A pseudo-likelihood estimator of the Ornstein–Uhlenbeck parameters from suprema observations

Abstract

In this paper, we propose an estimator for the Ornstein–Uhlenbeck parameters based on observations of its supremum. We derive an analytic expression for the supremum density. Making use of the pseudo-likelihood method based on the supremum density, our estimator is constructed as the maximal argument of this function. Using weak-dependency results, we prove some statistical properties on the estimator such as consistency and asymptotic normality. Finally, we apply our estimator to simulated and real data.

Appendix

1.1 Parabolic cylinder function

We recall the definition and some auxiliary results about the Parabolic cylinder function. Some of these results can be found in Blanchet-Scalliet et al. (2020, 2024); Lebedev (1965).

For all $x,\nu \in {\mathbb {R}}$, the Parabolic cylinder function $D_\nu \left( x\right)$ is a solution of the differential equation:

$$\begin{aligned} y''\left( x\right) +\left( \nu +\frac{1}{2}-\frac{1}{4}x^2\right) y\left( x\right) =0. \end{aligned}$$

Moreover for $\nu \in {\mathbb {C}}$ with $\text {Re}(\nu )>0$ and $z\in {\mathbb {C}}$, the function $(z,\nu )\mapsto D_{\nu }(z)$ is a holomorphic function ( Lebedev 1965, Chapter 10).

The Parabolic cylinder function satisfies the following relations:

$$\begin{aligned} \partial _x D_{\nu }(x)=\nu D_{\nu -1}(x)-\frac{x}{2}D_\nu (x), \end{aligned}$$

and for all $\nu ,a\in {\mathbb {R}}$,

$$\begin{aligned} \int _{-\infty }^a e^{-\frac{x^2}{2}}D_{\nu }(-x\sqrt{2})dx=\frac{e^{-\frac{a^2}{2}}}{\sqrt{2}}D_{\nu -1}(-a\sqrt{2}). \end{aligned}$$

(A.1)

We remind now some properties of the Parabolic cylinder function $\nu$-zeros. Let $n\in {\mathbb {N}}^\star$, we denote $\nu _{n,m}$, the positive (ordered) zeros of the function $\nu \mapsto D_{\nu }\left( -m\sqrt{2}\right)$. Then

$$\begin{aligned} \int _{-\infty }^m D^2_{\nu _{n,m}}(-x\sqrt{2})dx=-\frac{\nu _{n,m}}{\sqrt{2}}D_{\nu _{n,m}-1}(-m\sqrt{2})\partial _{\nu }D_{\nu _{n,m}}(-m \sqrt{2}). \end{aligned}$$

(A.2)

Furthermore, according to Blanchet-Scalliet et al. (2024):

$$\begin{aligned} \partial _m\nu _{n,m}&=\sqrt{2}\frac{\partial _x D_{\nu _{n,m}}(-m\sqrt{2})}{\partial _{\nu } D_{\nu _{n,m}}(-m\sqrt{2})} \end{aligned}$$

(A.3)

$$\begin{aligned}&=-\frac{2}{\sqrt{\pi }\int _0^\infty e^{-\left( 2\nu _{n,m}+1\right) u+m^2\tanh \left( u\right) }\text {erfc}\left( -m\sqrt{\tanh \left( u\right) } \right) \frac{du}{\sqrt{\sinh (u)\cosh (u)}}}. \end{aligned}$$

(A.4)

The following asymptotic expansion is verified:

$$\begin{aligned} \nu _{n,m}\underset{m\rightarrow -\infty }{=}\frac{m^2}{2}-\frac{1}{2}-|m|^\frac{2}{3} 2^{-\frac{1}{3}}a_n+O\left( |m|^{-\frac{2}{3}}\right) , \end{aligned}$$

(A.5)

where $a_n$, $n\in {\mathbb {N}}^\star$ are the zeros of the first kind Airy function. Furthermore, the following convergence is verified:

$$\begin{aligned} \nu _{n,m}\underset{m\rightarrow +\infty }{\longrightarrow }n-1. \end{aligned}$$

(A.6)

