT with normalizing constant cT .

(i) Show the CLT cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0, 1).

(ii) Show that that bias is o(c−1T ): cT (EŝT (x)− s(x))→ 0.

3 It seems that (i) and (ii) cannot be satisfied simultaneously:

• Var(ŝT (x)) ∼mTT

s(x)2

(b−a)2 suggesting thatc2T mT

T → 1 and σ̄(x) =s(x)b−a ;

• EŝT (x)− s(x) � m−αT suggesting that cT m

−αT → 0;

• Not possible to take cT = Tα

2α+1 ,

• but can take cT = Tβ , for any 0 < β < α2α+1 .

Confidence intervals for the Lévy density A pointwise CLT

Theorem ([F. 2008])

1 Sieve: regular piece-wise polynomials of degree k with m classes

2 s has smoothness α ≥ 1 on D with α ≤ k + 1

3 b2m(x) :=∑k

j=0(2j + 1)Q2j

(2x−(am+bm)

bm−am

), where Qj is the Legendre

polynomial of degree j on [−1,1] and (am,bm] is the class containing x.

Then, for any 0 < β < α2α+1 ,

T β

bmT(ŝT (x)− s(x))

D→ s(x)b − a

N (0,1), T →∞,

provided that mT =[T 1−2β

]and π̄ := maxk{tk − tk−1} ≤ T−γ with γ > 1− β.

Asymptotic 100(1− α)% confidence intervals for s(x):

ŝT (x)±bmT (x)

(b−a)1/2 · T−β · ŝ1/2

T(x) · zα/2

Confidence intervals for the Lévy density A pointwise CLT

Theorem ([F. 2008])

1 Sieve: regular piece-wise polynomials of degree k with m classes

2 s has smoothness α ≥ 1 on D with α ≤ k + 1

3 b2m(x) :=∑k

j=0(2j + 1)Q2j

(2x−(am+bm)

bm−am

), where Qj is the Legendre

polynomial of degree j on [−1,1] and (am,bm] is the class containing x.

Then, for any 0 < β < α2α+1 ,

T β

bmT(ŝT (x)− s(x))

D→ s(x)b − a

N (0,1), T →∞,

provided that mT =[T 1−2β

]and π̄ := maxk{tk − tk−1} ≤ T−γ with γ > 1− β.

Asymptotic 100(1− α)% confidence intervals for s(x):

ŝT (x)±bmT (x)

(b−a)1/2 · T−β · ŝ1/2

T(x) · zα/2

Confidence intervals for the Lévy density A pointwise CLT

Theorem ([F. 2008])

1 Sieve: regular piece-wise polynomials of degree k with m classes

2 s has smoothness α ≥ 1 on D with α ≤ k + 1

3 b2m(x) :=∑k

j=0(2j + 1)Q2j

(2x−(am+bm)

bm−am

), where Qj is the Legendre

polynomial of degree j on [−1,1] and (am,bm] is the class containing x.

Then, for any 0 < β < α2α+1 ,

T β

bmT(ŝT (x)− s(x))

D→ s(x)b − a

N (0,1), T →∞,

provided that mT =[T 1−2β

]and π̄ := maxk{tk − tk−1} ≤ T−γ with γ > 1− β.

Asymptotic 100(1− α)% confidence intervals for s(x):

ŝT (x)±bmT (x)

(b−a)1/2 · T−β · ŝ1/2

T(x) · zα/2

Confidence intervals for the Lévy density A pointwise CLT

Theorem ([F. 2008])

1 Sieve: regular piece-wise polynomials of degree k with m classes

2 s has smoothness α ≥ 1 on D with α ≤ k + 1

3 b2m(x) :=∑k

j=0(2j + 1)Q2j

(2x−(am+bm)

bm−am

), where Qj is the Legendre

polynomial of degree j on [−1,1] and (am,bm] is the class containing x.

Then, for any 0 < β < α2α+1 ,

T β

bmT(ŝT (x)− s(x))

D→ s(x)b − a

N (0,1), T →∞,

provided that mT =[T 1−2β

]and π̄ := maxk{tk − tk−1} ≤ T−γ with γ > 1− β.

