Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Fractional SDE

1. Introduction

Since the 1980’s, the statistical inference for stochastic differential equations (SDE) driven by a Brownian motion has been widely investigated by many authors in the parametric and in the nonparametric frameworks. Classically (see Kutoyants [18]), the estimators of the drift function are computed from one path of the solution to the SDE and converge when the time horizon $T > 0$ goes to infinity. The existence and the uniqueness of the stationary solution to the SDE are then required, and obtained thanks to restrictive conditions on the drift function.
Since few years, a new type of parametric and nonparametric estimators is investigated ; those computed from multiple independent observations on $[0, T]$ of the SDE solution. Indeed, this functional data analysis problem is already studied in the parametric framework (see Ditlevsen and De Gaetano [14], Overgaard et al. [21], Picchini, De Gaetano and Ditlevsen [22], Picchini and Ditlevsen [23], Comte, Genon-Catalot and Samson [6], Delattre and Lavielle [10], Delattre, Genon-Catalot and Samson [9], Dion and Genon-Catalot [13], Delattre, Genon-Catalot and Larédo [8], etc.) and more recently in the nonparametric framework (see Comte and Genon-Catalot [4, 5], Della Maestra and Hoffmann [11], and Marie and Rosier [19]). In [4, 5], the authors extend to the diffusion processes framework the projection least squares estimators already well studied in the regression framework (see Cohen et al. [2] and Comte and Genon-Catalot [3]). Our paper deals with a nonparametric estimation problem close to this last one.

Consider the stochastic process $Z = (Z_{t})_{t \in [0, T]}$ , defined by

(1)

Z_{t} = \int_{0}^{t} J_{0} (s) d ⟨ M ⟩_{s} + M_{t}; \forall t \in [0, T],

where $M = (M_{t})_{t \in [0, T]} \neq 0$ is a centered, continuous and square integrable martingale vanishing at $0$ , and $J_{0}$ is an unknown function which belongs to $L^{2} ([0, T], d ⟨ M ⟩_{t})$ . By assuming that the quadratic variation $⟨ M ⟩_{t}$ of $M$ is deterministic for every $t \in [0, T]$ , our paper deals with the estimator ${ˆ J}_{m, N}$ of $J_{0}$ minimizing the objective function

J ⟼ γ_{m, N} (J) := \frac{1}{N} N \sum i = 1 (\int_{0}^{T} J (s)^{2} d ⟨ M^{i} ⟩_{s} - 2 \int_{0}^{T} J (s) d Z_{s}^{i})

on a $m$ -dimensional function space $S_{m}$ , where $M^{1}, \dots, M^{N}$ (resp. $Z^{1}, \dots, Z^{N}$ ) are $N \in N^{*}$ independent copies of $M$ (resp. $Z$ ) and $m \in {1, \dots, N}$ . Precisely, risk bounds are established on ${ˆ J}_{m, N}$ and on the adaptive estimator ${ˆ J}_{ˆ m, N}$ , where

ˆ m = arg min m \in M_{N} {γ_{m, N} ({ˆ J}_{m, N}) + p e n (m)} w i t h p e n (.) := c_{c a l} \frac{.}{N} a n d M_{N} \subset {1, \dots, N} .

Now, consider the differential equation

(2)

X_{t} = x_{0} + \int_{0}^{t} V (X_{s}) (b_{0} (s) d s + σ (s) d B_{s}); t \in [0, T],

where $x_{0} \in R^{*}$ , $B$ is a fractional Brownian motion of Hurst parameter $H \in (1 / 2, 1)$ , the stochastic integral with respect to $B$ is taken pathwise (in Young’s sense), and $V : R \to R$ , $σ : [0, T] \to R^{*}$ and $b_{0} : [0, T] \to R$ are regular enough. An appropriate transformation (see Subsection 4.1) allows to rewrite Equation (2) as a model of type (1) driven by the Molchan martingale which quadratic variation is $t^{2 - 2 H}$ for every $t \in [0, T]$ . Our paper also deals with a risk bound on an estimator of $b_{0} / σ$ derived from ${ˆ J}_{m, N}$ .
Finally, let us consider a financial market model in which the prices of the risky asset are modeled by the following equation of type (2):

(3)

S_{t} = S_{0} + \int_{0}^{t} S_{u} (({¯ ¯ b}_{0} (u) - \frac{σ^{2}}{} 2 u^{2 H - 1}) d u + σ d B_{u}); t \in [0, T]

where $S_{0}, σ \in (0, \infty)$ and ${¯ ¯ b}_{0} \in C^{0} ([0, T]; R)$ . This is a non-autonomous extension of the fractional Black-Scholes model defined in Hu et al. [15]. An estimator of ${¯ ¯ b}_{0}$ is derived from ${ˆ J}_{m, N}$ at Subsection 4.3.
Up to our knowledge, only Comte and Marie [7] deals with a nonparametric estimator of the drift function computed from multiple independent observations on $[0, T]$ of the solution to a fractional SDE.