Recall that the zeros $a_n$ of the first kind Airy function are all real, negative and satisfy the following inequality ( Pittaluga and Sacripante 1991), for $n\ge 1$:

$$\begin{aligned} -\left( \frac{3\pi }{8}(4n-1)\right) ^\frac{2}{3}\left( 1+\frac{5}{48\left( \frac{3\pi }{8}(4n-1)\right) ^2} \right) <a_n\le -\left( \frac{3\pi }{8}(4n-1)\right) ^\frac{2}{3}. \end{aligned}$$

(A.7)

1.2 Smoothness of the cumulative distribution function

Some smoothness properties are needed to prove the set of derivations and continuities of the functions presented in the proof of consistency and asymptotic normality. These properties will be proved using results on holomorphic functions. Making use of (A.3), we can rewrite the cumulative distribution function as:

$$\begin{aligned} {\mathbb {P}}(S^{1,0}<m)=-\frac{e^{ -\lambda m^{2}}}{2\sqrt{{\lambda }{\pi }}}\sum _{n\ge 1} e^{-\lambda \nu _{n,m,\lambda }\Delta }\frac{\partial _m \nu _{n,m,\lambda }}{\nu _{n,m,\lambda }^{2}}. \end{aligned}$$

(A.8)

Proposition A.1

The cumulative distribution function of the supremum verifies the following properties:

• 1. $(m,\lambda )\mapsto {\mathbb {P}}(S^{1,0}<m)$ is a smooth function on ${\mathbb {R}}\times {\mathbb {R}}_+^\star$.

• 2. For any $k,j\in {\mathbb {N}}^\star$:

  $$\begin{aligned} \partial _{\lambda }^k\partial _m^j{\mathbb {P}}(S^{1,0}<m)= \sum _{n\ge 1} \partial _{\lambda }^k\partial _m^j f_n(\Delta ,m,\lambda ), \end{aligned}$$

  with:

  $$\begin{aligned} f_n(\Delta ,m,\lambda )= -\frac{e^{-m^2\lambda -\lambda \nu _{n,m,\lambda }\Delta }}{\sqrt{2\pi }}\frac{D_{\nu _{n,m,\lambda }-1}\left( -m\sqrt{2\lambda }\right) }{\nu _{n,m,\lambda }\partial _{\nu }D_{\nu _{n,m,\lambda }}\left( -m\sqrt{2\lambda }\right) }. \end{aligned}$$

  (A.9)

Proof

We introduce the following notation, ${\mathbb {C}}^{\theta _1\le \text {arg} \le \theta _2}_{M_1\le |.|\le M_2}=\{z\in {\mathbb {C}} \text { s.t. } \theta _1\le \text {arg}(z) \le \theta _2 \text {, } M_1\le |z|\le M_2\}$. We denote $(z,\lambda )\mapsto {\tilde{F}}_\Delta (z,\lambda )$ the continuation of the cumulative distribution function on ${\mathbb {C}}\times {\mathbb {C}}^{-\frac{\pi }{2}<\text {arg}<\frac{\pi }{2}}_ {|.|\ne 0}$. According to the Implicit Function Theorem, the function $(z,\lambda )\mapsto \nu _{n,z,\lambda }$ is holomorphic. Then by composition of holomorphic functions, we deduce that (A.9) is holomorphic.

We can write ${\tilde{F}}_\Delta$ as follows:

$$\begin{aligned} {\tilde{F}}_\Delta (z,\lambda )&={\tilde{G}}_1(z,\lambda )\mathbbm {1}_{{\mathbb {C}}^{\frac{\pi }{2}<arg<\frac{3}{2\pi }}_{|.|>M}\times {\mathbb {C}}^{-\frac{\pi }{2}\le \text {arg}<\frac{\pi }{2}}_ {|.|>M}}(z,\lambda )+{\tilde{G}}_2(z,\lambda )\mathbbm {1}_{{\mathbb {C}}_{|.|<M}\times {\mathbb {C}}_{0<|.|<M}^{-\frac{\pi }{2}\le arg<\frac{\pi }{2}}}(z,\lambda )\\&+{\tilde{G}}_3(z,\lambda )\mathbbm {1}_{{\mathbb {C}}^{-\frac{\pi }{2}<arg<\frac{\pi }{2}}_{|.|>M}\times {\mathbb {C}}^{-\frac{\pi }{2}\le \text {arg}<\frac{\pi }{2}}_{{|.|>M}}}(z,\lambda ), \end{aligned}$$

with M large enough.