Asymptotic 100(1− α)% confidence intervals for s(x):

ŝT (x)±bmT (x)

(b−a)1/2 · T−β · ŝ1/2

T(x) · zα/2

Confidence intervals for the Lévy density A pointwise CLT

Theorem ([F. 2008])

1 Sieve: regular piece-wise polynomials of degree k with m classes

2 s has smoothness α ≥ 1 on D with α ≤ k + 1

3 b2m(x) :=∑k

j=0(2j + 1)Q2j

(2x−(am+bm)

bm−am

), where Qj is the Legendre

polynomial of degree j on [−1,1] and (am,bm] is the class containing x.

Then, for any 0 < β < α2α+1 ,

T β

bmT(ŝT (x)− s(x))

D→ s(x)b − a

N (0,1), T →∞,

provided that mT =[T 1−2β

]and π̄ := maxk{tk − tk−1} ≤ T−γ with γ > 1− β.

Asymptotic 100(1− α)% confidence intervals for s(x):

ŝT (x)±bmT (x)

(b−a)1/2 · T−β · ŝ1/2

T(x) · zα/2

Confidence intervals for the Lévy density A pointwise CLT

Theorem ([F. 2008])

1 Sieve: regular piece-wise polynomials of degree k with m classes

2 s has smoothness α ≥ 1 on D with α ≤ k + 1

3 b2m(x) :=∑k

j=0(2j + 1)Q2j

(2x−(am+bm)

bm−am

), where Qj is the Legendre

polynomial of degree j on [−1,1] and (am,bm] is the class containing x.

Then, for any 0 < β < α2α+1 ,

T β

bmT(ŝT (x)− s(x))

D→ s(x)b − a

N (0,1), T →∞,

provided that mT =[T 1−2β

]and π̄ := maxk{tk − tk−1} ≤ T−γ with γ > 1− β.

Asymptotic 100(1− α)% confidence intervals for s(x):

ŝT (x)±bmT (x)

(b−a)1/2 · T−β · ŝ1/2

T(x) · zα/2

Confidence intervals for the Lévy density A pointwise CLT

On the tools behind the proof

1 cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0,1) will follow from the CLT for i.i.d.

2

∣∣∣∣Eϕ (X∆)∆ −∫ϕ(y)ν(dy)

∣∣∣∣ ≤ K (ϕ) supy∈[c,d ]

∣∣∣∣ 1∆P [X∆ ≥ y ]− ν([y ,∞))∣∣∣∣,

where K (ϕ) := (‖ϕ‖∞ + ‖ϕ′‖1) if ϕ ∈ C1([c,d ]) and supp(ϕ) ⊂ [c,d ].

3 Suppose that ν(dx) = s(x)dx with s bounded and Lipschitz on every

[c,d ] ⊂ R\{0}. Then, for some k 0,supy∈D

∣∣ 1t P [Xt ≥ y ]− ν([y ,∞))

∣∣ ≤ kt , whenever t < t0.4 Taylor’s expansions:

s(y)− s(x) =r∑

j=1

s(j)(x)j!

(y − x)j +∫ y

x

(s(r)(v)− s(r)(x)

) (y − v)r−1(r − 1)!

dv ,

where r := bαc for α > 1.

Confidence intervals for the Lévy density A pointwise CLT

On the tools behind the proof

1 cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0,1) will follow from the CLT for i.i.d.

2

∣∣∣∣Eϕ (X∆)∆ −∫ϕ(y)ν(dy)

∣∣∣∣ ≤ K (ϕ) supy∈[c,d ]

∣∣∣∣ 1∆P [X∆ ≥ y ]− ν([y ,∞))∣∣∣∣,

where K (ϕ) := (‖ϕ‖∞ + ‖ϕ′‖1) if ϕ ∈ C1([c,d ]) and supp(ϕ) ⊂ [c,d ].

3 Suppose that ν(dx) = s(x)dx with s bounded and Lipschitz on every

[c,d ] ⊂ R\{0}. Then, for some k 0,supy∈D

∣∣ 1t P [Xt ≥ y ]− ν([y ,∞))

∣∣ ≤ kt , whenever t < t0.4 Taylor’s expansions:

s(y)− s(x) =r∑

j=1

s(j)(x)j!

(y − x)j +∫ y

x

(s(r)(v)− s(r)(x)

) (y − v)r−1(r − 1)!

dv ,

where r := bαc for α > 1.

Confidence intervals for the Lévy density A pointwise CLT

On the tools behind the proof

1 cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0,1) will follow from the CLT for i.i.d.