At Section 2, a detailed definition of the projection least square estimator of $J_{0}$ is provided. Section 3 deals with risk bounds on ${ˆ J}_{m, N}$ and on the adaptive estimator ${ˆ J}_{ˆ m, N}$ . At Section 4, the results of Section 3 on the estimator of $J_{0}$ are applied to the estimation of $b_{0}$ in Equation (2) and then of ${¯ ¯ b}_{0}$ in Equation (3). Finally, at Section 5, some numerical experiments on Model (1) are provided when $M$ is the Molchan martingale.

2. A projection least square estimator of the map $J_{0}$

In the sequel, the quadratic variation $⟨ M ⟩ = (⟨ M ⟩_{t})_{t \in [0, T]}$ of $M$ fulfills the following assumption.

Assumption 2.1.

The (nonnegative, increasing and continuous) process $⟨ M ⟩$ is a deterministic function.

For some results, $⟨ M ⟩$ fulfills the following stronger assumption.

Assumption 2.2.

There exists $μ \in C^{0} ((0, T]; R_{+})$ such that $1 / μ$ is continuous from $[0, T]$ into $R_{+}$ , and such that

⟨ M ⟩_{t} = \int_{0}^{t} μ (s) d s; \forall t \in [0, T] .

2.1. The objective function

In order to define a least square projection estimator of $J_{0}$ , let us consider $N \in N^{*}$ independent copies $M^{1}, \dots, M^{N}$ (resp. $Z^{1}, \dots, Z^{N}$ ) of $M$ (resp. $Z$ ), and the objective function $γ_{m, N}$ defined by

γ_{m, N} (J) := \frac{1}{N} N \sum i = 1 (\int_{0}^{T} J (s)^{2} d ⟨ M^{i} ⟩_{s} - 2 \int_{0}^{T} J (s) d Z_{s}^{i})

for every $J \in S_{m}$ , where $m \in {1, \dots, N}$ , $S_{m} := s p a n {φ_{1}, \dots, φ_{m}}$ and $φ_{1}, \dots, φ_{N}$ are continuous functions from $[0, T]$ into $R$ such that $(φ_{1}, \dots, φ_{N})$ is an orthonormal family in $L^{2} ([0, T], d t)$ .

Remark. Note that since $t \in [0, T] \mapsto ⟨ M ⟩_{t}$ is nonnegative, increasing and continuous, and since the $φ_{j}$ ’s are continuous from $[0, T]$ into $R$ , the objective function $γ_{m, N}$ is well-defined.

For any $J \in S_{m}$ ,

	$E (γ_{m, N} (J))$	$=$	$\int_{0}^{T} J (s)^{2} d ⟨ M ⟩_{s} - 2 \int_{0}^{T} J (s) J_{0} (s) d ⟨ M ⟩_{s} - 2 E (\int_{0}^{T} J (s) d M_{s})$
		$=$	$\int_{0}^{T} (J (s) - J_{0} (s))^{2} d ⟨ M ⟩_{s} - \int_{0}^{T} J_{0} (s)^{2} d ⟨ M ⟩_{s} .$

Then, the more $J$ is close to $J_{0}$ , the more $E (γ_{m, N} (J))$ is small. For this reason, the estimator of $J_{0}$ minimizing $γ_{m, N}$ is studied in this paper.

2.2. The projection least square estimator

Consider the estimator

(4)

{ˆ J}_{m, N} := arg min J \in S_{m} γ_{m, N} (J)

of $J_{0}$ . Since $S_{m} = s p a n {φ_{1}, \dots, φ_{m}}$ , there exist $m$

square integrable random variables

{ˆ θ}_{1}, \dots, {ˆ θ}_{m}

such that

{ˆ J}_{m, N} = m \sum j = 1 {ˆ θ}_{j} φ_{j} .