For ${\tilde{G}}_1$ and ${\tilde{G}}_3$, using the asymptotic expansion (A.5) and the limit (A.6), we easily prove the normal convergence of the associate series.

For ${\tilde{G}}_2$, the normal convergence can be obtained using the following equivalence from Alili et al. (2005):

$$\begin{aligned} \nu _{n,z,\lambda }\underset{n\rightarrow +\infty }{\sim }2n-1+\frac{4\lambda z^2}{\pi ^2}-2\frac{\sqrt{\lambda z^2}}{\pi }\sqrt{4n-1+4\frac{\lambda z^2}{\pi ^2}} \end{aligned}$$

and

$$\begin{aligned} D_{\nu }(z)\underset{\nu \rightarrow +\infty }{=}\sqrt{2}\left( \nu +\frac{1}{2}\right) ^{\frac{\nu }{2}}e^{-\left( \frac{\nu }{2}+\frac{1}{4}\right) }\cos \left( z\sqrt{\nu +\frac{1}{2}}-\frac{\pi \nu }{2}\right) \left( 1+O(\nu ^{-\frac{1}{2}})\right) . \end{aligned}$$

From Theorem 3.2 in Palka (1991), the conclusion holds. $\square$

1.3 Asymptotic expansions

For the integrability conditions required in the Ergodic Theorem, some asymptotic expansions on the cumulative distribution and the probability density of the supremum $S^{1,0}$ are provided. In the following proofs, without loss of generality we assume $\theta =(0,\lambda ,1)$. To return to the three parameters case, we replace m, $\Delta$, $\lambda$ respectively by $m-\frac{\mu }{\lambda }$, $\beta \Delta$ and $\frac{\lambda }{\beta }$.

1.3.1 For large positive m

Since the zeros $\nu _{n,m,\lambda }$ tend to positive integers when m goes to infinity, then we are able to give an asymptotic expansion for (5.1).

Proposition A.2

For large positive m, the cumulative distribution function of $S^{1,0}$ has the following asymptotic expansion:

$$\begin{aligned} {\mathbb {P}}(S^{1,0}<m)\underset{m\rightarrow +\infty }{=}1-\frac{e^{-\lambda m^2}}{2\sqrt{\pi \lambda }m}\left( 1+o(m^{-2+\delta })\right) , \end{aligned}$$

(A.10)

with $0<\delta <2$.

Proof

Recall that the cumulative distribution function is given by (5.1). Using Formula (A.6), we obtain:

$$\begin{aligned} {\mathbb {P}}(S^{1,0}<m)\underset{m\rightarrow +\infty }{=}\frac{1}{2\sqrt{\pi }}e^{-\lambda m^2}\left[ \frac{D^2_{-1}(-m\sqrt{2\lambda })}{\int _{-\infty }^{m\sqrt{\lambda }} D^2_{0}(-x\sqrt{2})dx}+ \sum _{n\ge 1}e^{-\lambda n\Delta }\frac{D^2_{n-1}(-m\sqrt{2\lambda })}{\int _{-\infty }^{m\sqrt{\lambda }}D^2_{n}(-x\sqrt{2})dx}\right] . \nonumber \\ \end{aligned}$$

(A.11)