2

∣∣∣∣Eϕ (X∆)∆ −∫ϕ(y)ν(dy)

∣∣∣∣ ≤ K (ϕ) supy∈[c,d ]

∣∣∣∣ 1∆P [X∆ ≥ y ]− ν([y ,∞))∣∣∣∣,

where K (ϕ) := (‖ϕ‖∞ + ‖ϕ′‖1) if ϕ ∈ C1([c,d ]) and supp(ϕ) ⊂ [c,d ].

3 Suppose that ν(dx) = s(x)dx with s bounded and Lipschitz on every

[c,d ] ⊂ R\{0}. Then, for some k 0,supy∈D

∣∣ 1t P [Xt ≥ y ]− ν([y ,∞))

∣∣ ≤ kt , whenever t < t0.4 Taylor’s expansions:

s(y)− s(x) =r∑

j=1

s(j)(x)j!

(y − x)j +∫ y

x

(s(r)(v)− s(r)(x)

) (y − v)r−1(r − 1)!

dv ,

where r := bαc for α > 1.

Confidence intervals for the Lévy density A pointwise CLT

On the tools behind the proof

1 cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0,1) will follow from the CLT for i.i.d.

2

∣∣∣∣Eϕ (X∆)∆ −∫ϕ(y)ν(dy)

∣∣∣∣ ≤ K (ϕ) supy∈[c,d ]

∣∣∣∣ 1∆P [X∆ ≥ y ]− ν([y ,∞))∣∣∣∣,

where K (ϕ) := (‖ϕ‖∞ + ‖ϕ′‖1) if ϕ ∈ C1([c,d ]) and supp(ϕ) ⊂ [c,d ].

3 Suppose that ν(dx) = s(x)dx with s bounded and Lipschitz on every

[c,d ] ⊂ R\{0}. Then, for some k 0,supy∈D

∣∣ 1t P [Xt ≥ y ]− ν([y ,∞))

∣∣ ≤ kt , whenever t < t0.4 Taylor’s expansions:

s(y)− s(x) =r∑

j=1

s(j)(x)j!

(y − x)j +∫ y

x

(s(r)(v)− s(r)(x)

) (y − v)r−1(r − 1)!

dv ,

where r := bαc for α > 1.

Confidence intervals for the Lévy density A pointwise CLT

On the tools behind the proof

1 cT (ŝT (x)− EŝT (x))D→ σ̄(x)N (0,1) will follow from the CLT for i.i.d.

2

∣∣∣∣Eϕ (X∆)∆ −∫ϕ(y)ν(dy)

∣∣∣∣ ≤ K (ϕ) supy∈[c,d ]

∣∣∣∣ 1∆P [X∆ ≥ y ]− ν([y ,∞))∣∣∣∣,

where K (ϕ) := (‖ϕ‖∞ + ‖ϕ′‖1) if ϕ ∈ C1([c,d ]) and supp(ϕ) ⊂ [c,d ].

3 Suppose that ν(dx) = s(x)dx with s bounded and Lipschitz on every

[c,d ] ⊂ R\{0}. Then, for some k 0,supy∈D

∣∣ 1t P [Xt ≥ y ]− ν([y ,∞))

∣∣ ≤ kt , whenever t < t0.4 Taylor’s expansions:

s(y)− s(x) =r∑

j=1

s(j)(x)j!

(y − x)j +∫ y

x

(s(r)(v)− s(r)(x)

) (y − v)r−1(r − 1)!

dv ,

where r := bαc for α > 1.

Confidence bands for the Lévy density A uniform CLT

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Confidence bands for the Lévy density A uniform CLT

General ideas

• Tools coming from [Bickel & Rosenblatt 1973] on confidence bands for

kernel density estimators.

• Set-up: Regular piece-wise polynomials as sieves and regular sampling

{tnk := kδn}nk=0, where δn :=tnn → 0 and tn →∞.

• Express ŝn(x) in terms of the empirical distribution F n of {Xtnk+1 − Xtnk }nk=0:

ŝn(x) :=m∑

i=1

k∑j=0

β̂n(ϕi,j )ϕi,j (x) = L(x ; m, δ−1n , F̄

n) ,where F̄ n := 1− F n and

L(x ; m, κ,F ) := κm∑

i=1

k∑j=0

ϕi,j (x)∫ϕi,j (u)dF̄ (u).