Then,

\nabla γ_{m, N} ({ˆ J}_{m, N}) = {(\frac{1}{N} N \sum i = 1 (2 m \sum k = 1 {ˆ θ}_{k} \int_{0}^{T} φ_{j} (s) φ_{k} (s) d ⟨ M^{i} ⟩_{s} - 2 \int_{0}^{T} φ_{j} (s) d Z_{s}^{i}))}_{j \in {1, \dots, m}} .

Therefore, by (4), necessarily

{ˆ θ}_{m, N} := ({ˆ θ}_{1}, \dots, {ˆ θ}_{m})^{*} = Ψ_{m}^{- 1} z_{m, N},

where

Ψ_{m} := {(\int_{0}^{T} φ_{j} (s) φ_{k} (s) d ⟨ M ⟩_{s})}_{j, k \in {1, \dots, m}} a n d z_{m, N} := {(\frac{1}{N} N \sum i = 1 \int_{0}^{T} φ_{j} (s) d Z_{s}^{i})}_{j \in {1, \dots, m}} .

3. Risk bound and model selection

In the sequel, the space $L^{2} ([0, T], d ⟨ M ⟩_{t})$ is equipped with the scalar product $⟨ ., . ⟩_{⟨ M ⟩}$ defined by

⟨ φ, ψ ⟩_{⟨ M ⟩} := \int_{0}^{T} φ (s) ψ (s) d ⟨ M ⟩_{s}

for every $φ, ψ \in L^{2} ([0, T], d ⟨ M ⟩_{t})$ . The associated norm is denoted by $∥ . ∥_{⟨ M ⟩}$ .

First, the following proposition provides a risk bound on ${ˆ J}_{m, N}$ for a fixed $m \in {1, \dots, N}$ .

Proposition 3.1.

Under Assumption 2.1,

(5)

E (∥ {ˆ J}_{m, N} - J_{0} ∥_{⟨ M ⟩}^{2}) ⩽ min J \in S_{m} ∥ J - J_{0} ∥_{⟨ M ⟩}^{2} + \frac{2 m}{N} .

Note that Inequality (5

) says first that the bound on the variance of our least square estimator of

J_{0}

is of order

m / N

, as in the usual nonparametric regression framework. Under Assumption 2.2, the following corollary provides a more understandable expression of the bound on the bias in Inequality (5).

Corollary 3.2.

Under Assumption 2.2,

E (∥ {ˆ J}_{m, N} - J_{0} ∥_{2}^{2}) ⩽ ∥ μ^{- 1} ∥_{\infty, T} ∥ p_{S_{m} (μ)}^{⊥} (μ^{1 / 2} J_{0}) - μ^{1 / 2} J_{0} ∥_{2}^{2} + 2 ∥ μ^{- 1} ∥_{\infty, T} \frac{m}{N}

where

S_{m} (μ) := {ι \in L^{2} ([0, T], d t) : \exists φ \in S_{m}, \forall t \in (0, T], ι (t) = μ (t)^{1 / 2} φ (t)} .

For instance, assume that $S_{m} = s p a n {{¯ ¯¯ ¯ φ}_{1}, \dots, {¯ ¯¯ ¯ φ}_{m}}$ , where

{¯ ¯¯ ¯ φ}_{1} (t) := \sqrt{\frac{1}{μ (t) T}}, {¯ ¯¯ ¯ φ}_{2 j} (t) := \sqrt{\frac{2}{μ (t) T}} cos (2 π j \frac{t}{T}) a n d {¯ ¯¯ ¯ φ}_{2 j + 1} (t) := \sqrt{\frac{2}{μ (t) T}} sin (2 π j \frac{t}{T})

for every $t \in [0, T]$ and $j \in N^{*}$ satisfying $2 j + 1 ⩽ m$ . The basis $(φ_{1}, \dots, φ_{m})$ of $S_{m}$ , orthonormal in $L^{2} ([0, T], d t)$ , is obtained from $({¯ ¯¯ ¯ φ}_{1}, \dots, {¯ ¯¯ ¯ φ}_{m})$

via the Gram-Schmidt process. Consider also the Sobolev space

W_{2}^{β} ([0, T]) := {ι \in C^{β - 1} ([0, T]; R) : \int_{0}^{T} ι^{(β)} (t)^{2} d t < \infty}; β \in N^{*},

and assume that there exists $ι_{0} \in W_{2}^{β} ([0, T])$ such that $ι_{0} (t) = μ (t)^{1 / 2} J_{0} (t)$ for every $t \in (0, T]$ . Then, by DeVore and Lorentz [12], Theorem 2.3 p. 205, there exists a constant $c_{β, T} > 0$ , not depending on $m$ , such that