According to Formula (10.5.4) in Lebedev (1965):

$$\begin{aligned} \frac{e^{-\lambda m^2}}{2\sqrt{\pi }}\frac{D^2_{-1}(-m\sqrt{2\lambda })}{\int _{-\infty }^{m\sqrt{\lambda }}D^2_{0}(-x\sqrt{2})dx}=\frac{1}{2}\left( 1+erf(m\sqrt{\lambda }) \right) , \end{aligned}$$

where $erf(m)=\frac{2}{\sqrt{\pi }}\int _0^me^{-t^2}dt$ is the Error function. Then applying Formulas (4.9.6) and (4.13.4) in Lebedev (1965), we get:

$$\begin{aligned} \sum _{n=1}^\infty e^{-\lambda n \Delta }\frac{D^2_{n-1}(-m\sqrt{2\lambda })}{\int _{-\infty }^{m\sqrt{\lambda }}D^2_{n}(-x\sqrt{2\lambda })dx}\underset{m\rightarrow +\infty }{=}e^{-\lambda m^2\frac{1-e^{-\lambda \Delta }}{1+e^{-\lambda \Delta }}}O(1). \end{aligned}$$

As we combine these two results, we get:

$$\begin{aligned} {\mathbb {P}}(S^{1,0}<m)\underset{m\rightarrow +\infty }{=}\frac{1}{2}\left( 1+erf(m\sqrt{\lambda }) \right) +e^{-\lambda m^2\frac{1-e^{-\lambda \Delta }}{1+e^{-\lambda \Delta }}}O(1). \end{aligned}$$

Since $1-erf(m)\underset{m\rightarrow +\infty }{=}\frac{e^{-m^2}}{\sqrt{\pi }m}(1+O(m^{-2}))$, the conclusion holds. $\square$

Proposition A.3

For large positive m, the asymptotic expansion of the probability density of $S^{1,0}$ is given by:

$$\begin{aligned} f_\Delta (m,\lambda )\underset{m\rightarrow +\infty }{=}\sqrt{\frac{\lambda }{\pi }}e^{{-\lambda m^{2}}}\left( 1+me^{-\lambda {m^{2}}\frac{1-{e^{- \lambda \Delta }}}{1+{e^{- \lambda \Delta }}}}O(1)\right) , \end{aligned}$$

(A.12)

with $\left| O(1)\right| \le \frac{4\sqrt{\lambda } e^{-\lambda \Delta }}{\sqrt{\pi (1-e^{-2\lambda \Delta })}}$.

Proof

When m goes to infinity, using (A.11) the derivative of the cumulative distribution function satisfies:

$$\begin{aligned} f_\Delta (m,\lambda )\underset{m\rightarrow +\infty }{=}\sqrt{\frac{\lambda }{\pi }}e^{-\lambda m^{2}}+\frac{1}{2\sqrt{\pi }}\partial _m\left[ e^{-\lambda m^2}\sum _{n\ge 1}e^{-\lambda n\Delta }\frac{D^{2}_{n-1}(-m\sqrt{2\lambda })}{\int _{-\infty }^{m\sqrt{\lambda }}D^2_{n}(-x\sqrt{2})dx}\right] . \end{aligned}$$

Using Formulas (4.9.5) and (4.13.4) in Lebedev (1965), one can prove that:

$$\begin{aligned} \partial _m\left[ e^{-\lambda m^2}\sum _{n\ge 1}e^{-\lambda n\Delta }\frac{D^2_{n-1}(-m\sqrt{2\lambda })}{\int _{-\infty }^{m\sqrt{\lambda }}D^2_{n}(-x\sqrt{2})dx}\right] \underset{m\rightarrow +\infty }{=}me^{-\frac{2\lambda m^2}{1+e^{-\lambda \Delta }}}O(1), \end{aligned}$$

with $\left| O(1)\right| \le \frac{8\lambda e^{-\lambda \Delta }}{\sqrt{\pi (1-e^{-2\lambda \Delta })}}$. $\square$

Remark A.4

We have:

$$\begin{aligned} f_\Delta (m,\lambda )\underset{m\rightarrow +\infty }{=}e^{-\lambda m^2}\sqrt{\frac{\lambda }{\pi }}\left( 1+o(m^{-\alpha })\right) , \end{aligned}$$

with $\alpha >0$.