Confidence bands for the Lévy density A uniform CLT

General ideas

• Tools coming from [Bickel & Rosenblatt 1973] on confidence bands for

kernel density estimators.

• Set-up: Regular piece-wise polynomials as sieves and regular sampling

{tnk := kδn}nk=0, where δn :=tnn → 0 and tn →∞.

• Express ŝn(x) in terms of the empirical distribution F n of {Xtnk+1 − Xtnk }nk=0:

ŝn(x) :=m∑

i=1

k∑j=0

β̂n(ϕi,j )ϕi,j (x) = L(x ; m, δ−1n , F̄

n) ,where F̄ n := 1− F n and

L(x ; m, κ,F ) := κm∑

i=1

k∑j=0

ϕi,j (x)∫ϕi,j (u)dF̄ (u).

Confidence bands for the Lévy density A uniform CLT

General ideas

• Tools coming from [Bickel & Rosenblatt 1973] on confidence bands for

kernel density estimators.

• Set-up: Regular piece-wise polynomials as sieves and regular sampling

{tnk := kδn}nk=0, where δn :=tnn → 0 and tn →∞.

• Express ŝn(x) in terms of the empirical distribution F n of {Xtnk+1 − Xtnk }nk=0:

ŝn(x) :=m∑

i=1

k∑j=0

β̂n(ϕi,j )ϕi,j (x) = L(x ; m, δ−1n , F̄

n) ,where F̄ n := 1− F n and

L(x ; m, κ,F ) := κm∑

i=1

k∑j=0

ϕi,j (x)∫ϕi,j (u)dF̄ (u).

Confidence bands for the Lévy density A uniform CLT

General ideas

• Tools coming from [Bickel & Rosenblatt 1973] on confidence bands for

kernel density estimators.

• Set-up: Regular piece-wise polynomials as sieves and regular sampling

{tnk := kδn}nk=0, where δn :=tnn → 0 and tn →∞.

• Express ŝn(x) in terms of the empirical distribution F n of {Xtnk+1 − Xtnk }nk=0:

ŝn(x) :=m∑

i=1

k∑j=0

β̂n(ϕi,j )ϕi,j (x) = L(x ; m, δ−1n , F̄

n) ,where F̄ n := 1− F n and

L(x ; m, κ,F ) := κm∑

i=1

k∑j=0

ϕi,j (x)∫ϕi,j (u)dF̄ (u).

Confidence bands for the Lévy density A uniform CLT

• Similarly, ŝn(x)− Eŝn(x) = L(

x ; m, δ−1n , F̄ n − F̄δn)

, where Fδn is the

distribution of Xδn .

• [Bickel & Rosenblatt]: Let Z 0n (x) = n1/2 (F ∗n (x)− x), where F ∗n is theempirical distribution of

{Fδn(

Xtnk+1 − Xtnk)}n

k=1(uniform random sample).

• Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)• Approximate Z 0n by the Brownian bridge Z 0(x) := Z (x)− xZ (1), where{Z (x)}x∈[0,1] is a B.M. If Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0 (Fδn (·))

),

‖Y n0 − Y n1 ‖∞ =mtn

Op(

n1/4(log n)1/2(log log n)1/4).

• Strategy: Devise successive approximation Y n2 , . . . ,Ynl such that

• For some an and bn an (‖Y nl ‖∞ − bn)D→ G, non-degenerate distribution,

• an‖Y np+1 − Y np ‖∞ = op(1), for p = 1, . . . , l − 1.

Confidence bands for the Lévy density A uniform CLT

• Similarly, ŝn(x)− Eŝn(x) = L(

x ; m, δ−1n , F̄ n − F̄δn)

, where Fδn is the

distribution of Xδn .

• [Bickel & Rosenblatt]: Let Z 0n (x) = n1/2 (F ∗n (x)− x), where F ∗n is theempirical distribution of

{Fδn(

Xtnk+1 − Xtnk)}n

k=1(uniform random sample).

• Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)• Approximate Z 0n by the Brownian bridge Z 0(x) := Z (x)− xZ (1), where{Z (x)}x∈[0,1] is a B.M. If Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0 (Fδn (·))

),

‖Y n0 − Y n1 ‖∞ =mtn

Op(

n1/4(log n)1/2(log log n)1/4).

• Strategy: Devise successive approximation Y n2 , . . . ,Ynl such that

• For some an and bn an (‖Y nl ‖∞ − bn)D→ G, non-degenerate distribution,

• an‖Y np+1 − Y np ‖∞ = op(1), for p = 1, . . . , l − 1.