∥ p_{S_{m} (μ)}^{⊥} (μ^{1 / 2} J_{0}) - μ^{1 / 2} J_{0} ∥_{2}^{2} = ∥ p_{S_{m} (μ)}^{⊥} (ι_{0}) - ι_{0} ∥_{2}^{2} ⩽ c_{β, T} m^{- 2 β} .

Therefore, by Corollary 3.2,

E (∥ {ˆ J}_{m, N} - J_{0} ∥_{2}^{2}) ⩽ ∥ μ^{- 1} ∥_{\infty, T} (c_{β, T} m^{- 2 β} + \frac{2 m}{N}) .

Finally, consider $m_{N} \in {1, \dots, N}$ , $M_{N} := {1, \dots, m_{N}}$ and

ˆ m = arg min m \in M_{N} {γ_{m, N} ({ˆ J}_{m, N}) + p e n (m)} w i t h p e n (.) := c_{c a l} \frac{.}{N},

where $c_{c a l} > 0$ is a constant to calibrate in practice via, for instance, the

slope heuristic

. In the sequel, the

φ_{j}

’s fulfill the following assumption.

Assumption 3.3.

For every $m, m^{'} \in {1, \dots, N}$ , if $m > m^{'}$ , then $S_{m^{'}} \subset S_{m}$ .

The following theorem provides a risk bound on the adaptive estimator ${ˆ J}_{ˆ m, N}$ .

Theorem 3.4.

Under Assumptions 2.1 and 3.3, there exists a deterministic constant $c_{???} > 0$ , not depending on $N$ , such that

E (∥ {ˆ J}_{ˆ m, N} - J_{0} ∥_{⟨ M ⟩}^{2}) ⩽ c_{???} (min m \in M_{N} {E (∥ {ˆ J}_{m, N} - J_{0} ∥_{⟨ M ⟩}^{2}) + p e n (m)} + \frac{1}{N}) .

Moreover, under Assumption 2.2,

E (∥ {ˆ J}_{ˆ m, N} - J_{0} ∥_{2}^{2}) ⩽ c_{???} ∥ μ^{- 1} ∥_{\infty, T} (min m \in M_{N} {∥ p_{S_{m} (μ)}^{⊥} (μ^{1 / 2} J_{0}) - μ^{1 / 2} J_{0} ∥_{2}^{2} + (2 + c_{c a l}) \frac{m}{N}} + \frac{1}{N}) .

As in the usual nonparametric regression framework, since $p e n (m)$ is of same order than the bound on the variance term of ${ˆ J}_{m, N}$ for every $m \in M_{N}$ , Theorem 3.4 says that the risk of our adaptive estimator is controlled by the minimal risk of ${ˆ J}_{., N}$ on $M_{N}$ up to a multiplicative constant not depending on $N$ .

4. Application to differential equations driven by the fractional Brownian motion

Throughout this section, $V : R \to R$ is twice continuously differentiable with bounded derivatives, $σ : [0, T] \to R^{*}$ is $γ$ -Hölder continuous with $γ \in (1 - H, 1]$ , and $b_{0} : [0, T] \to R$ is continuous. Under these conditions on $V$ , $σ$ and $b_{0}$ , Equation (2) has a unique solution which paths are $α$ -Hölder continuous from $[0, T]$ into $R$ for every $α \in (1 / 2, H)$ (see Kubilius et al. [17], Theorem 1.42). The maps $V$ and $σ$ are known and our purpose is to provide a nonparametric estimator of $b_{0}$ .