Corollary A.4.1

For large positive m,

$$\begin{aligned} \log \left( f_\Delta (m,\lambda )\right) \underset{m\rightarrow +\infty }{=}-\lambda m^2 +\frac{1}{2}\log \left( \frac{\lambda }{\pi }\right) +me^{-m^2\lambda \frac{1-e^{-\Delta \lambda }}{1+e^{-\Delta \lambda }}}O(1), \end{aligned}$$

(A.13)

with $|O(1)|\le \frac{16\sqrt{\lambda } e^{-\lambda \Delta }}{3\sqrt{\pi (1-e^{-2\lambda \Delta )}}}$.

A similar reasoning as the one in the proof of Propositions A.2 and A.3 may be applied to prove the following results:

Proposition A.5

For large positive m, the following asymptotic expansions are satisfied:

• 1. $\partial _\lambda \log \left( f_\Delta (m,\lambda )\right) \underset{m\rightarrow +\infty }{=} -m^2+\frac{1}{2\lambda }+m^3e^{-m^2\lambda \frac{1-e^{-\lambda \Delta }}{1+e^{-\lambda \Delta }}}O(1).$

• 2. $\partial ^2_\lambda \log \left( f_\Delta (m,\lambda )\right) \underset{m\rightarrow +\infty }{=} -\frac{1}{2\lambda ^2}+m^5e^{-m^2\lambda \frac{1-e^{-\lambda \Delta }}{1+e^{-\lambda \Delta }}}O(1).$

• 3. $\partial ^3_\lambda \log \left( f_\Delta (m,\lambda )\right) \underset{m\rightarrow +\infty }{=} \frac{1}{\lambda ^3}+m^7e^{-m^2\lambda \frac{1-e^{-\lambda \Delta }}{1+e^{-\lambda \Delta }}}O(1).$

1.3.2 For large negative m

Using (A.4) and (A.5), we can give an asymptotic expansion for the cumulative distribution function of $S^{1,0}$ for large negative m.

Proposition A.6

For large negative m, the cumulative distribution function of $S^{1,0}$ has the following asymptotic expansion:

$$\begin{aligned} {\mathbb {P}}(S^{1,0}<m)=2|m\sqrt{\lambda }|^{-3}\frac{e^{-\lambda m^2-\left( \frac{m^2\lambda }{2}-\frac{1}{2}\right) \lambda \Delta +|m|^\frac{2}{3}\lambda ^\frac{4}{3}2^{-\frac{1}{3}}a_1\Delta }}{\sqrt{\pi }}(1+o(1)), \end{aligned}$$

(A.14)

where $a_1$ is the first zero of the Airy function of the first kind.

Sketch of the Proof

Formula (A.8) gives:

$$\begin{aligned} {\mathbb {P}}\left( S^{1,0}<m\right) =-\frac{e^{-\lambda m^2-\lambda \Delta \nu _{1,m,\lambda }}}{2\sqrt{\pi }}\frac{\partial _m\nu _{1,m,\lambda }}{\sqrt{\lambda }\nu _{1,m,\lambda }^2}\left( 1+\sum _{n\ge 2}e^{-\lambda \Delta (\nu _{n,m,\lambda }-\nu _{1,m,\lambda })}\frac{\nu _{1,m,\lambda }^2\partial _m\nu _{n,m,\lambda } }{\nu _{n,m,\lambda }^2 \partial _m \nu _{1,m,\lambda } }\right) . \end{aligned}$$

From the asymptotic expansion of $\nu$-zeros for large negative m (A.5) and (A.4), it follows that:

$$\begin{aligned} \frac{\partial _m\nu _{1,m,\lambda }}{\sqrt{\lambda }\nu _{1,m,\lambda }^2}&\underset{m\rightarrow -\infty }{=}\ -4|m\sqrt{\lambda }|^{-3}(1+o(1)),\\ \frac{\nu _{1,m,\lambda }^2\partial _m\nu _{n,m,\lambda } }{\nu _{n,m,\lambda }^2 \partial _m \nu _{1,m,\lambda } }&\underset{m\rightarrow -\infty }{\longrightarrow }\ 1 \hbox {~~if~~} n<N(m,\lambda )=\lfloor \frac{2\lambda m^2}{3\pi }+\frac{1}{4} \rfloor +1,\\ \frac{\nu _{1,m,\lambda }^2\partial _m\nu _{n,m,\lambda } }{\nu _{n,m,\lambda }^2 \partial _m \nu _{1,m,\lambda } }&\underset{m\rightarrow -\infty , n\rightarrow \infty }{\longrightarrow }0 \hbox {~~if~~} n>N(m,\lambda ). \end{aligned}$$