Confidence bands for the Lévy density A uniform CLT

• Similarly, ŝn(x)− Eŝn(x) = L(

x ; m, δ−1n , F̄ n − F̄δn)

, where Fδn is the

distribution of Xδn .

• [Bickel & Rosenblatt]: Let Z 0n (x) = n1/2 (F ∗n (x)− x), where F ∗n is theempirical distribution of

{Fδn(

Xtnk+1 − Xtnk)}n

k=1(uniform random sample).

• Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)• Approximate Z 0n by the Brownian bridge Z 0(x) := Z (x)− xZ (1), where{Z (x)}x∈[0,1] is a B.M. If Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0 (Fδn (·))

),

‖Y n0 − Y n1 ‖∞ =mtn

Op(

n1/4(log n)1/2(log log n)1/4).

• Strategy: Devise successive approximation Y n2 , . . . ,Ynl such that

• For some an and bn an (‖Y nl ‖∞ − bn)D→ G, non-degenerate distribution,

• an‖Y np+1 − Y np ‖∞ = op(1), for p = 1, . . . , l − 1.

Confidence bands for the Lévy density A uniform CLT

• Similarly, ŝn(x)− Eŝn(x) = L(

x ; m, δ−1n , F̄ n − F̄δn)

, where Fδn is the

distribution of Xδn .

• [Bickel & Rosenblatt]: Let Z 0n (x) = n1/2 (F ∗n (x)− x), where F ∗n is theempirical distribution of

{Fδn(

Xtnk+1 − Xtnk)}n

k=1(uniform random sample).

• Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)• Approximate Z 0n by the Brownian bridge Z 0(x) := Z (x)− xZ (1), where{Z (x)}x∈[0,1] is a B.M. If Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0 (Fδn (·))

),

‖Y n0 − Y n1 ‖∞ =mtn

Op(

n1/4(log n)1/2(log log n)1/4).

• Strategy: Devise successive approximation Y n2 , . . . ,Ynl such that

• For some an and bn an (‖Y nl ‖∞ − bn)D→ G, non-degenerate distribution,

• an‖Y np+1 − Y np ‖∞ = op(1), for p = 1, . . . , l − 1.

Confidence bands for the Lévy density A uniform CLT

• Similarly, ŝn(x)− Eŝn(x) = L(

x ; m, δ−1n , F̄ n − F̄δn)

, where Fδn is the

distribution of Xδn .

• [Bickel & Rosenblatt]: Let Z 0n (x) = n1/2 (F ∗n (x)− x), where F ∗n is theempirical distribution of

{Fδn(

Xtnk+1 − Xtnk)}n

k=1(uniform random sample).

• Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)• Approximate Z 0n by the Brownian bridge Z 0(x) := Z (x)− xZ (1), where{Z (x)}x∈[0,1] is a B.M. If Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0 (Fδn (·))

),

‖Y n0 − Y n1 ‖∞ =mtn

Op(

n1/4(log n)1/2(log log n)1/4).

• Strategy: Devise successive approximation Y n2 , . . . ,Ynl such that

• For some an and bn an (‖Y nl ‖∞ − bn)D→ G, non-degenerate distribution,

• an‖Y np+1 − Y np ‖∞ = op(1), for p = 1, . . . , l − 1.

Confidence bands for the Lévy density A uniform CLT

• Similarly, ŝn(x)− Eŝn(x) = L(

x ; m, δ−1n , F̄ n − F̄δn)

, where Fδn is the

distribution of Xδn .

• [Bickel & Rosenblatt]: Let Z 0n (x) = n1/2 (F ∗n (x)− x), where F ∗n is theempirical distribution of

{Fδn(

Xtnk+1 − Xtnk)}n

k=1(uniform random sample).

• Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)• Approximate Z 0n by the Brownian bridge Z 0(x) := Z (x)− xZ (1), where{Z (x)}x∈[0,1] is a B.M. If Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0 (Fδn (·))

),

‖Y n0 − Y n1 ‖∞ =mtn

Op(

n1/4(log n)1/2(log log n)1/4).

• Strategy: Devise successive approximation Y n2 , . . . ,Ynl such that

• For some an and bn an (‖Y nl ‖∞ − bn)D→ G, non-degenerate distribution,

• an‖Y np+1 − Y np ‖∞ = op(1), for p = 1, . . . , l − 1.