4.1. Auxiliary model

The model transformation used in the sequel has been introduced in Kleptsyna and Le Breton [16] in the parametric estimation framework. Consider the function space

Q := {Q : the function t \mapsto t^{1 / 2 - H} Q (t) belongs % to L^{1} ([0, T], d t)},

let $Q_{0} : [0, T] \to R$ be the map defined by

Q_{0} (t) := \frac{b_{0} (t)}{σ (t)}; \forall t \in [0, T],

and assume that $Q_{0} \in Q$ . Consider also the Molchan martingale $M$ defined by

M_{t} := \int_{0}^{t} ℓ (t, s) d B_{s}; \forall t \in [0, T],

where

ℓ (t, s) := c_{H} s^{1 / 2 - H} (t - s)^{1 / 2 - H} 1_{(0, t)} (s) %; \forall s, t \in [0, T]

with

c_{H} = {(\frac{Γ (3 - 2 H)}{2 H Γ (3 / 2 - H)^{3} Γ (H + 1 / 2)})}^{1 / 2},

and the process $Z$ defined by

Z_{t} := \int_{0}^{t} ℓ (t, s) d Y_{s} w i t h Y_{t} := \int_{0}^{t} \frac{d X_{s}}{V (X_{s}) σ (s)}

for every $t \in [0, T]$ . Then, Equation (2) leads to

(6)		$Z_{t}$	$=$	$j (Q_{0}) (t) + M_{t}$
(6)			$=$	$\int_{0}^{t} J (Q_{0}) (s) d ⟨ M ⟩_{s} + M_{t},$

where

j (Q) (t) := \int_{0}^{t} ℓ (t, s) Q (s) d s a n d J (Q) (t) := (2 - 2 H)^{- 1} t^{2 H - 1} j (Q)^{'} (t)

for every $Q \in Q$ and almost every $t \in [0, T]$ . Note that the Molchan martingale $M$ fulfills Assumption 2.2 with $μ (t) = t^{1 - 2 H}$ for every $t \in (0, T]$ .

4.2. An estimator of $Q_{0}$

In Model (6), for any $m \in {1, \dots, N}$ , the solution ${ˆ J}_{m, N}$ to Problem (4) is a nonparametric estimator of $J (Q_{0})$ . So, this subsection deals with an estimator of $Q_{0}$ solving the inverse problem

(7)

J (ˆ Q) = {ˆ J}_{m, N} .

Let us consider the function space

J := {ι : the function t \in [0, T] ⟼ \int_{0}^{t} s^{1 - 2 H} ι (s) d s belongs to I_{0 +}^{3 / 2 - H} (L^{1} ([0, T], d t))},

where $I_{0 +}^{3 / 2 - H} (.)$ is the Riemann-Liouville left-sided fractional integral of order $3 / 2 - H$ . The reader can refer to Samko et al. [25] on fractional calculus.

In order to provide an estimator of $Q_{0}$ with a closed-form expression, let us establish first the following technical proposition.

Proposition 4.1.

The map $J : Q \mapsto J (Q)$ is one-to-one from $Q$ into $J$ . Moreover, for every $ι \in J$ and almost every $t \in [0, T]$ ,

J^{- 1} (φ) (t) = {¯ c}_{H} t^{H - 1 / 2} \int_{0}^{t} (t - s)^{H - 3 / 2} s^{1 - 2 H} ι (s) d s

with

{¯ c}_{H} = \frac{2 - 2 H}{c_{H} Γ (3 / 2 - H) Γ (H - 1 / 2)} .

By Proposition 4.1, if $φ_{1}, \dots, φ_{m} \in J$ , then

{ˆ Q}_{m, N} (t) := {¯ c}_{H} t^{H - 1 / 2} \int_{0}^{t} (t - s)^{H - 3 / 2} s^{1 - 2 H} {ˆ J}_{m, N} (s) d s; t \in [0, T]

is the solution to Problem (7) in $Q$ . Note that even if the $φ_{j}$ ’s don’t belong to $J$ , since these functions are continuous from $[0, T]$ into $R$ , ${ˆ Q}_{m, N}$ is well-defined but not necessarily a solution to Problem (7

). A simple vector subspace of

J

is provided at the end of this subsection.

The following proposition provides risk bounds on

{ˆ Q}_{m, N}

m \in {1, \dots, N}

, and on the adaptive estimator

{ˆ Q}_{ˆ m, N}

Proposition 4.2.

If the $φ_{j}$ ’s belong to $J$ , then there exists a deterministic constant $c_{???, 1} > 0$ , not depending on $N$ , such that

E (∥ {ˆ Q}_{m, N} - Q_{0} ∥_{2}^{2}) ⩽ c_{???, 1} (min ι \in S_{m} ∥ ι - J (Q_{0}) ∥_{⟨ M ⟩}^{2} + \frac{m}{N}); \forall m \in {1, \dots, N} .