Therefore for all $c>1$, there exists $M>0$ such that for all $m<-M$, we get:

$$\begin{aligned} \sum _{n\ge 2}e^{-\lambda \Delta (\nu _{n,m,\lambda }-\nu _{1,m,\lambda })}\frac{\nu _{1,m,\lambda }^2\partial _m\nu _{n,m,\lambda } }{\nu _{n,m,\lambda }^2 \partial _m \nu _{1,m,\lambda } }<c\sum _{n\ge 2}e^{-\lambda \Delta (\nu _{n,m,\lambda }-\nu _{1,m,\lambda })}. \end{aligned}$$

Using (A.7), we easily prove that for large negative m,

$$\begin{aligned} \sum _{n\ge 2}e^{-\lambda \Delta (\nu _{n,m,\lambda }-\nu _{1,m,\lambda })}=e^{-\lambda \Delta (\nu _{2,m,\lambda }-\nu _{1,m,\lambda })}O(1)=o(1). \end{aligned}$$

We then conclude the proof with (A.5). $\square$

Using similar arguments, we can prove the following result:

Proposition A.7

For large negative m, the asymptotic expansion of the probability density of $S^{1,0}$ is given by:

$$\begin{aligned} f_\Delta (m,\lambda )=2|m|^{-2}\frac{e^{-\lambda m^2-\left( \frac{m^2\lambda }{2}-\frac{1}{2}\right) \lambda \Delta +|m|^\frac{2}{3}\lambda ^\frac{4}{3}2^{-\frac{1}{3}}a_1\Delta }}{\sqrt{\pi }}(2+\Delta \lambda )(1+O(|m|^{-\frac{2}{3}})). \end{aligned}$$

(A.15)

Corollary A.7.1

For large negative m,

$$\begin{aligned}{} & {} \log f_\Delta (m,\lambda )=-\lambda m^2-\left( \frac{m^2\lambda }{2}-\frac{1}{2}\right) \lambda \Delta +|m|^\frac{2}{3}\lambda ^\frac{4}{3}2^{-\frac{1}{3}}a_1\Delta \\{} & {} +\log \frac{2(2+\Delta \lambda )}{\sqrt{\pi }}-2\log (|m|)+O(|m|^{-\frac{2}{3}}). \end{aligned}$$

A similar reasoning as the one in the proof of Proposition A.6 may be applied to prove the following:

Proposition A.8

For large negative m, the following asymptotic expansions are satisfied:

• 1. $\partial _\lambda \log (f_\Delta (m,\lambda ))=-m^2-\Delta \left( m^2\lambda -\frac{1}{2}-|m|^\frac{2}{3}\frac{2^{\frac{2}{3}}\lambda ^{-\frac{2}{3}}}{3}a_1\right) +\frac{\Delta }{2+\Delta \lambda }+|m|^{-\frac{2}{3}}O(1).$

• 2. $\partial ^2_\lambda \log (f_\Delta (m,\lambda ))=-\Delta \left( m^2+|m|^\frac{2}{3}\frac{2^{\frac{5}{3}}\lambda ^{-\frac{5}{3}}}{9}a_1\right) -\frac{\Delta ^2}{(2+\Delta \lambda )^2}+|m|^{-\frac{2}{3}}O(1).$

• 3. $\partial ^3_\lambda \log (f_\Delta (m,\lambda ))= \Delta |m|^\frac{2}{3}\frac{5}{27}2^{\frac{5}{3}}\lambda ^{-\frac{8}{3}}a_1+\frac{2\Delta ^3}{(2+\Delta \lambda )^3}+|m|^{-\frac{2}{3}}O(1).$