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Successive approximation

1 Y n0 (x) := ŝn(x)− Eŝn(x) = L(

x ; m, n1/2

tn,−Z 0n (Fδn (·))

)2 Y n1 (x) := L

(x ; m, n

1/2

tn,−Z 0

(F̄δn (·)

)), ({Z 0(x)}x≤1 Brownian bridge);

3 Y n2 (x) := L(

x ; m, n1/2

tn,Z(F̄δn (·)

))({Z (x)}x≤1 Brownian motion);

4 Y n3 (x) := L(

x ; m, t−1/2n ,Z(

1δn

F̄δn (·)))

;

5 Y n4 (x) := L(

x ; m, t−1/2n ,Z(∫∞· s(u)du

));

6 Y n5 (x) := L(

x ; m, t−1/2n ,∫∞· s

1/2(u)dZ (u))

=

t−1/2n∑m

i=1∑k

j=0 ϕi,j (x)∫ xi

xi−1s1/2(u)ϕi,j (u)dZ (u);

7 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u) ≈ s−1/2(x)Y n5 (x);

Confidence bands for the Lévy density A uniform CLT

Maxima of the last approximation

1 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u)

2 supx∈D |Y n6 (x)|D= t−1/2n m1/2 max1≤i≤m ζi , where, given Zj

iid∼ N (0,1),

ζiiid∼ ζ := sup

x∈[−1,1]

∣∣∣∣∣∣k∑

j=0

√2j + 1

2Qj (x)Zj

∣∣∣∣∣∣ .3 Question: What is the extreme value distribution and normalizing

constants of Mm := max{ζi : 1 ≤ i ≤ m}?

4 If k = 0, then

limm→∞

P(

Mm ≤y

am+ bm

)= e−2e

−y,

with am = (2 log m)1/2 and

bm = (2 log m)1/2 − 12 (2 log m)−1/2 (log log m + log 4π).

5 The case of k = 1 can also be worked out. The extreme value distribution

will Gumbel: e−4e−y

.

Confidence bands for the Lévy density A uniform CLT

Maxima of the last approximation

1 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u)

2 supx∈D |Y n6 (x)|D= t−1/2n m1/2 max1≤i≤m ζi , where, given Zj

iid∼ N (0,1),

ζiiid∼ ζ := sup

x∈[−1,1]

∣∣∣∣∣∣k∑

j=0

√2j + 1

2Qj (x)Zj

∣∣∣∣∣∣ .3 Question: What is the extreme value distribution and normalizing

constants of Mm := max{ζi : 1 ≤ i ≤ m}?

4 If k = 0, then

limm→∞

P(

Mm ≤y

am+ bm

)= e−2e

−y,

with am = (2 log m)1/2 and

bm = (2 log m)1/2 − 12 (2 log m)−1/2 (log log m + log 4π).

5 The case of k = 1 can also be worked out. The extreme value distribution

will Gumbel: e−4e−y

.

Confidence bands for the Lévy density A uniform CLT

Maxima of the last approximation

1 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u)

2 supx∈D |Y n6 (x)|D= t−1/2n m1/2 max1≤i≤m ζi , where, given Zj

iid∼ N (0,1),

ζiiid∼ ζ := sup

x∈[−1,1]

∣∣∣∣∣∣k∑

j=0

√2j + 1

2Qj (x)Zj

∣∣∣∣∣∣ .3 Question: What is the extreme value distribution and normalizing

constants of Mm := max{ζi : 1 ≤ i ≤ m}?

4 If k = 0, then

limm→∞

P(

Mm ≤y

am+ bm

)= e−2e

−y,

with am = (2 log m)1/2 and

bm = (2 log m)1/2 − 12 (2 log m)−1/2 (log log m + log 4π).

5 The case of k = 1 can also be worked out. The extreme value distribution

will Gumbel: e−4e−y

.

Confidence bands for the Lévy density A uniform CLT

Maxima of the last approximation

1 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u)

2 supx∈D |Y n6 (x)|D= t−1/2n m1/2 max1≤i≤m ζi , where, given Zj

iid∼ N (0,1),

ζiiid∼ ζ := sup

x∈[−1,1]

∣∣∣∣∣∣k∑

j=0

√2j + 1

2Qj (x)Zj

∣∣∣∣∣∣ .3 Question: What is the extreme value distribution and normalizing

constants of Mm := max{ζi : 1 ≤ i ≤ m}?