If in addition the $φ_{j}$ ’s fulfill Assumption 3.3, then there exists a deterministic constant $c_{???, 2} > 0$ , not depending on $N$ , such that

E (∥ {ˆ Q}_{ˆ m, N} - Q_{0} ∥_{2}^{2}) ⩽ c_{???, 2} (min m \in M_{N} {min ι \in S_{m} ∥ ι - J (Q_{0}) ∥_{⟨ M ⟩}^{2} + \frac{m}{N}} + \frac{1}{N}) .

Proposition 4.2 says that the MISE of ${ˆ Q}_{m, N}$ , $m \in {1, \dots, N}$ , (resp. ${ˆ Q}_{ˆ m, N}$ ) has at most the same bound than the MISE of ${ˆ J}_{m, N}$ (resp. ${ˆ J}_{ˆ m, N}$ ).

Finally, the following proposition provides a simple vector subspace of $J$ .

Proposition 4.3.

The function space

J := {ι \in C^{1} ([0, T]; R) : lim t \to 0^{+} t^{- 2 H} ι (t) and lim t \to 0^{+} t^{1 - 2 H} ι^{'} (t) exist and are finite}

is a subset of $J$ .

Consider $m \in {1, \dots, N}$ and $ψ_{1}, \dots, ψ_{m} \in C^{1} ([0, T]; R)$ such that $(ψ_{1}, \dots, ψ_{m})$ is an orthonormal family of $L^{2} ([0, T], d t)$ . In particular, note that $ψ_{1}, \dots, ψ_{m}$ are linearly independent. Moreover, assume that $S_{m} = s p a n {{¯ ¯¯ ¯ φ}_{1}, \dots, {¯ ¯¯ ¯ φ}_{m}}$ with ${¯ ¯¯ ¯ φ}_{j} (t) := t^{2 H} ψ_{j} (t)$ for every $j \in {1, \dots, m}$ and $t \in [0, T]$ . The basis $(φ_{1}, \dots, φ_{m})$ of $S_{m}$ , orthonormal in $L^{2} ([0, T], d t)$ , is obtained from $({¯ ¯¯ ¯ φ}_{1}, \dots, {¯ ¯¯ ¯ φ}_{m})$ via the Gram-Schmidt process, and the $φ_{j}$ ’s belong to $J \subset J$ . For every $ι \in S_{m}$ , there exist $α_{1} (ι), \dots, α_{m} (ι) \in R$ such that

ι = m \sum j = 1 α_{j} (ι) {¯ ¯¯ ¯ φ}_{j} = υ μ^{- 1 / 2} ¯ ι,

where $υ (t) := t^{H + 1 / 2}$ for every $t \in [0, T]$ , and

¯ ι := m \sum j = 1 α_{j} (ι) ψ_{j} \in S_{m} w i t h S_{m} := s p a n {ψ_{1}, \dots, ψ_{m}} .

So, by assuming that $J (Q_{0}) \in J$ , the bound on the bias term of ${ˆ J}_{m, N}$ in the inequalities of Proposition 4.2 can be controlled the following way:

	$min ι \in S_{m} ∥ ι - J (Q_{0}) ∥_{⟨ M ⟩}^{2}$	$=$	$min ¯ ι \in S_{m} ∥ υ (¯ ι - υ^{- 1} μ^{1 / 2} J (Q_{0})) ∥_{2}^{2}$
		$⩽$	$T^{2 H + 1} min ¯ ι \in S_{m} ∥ ¯ ι - ¯ ¯ ¯ μ J (Q_{0}) ∥_{2}^{2} = ∥ p_{S_{m}}^{⊥} (¯ μ J (Q_{0})) - ¯ ¯ ¯ μ J (Q_{0}) ∥_{2}^{2}$

where, for every $t \in (0, T]$ ,

¯ ¯ ¯ μ (t) := υ (t)^{- 1} μ (t)^{1 / 2} = t^{- 2 H} .

If $(ψ_{1}, \dots, ψ_{m})$ is the $m$ -dimensional trigonometric basis, and if there exists $ι_{0} \in W_{2}^{β} ([0, T])$ such that $ι_{0} (t) = t^{- 2 H} J (Q_{0}) (t)$ for every $t \in (0, T]$ , then

min ι \in S_{m} ∥ ι - J (Q_{0}) ∥_{⟨ M ⟩}^{2} ⩽ T^{2 H + 1} ∥ p_{S_{m}}^{⊥} (ι_{0}) - ι_{0} ∥_{2}^{2} ⩽ c_{β, T} T^{2 H + 1} m^{- 2 β} .

4.3. Example: drift estimation in a non-autonomous fractional Black-Scholes model

Let us consider a financial market model in which the prices process $S = (S_{t})_{t \in [0, T]}$ of the risky asset is defined by

S_{t} := S_{0} exp (\int_{0}^{t} ({¯ ¯ b}_{0} (u) - \frac{σ^{2}}{2} u^{2 H - 1}) d u + σ B_{t}); \forall t \in [0, T],

where $S_{0}, σ \in (0, \infty)$ and ${¯ ¯ b}_{0} \in C^{0} ([0, T]; R)$ . This is a non-autonomous extension of the fractional Black-Scholes model defined in Hu et al. [15]. Thanks to the change of variable formula for Young’s integral,

S_{t} = S_{0} + \int_{0}^{t} S_{u} (({¯ ¯ b}_{0} (u) - \frac{σ^{2}}{} 2 u^{2 H - 1}) d u + σ d B_{u}); \forall t \in [0, T] .

Then, $S$ is the solution to Equation (2) with $V = {I d}_{R}$ , $σ (.) \equiv σ > 0$ and

b_{0} (t) = {¯ ¯ b}_{0} (t) - \frac{σ^{2}}{2} t^{2 H - 1}; \forall t \in [0, T] .

Consider $N$ independent copies $S^{1}, \dots, S^{N}$ of $S$ . For every $i \in {1, \dots, N}$ and $t \in [0, T]$ , consider also

Z_{t}^{i} := \frac{c_{H}}{σ} \int_{0}^{t} \frac{u^{1 / 2 - H} (t - u)^{1 / 2 - H}}{S_{u}^{i}} d S_{u}^{i} .

If the volatility constant $σ$ is known, thanks to Subsection 4.2, a nonparametric estimator of ${¯ ¯ b}_{0}$ is given by

{ˆ ¯ ¯ b}_{0} (m, N; t) := \frac{σ^{2}}{2} t^{2 H - 1} + σ ˆ Q_{m, N} (t); t \in [0, T],

where

{ˆ Q}_{m, N} (t) := {¯ c}_{H} t^{H - 1 / 2} \int_{0}^{t} (t - s)^{H - 3 / 2} s^{1 - 2 H} {ˆ J}_{m, N} (s) d s; t \in [0, T]

and

{ˆ J}_{m, N} = arg min J \in S_{m} {\frac{1}{N} N \sum i = 1 (\int_{0}^{T} J (s)^{2} s^{1 - 2 H} d s - 2 \int_{0}^{T} J (s) d Z_{s}^{i})} .

Since

∥ {ˆ ¯ ¯ b}_{0} (m, N; .) - {¯ ¯ b}_{0} ∥_{2}^{2} = σ^{2} ∥ {ˆ Q}_{m, N} - Q_{0} ∥_{2}^{2} w i t h Q_{0} = \frac{{¯ ¯ b}_{0}}{σ},

if the $φ_{j}$ ’s belong to $J$ , then Proposition 4.2 provides risk bounds on ${ˆ ¯ ¯ b}_{0} (m, N; .)$ , $m \in {1, \dots, N}$ , and on the adaptive estimator ${ˆ ¯ ¯ b}_{0} (ˆ m, N; .)$ .

5. Numerical experiments

Some numerical experiments on our estimation method of $J_{0}$ in Equation (1) are presented in this subsection when $M$ is the Molchan martingale:

M_{t} = \int_{0}^{t} ℓ (t, s) d B_{s} = (2 - 2 H)^{1 / 2} \int_{0}^{t} s^{1 / 2 - H} d W_{s}; t \in [0, 1]

with $H \in {0.6, 0.9}$ and $W$ the Brownian motion driving the Mandelbrot-Van Ness representation of the fractional Brownian motion $B$ . The estimation method investigated on the theoretical side at Section 3 is implemented here for the three following examples of functions $J_{0}$ :

J_{0, 1} : t \in [0, 1] \mapsto 10 t^{2}, J_{0, 2} : t \in (0, 1] \mapsto 10 (- log (t))^{1 / 2} a n d J_{0, 3} : t \in (0, 1] \mapsto 20 t^{- 0.05} .