4 If k = 0, then

limm→∞

P(

Mm ≤y

am+ bm

)= e−2e

−y,

with am = (2 log m)1/2 and

bm = (2 log m)1/2 − 12 (2 log m)−1/2 (log log m + log 4π).

5 The case of k = 1 can also be worked out. The extreme value distribution

will Gumbel: e−4e−y

.

Confidence bands for the Lévy density A uniform CLT

Maxima of the last approximation

1 Y n6 (x) := t−1/2n

∑i∑

j ϕi,j (x)∫ xi

xi−1ϕi,j (u)dZ (u)

2 supx∈D |Y n6 (x)|D= t−1/2n m1/2 max1≤i≤m ζi , where, given Zj

iid∼ N (0,1),

ζiiid∼ ζ := sup

x∈[−1,1]

∣∣∣∣∣∣k∑

j=0

√2j + 1

2Qj (x)Zj

∣∣∣∣∣∣ .3 Question: What is the extreme value distribution and normalizing

constants of Mm := max{ζi : 1 ≤ i ≤ m}?

4 If k = 0, then

limm→∞

P(

Mm ≤y

am+ bm

)= e−2e

−y,

with am = (2 log m)1/2 and

bm = (2 log m)1/2 − 12 (2 log m)−1/2 (log log m + log 4π).

5 The case of k = 1 can also be worked out. The extreme value distribution

will Gumbel: e−4e−y

.

Confidence bands for the Lévy density A uniform CLT

Confidence bands for the Lévy density

1 Suppose that

• s is positive and has smoothness α > 2 on [a, b];

• s is bounded away from the origin

• s is C1 on an open interval (c, d) containing [a, b].

2 Thus, there exists ε0 > 0 and κ > 0 (explicitly computable) such that for

any 0 < ε < ε0,

limn→∞

P

(an

{κn

13−ε sup

x∈[a,b]s−1/2(x) |ŝn(x)− s(x)| − bn

}≤ y

)= e−2e

−y,

Asymptotic 100(1− α)% confidence bands for s(x):

s(x) ∈(

ŝn(x)± 1κ(

y∗αan

+ bn)

n−13 +εŝ1/2n (x)

)e−2e

−y∗α = 1− α

Confidence bands for the Lévy density A uniform CLT

Confidence bands for the Lévy density

1 Suppose that

• s is positive and has smoothness α > 2 on [a, b];

• s is bounded away from the origin

• s is C1 on an open interval (c, d) containing [a, b].

2 Thus, there exists ε0 > 0 and κ > 0 (explicitly computable) such that for

any 0 < ε < ε0,

limn→∞

P

(an

{κn

13−ε sup

x∈[a,b]s−1/2(x) |ŝn(x)− s(x)| − bn

}≤ y

)= e−2e

−y,

Asymptotic 100(1− α)% confidence bands for s(x):

s(x) ∈(

ŝn(x)± 1κ(

y∗αan

+ bn)

n−13 +εŝ1/2n (x)

)e−2e

−y∗α = 1− α

Confidence bands for the Lévy density A uniform CLT

Confidence bands for the Lévy density

1 Suppose that

• s is positive and has smoothness α > 2 on [a, b];

• s is bounded away from the origin

• s is C1 on an open interval (c, d) containing [a, b].

2 Thus, there exists ε0 > 0 and κ > 0 (explicitly computable) such that for

any 0 < ε < ε0,

limn→∞

P

(an

{κn

13−ε sup

x∈[a,b]s−1/2(x) |ŝn(x)− s(x)| − bn

}≤ y

)= e−2e

−y,

Asymptotic 100(1− α)% confidence bands for s(x):

s(x) ∈(

ŝn(x)± 1κ(

y∗αan

+ bn)

n−13 +εŝ1/2n (x)

)e−2e

−y∗α = 1− α

Confidence bands for the Lévy density A uniform CLT

Confidence bands for the Lévy density

1 Suppose that

• s is positive and has smoothness α > 2 on [a, b];

• s is bounded away from the origin

• s is C1 on an open interval (c, d) containing [a, b].