These functions belong to $L^{2} ([0, 1], d ⟨ M ⟩_{t})$ as required. Indeed, one the one hand, $J_{0, 1}$ is continuous on $[0, 1]$ and

	$- \int_{0}^{1} log (t) d ⟨ M ⟩_{t}$	$=$	$- \int_{0}^{1} log (t) t^{1 - 2 H} d t$
		$=$	$\frac{1}{2 - 2 H} (lim ε \to 0^{+} log (ε) ε^{2 - 2 H} + \int_{0}^{1} t^{1 - 2 H} d t) = \frac{1}{(2 - 2 H)^{2}} < \infty .$

On the other hand, for every $α \in (0, 1 / 2)$ such that $H \in (1 / 2, 1 - α)$ ,

	$\int_{0}^{1} t^{- 2 α} d ⟨ M ⟩_{t}$	$=$	$\int_{0}^{1} t^{1 - 2 α - 2 H} d t$
		$=$	$\frac{1}{2 (1 - α - H)} (1 - lim ε \to 0^{+} ε^{2 (1 - α - H)}) = \frac{1}{2 (1 - α - H)} < \infty .$

Since for every $t \in (0, 1]$ , $J_{0, 3} (t) = 20 t^{- α}$ with $α = 0.05$ , and since $H \in {0.6, 0.9} \subset (0.5, 0.95)$ in our numerical experiments, $J_{0, 3}$ is square-integrable with respect to $d ⟨$

Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Fractional SDE

Authors

Nonparametric Estimation for I.I.D. Paths of Fractional SDE

Nadaraya-Watson Estimator for I.I.D. Paths of Diffusion Processes

Projection Estimators of the Stationary Density of a Differential Equation Driven by the Fractional Brownian Motion

Nonparametric Estimation of the Trend in Reflected Fractional SDE

Adaptive estimation of the stationary density of a stochastic differential equation driven by a fractional Brownian motion

Parameter estimation for an Ornstein-Uhlenbeck Process driven by a general Gaussian noise with Hurst Parameter H∈ (0,1/2)

Limit distribution of the least square estimator with observations sampled at random times driven by standard Brownian motion

1. Introduction

2. A projection least square estimator of the map $J_{0}$

Assumption 2.1.

Assumption 2.2.

2.1. The objective function

2.2. The projection least square estimator

3. Risk bound and model selection

Proposition 3.1.

Corollary 3.2.

Assumption 3.3.

Theorem 3.4.

4. Application to differential equations driven by the fractional Brownian motion

4.1. Auxiliary model

4.2. An estimator of $Q_{0}$

Proposition 4.1.

Proposition 4.2.

Proposition 4.3.

4.3. Example: drift estimation in a non-autonomous fractional Black-Scholes model

5. Numerical experiments

Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Fractional SDE

Authors

Related Research

Nonparametric Estimation for I.I.D. Paths of Fractional SDE

Nadaraya-Watson Estimator for I.I.D. Paths of Diffusion Processes

Projection Estimators of the Stationary Density of a Differential Equation Driven by the Fractional Brownian Motion

Nonparametric Estimation of the Trend in Reflected Fractional SDE

Adaptive estimation of the stationary density of a stochastic differential equation driven by a fractional Brownian motion

Parameter estimation for an Ornstein-Uhlenbeck Process driven by a general Gaussian noise with Hurst Parameter H∈ (0,1/2)

Limit distribution of the least square estimator with observations sampled at random times driven by standard Brownian motion

This week in AI

1. Introduction

2. A projection least square estimator of the map J0

Assumption 2.1.

Assumption 2.2.

2.1. The objective function

2.2. The projection least square estimator

3. Risk bound and model selection

Proposition 3.1.

Corollary 3.2.

Assumption 3.3.

Theorem 3.4.

4. Application to differential equations driven by the fractional Brownian motion

4.1. Auxiliary model

4.2. An estimator of Q0

Proposition 4.1.

Proposition 4.2.

Proposition 4.3.

4.3. Example: drift estimation in a non-autonomous fractional Black-Scholes model

5. Numerical experiments

2. A projection least square estimator of the map $J_{0}$

4.2. An estimator of $Q_{0}$