2 Thus, there exists ε0 > 0 and κ > 0 (explicitly computable) such that for

any 0 < ε < ε0,

limn→∞

P

(an

{κn

13−ε sup

x∈[a,b]s−1/2(x) |ŝn(x)− s(x)| − bn

}≤ y

)= e−2e

−y,

Asymptotic 100(1− α)% confidence bands for s(x):

s(x) ∈(

ŝn(x)± 1κ(

y∗αan

+ bn)

n−13 +εŝ1/2n (x)

)e−2e

−y∗α = 1− α

Confidence bands for the Lévy density A uniform CLT

An example: Variance Gamma process

Figure: Confidence bands for the right-tail of the variance Gamma Levy density.

Final remarks Feasibility in realty

Outline

1 Introduction

Some Lévy-based financial models

Formulation of the problems

2 Estimation of Lévy densities

Minimax rate of convergence

Nonparametric sieve estimators

The rate of convergence of the risk

3 Confidence intervals for the Lévy density

A pointwise CLT

4 Confidence bands for the Lévy density

A uniform CLT

5 Final remarks

Feasibility in realty

Final remarks Feasibility in realty

Feasibility and robustness

• High-frequency real data will recover the tick-by-tick data, which exhibit

very particular features called microstructure noise.

• How frequent to sample? There is a tradeoff: The higher sampling

frequency, the smaller the error of the non-parametric methods (under

absence of noise), but the higher the microstructure noise.

• Need to analyze robustness of the methods towards “microstructure

noise” (or other kind of noise).

Final remarks Feasibility in realty

Feasibility and robustness

• High-frequency real data will recover the tick-by-tick data, which exhibit

very particular features called microstructure noise.

• How frequent to sample? There is a tradeoff: The higher sampling

frequency, the smaller the error of the non-parametric methods (under

absence of noise), but the higher the microstructure noise.

• Need to analyze robustness of the methods towards “microstructure

noise” (or other kind of noise).

Final remarks Feasibility in realty

Feasibility and robustness

• High-frequency real data will recover the tick-by-tick data, which exhibit

very particular features called microstructure noise.

• How frequent to sample? There is a tradeoff: The higher sampling

frequency, the smaller the error of the non-parametric methods (under

absence of noise), but the higher the microstructure noise.

• Need to analyze robustness of the methods towards “microstructure

noise” (or other kind of noise).

Appendix Bibliography

For Further Reading I

Figueroa-López.

Model selection for Lévy processes based on discrete-sampling.

To appear IMS volume of the 3rd Erich L. Lehmann Symposium, 2008.

Figueroa-López.

Sieve-based confidence intervals and bands for Lévy densities.

Preprint, 2009.

Figueroa-López.

Small-time moment asymptotics for Lévy processes.

Statistics and Probability Letters, 78, 3355-3365, 2008.

Appendix Bibliography

For Further Reading II

Figueroa-Lopez & Houdré.

Risk bounds for the non-parametric estimation of Lévy processes.

IMS Lecture Notes - Monograph Series. High Dimensional Probability,

51:96–116, 2006.

Appendix Additional details

Small-time behavior of moments

Regularity Condition (?)

1 ϕ is continuous (ν-almost everywhere).

2 ϕ(x) = o(x2), as x → 0.

3 ϕ(x) = O(g(x)), as |x | → ∞, where g is submultiplicative or subadditivesuch that

∫|x|>1

g(x)s(x)dx 0, or g(x) = ec|x|, c > 0).

Theorem (Woerner 02, Jacod 06, Figueroa 08)

limt→0

1tE {ϕ (Xt )} =

∫ϕ(x)s(x)dx .

Back

Appendix Additional details

Small-time behavior of moments

Regularity Condition (?)

1 ϕ is continuous (ν-almost everywhere).

2 ϕ(x) = o(x2), as x → 0.

3 ϕ(x) = O(g(x)), as |x | → ∞, where g is submultiplicative or subadditivesuch that

∫|x|>1

g(x)s(x)dx 0, or g(x) = ec|x|, c > 0).

Theorem (Woerner 02, Jacod 06, Figueroa 08)

limt→0

1tE {ϕ (Xt )} =

∫ϕ(x)s(x)dx .

Back

IntroductionSome Lévy-based financial modelsFormulation of the problems

Estimation of Lévy densitiesMinimax rate of convergenceNonparametric sieve estimatorsThe rate of convergence of the risk

Confidence intervals for the Lévy densityA pointwise CLT

Confidence bands for the Lévy densityA uniform CLT

Final remarksFeasibility in realty

AppendixAppendix