Bayesian inference of the fractional Ornstein–Uhlenbeck process under a flow sampling scheme

Abstract

Using recent developments in econometrics and computational statistics we consider the estimation of the fractional Ornstein–Uhlenbeck process under a flow sampling scheme. To address the problem, we adopt throughout the paper an exact discretization approach. A flow sampling scheme arises, for example, naturally in modelling asset prices in continuous time since the time integral over successive observations defines the observable increments of asset log-prices. Exact discretization delivers an ARIMA(1,1,1) model for log-prices with a fractional driving noise. Building on the resulting exact discretization formulae and covariance function, a new Markov Chain Monte Carlo scheme is proposed and apply it to investigate the properties of both the time and frequency domain likelihoods/posteriors. For the exact discrete model, we adopt a general sampling interval of length h. This allows us to determine the optimal choice of h independent of the sample size. To illustrate the methods, with no ambition to a comprehensive data analysis, we use high frequency stock price data showing the relevance of aggregation over time issues in modelling asset prices.

Appendix

1.1 Proof of Lemma 1

First, using properties of the Gamma and Beta special functions, it is a straightforward algebraic exercise to prove that $\frac{{\varLambda^{2} (H)}}{{\varGamma^{2} \left( {H + \frac{1}{2}} \right)}} = \frac{1}{2\pi }V(H)\frac{1}{H(2H - 1)}$, where $\varLambda^{2} (H) = - \frac{{2\varGamma (H + \frac{1}{2})\varGamma ( - 2H)}}{{\varGamma (\frac{1}{2} - H)}}$ and $V(H) = 2\varGamma (2 - 2H)\sin \left[ {\frac{\pi }{2}\left( {2H - 1} \right)} \right]$.

Next, define the increments of fBm by $B^{H} (\Delta_{t} ) = B^{H} (t) - B^{H} (t - h)$, $\Delta_{t} = \left( {t - h,\,\,t} \right)$. Then by Assumption 2 we obtain

$$ \frac{1}{{h^{2} }}E\left[ {B^{H} (\Delta_{r} )B^{H} (\Delta_{s} )} \right] = \frac{{\sigma^{2} }}{{h^{2} }}\frac{{\varLambda^{2} (H)}}{{2\varGamma^{2} \left( {H + \frac{1}{2}} \right)}}\left[ {\left| {r - s + h} \right|^{2H} + \left| {r - s - h} \right|^{2H} - 2\left| {r - s} \right|^{2H} } \right] $$

However, $\frac{{\left| {r - s + h} \right|^{2H} + \left| {r - s - h} \right|^{2H} - 2\left| {r - s} \right|^{2H} }}{{h^{2} }} \to 2H(2H - 1)\left| {r - s} \right|^{2H - 2}$ as $h \to 0$. Thus, we obtain (see also proposition 7.2.10 of Samorodnitsky and Taqqu 1994) $\frac{1}{{h^{2} }}E\left[ {B^{H} (\Delta_{r} )B^{H} (\Delta_{s} )} \right] \to \frac{{\sigma^{2} }}{2\pi }V(H)\left| {r - s} \right|^{2H - 2}$ as $h \to 0$. Divide $\left[ {a,\,\,b} \right]$ into equal subintervals $\Delta_{n1} , \cdots ,\Delta_{nn}$ of length $h_{n}$ and let $\phi_{n1} , \ldots ,\phi_{nn}$, be numbers defining $\phi$ such as $\phi_{n} (t) = \phi_{nk} ,\,\,t \in \Delta_{nk} ,\,k = 1, \ldots ,n$. We similarly divide $\left[ {c,\,\,d} \right]$ into subintervals and set $\psi_{n} (t) = \psi_{nk} ,\,\,t \in \Delta_{nk} ,\,k = 1, \ldots ,n$. Then, we can define the integral of a deterministic function with respect to the fBm, namely $\int_{a}^{b} {\phi (r)dB^{H} (r)}$, as the mean square limit of Riemann–Stieltjes approximating sums $\sum\nolimits_{k = 1}^{n} {\varphi_{nk} \frac{1}{{h_{n} }}B^{H} (\Delta_{nk} )}$ (see Cramer and Leadbetter 1967, p. 87 and Arnold 1974, Sect. 4.4) so that

$$ E\left( {\sum\nolimits_{k = 1}^{n} {\varphi_{nk} \frac{1}{{h_{n} }}B^{H} (\Delta_{nk} )} } \right)\left( {\sum\nolimits_{j = 1}^{m} {\psi_{mj} \frac{1}{{h_{m} }}B^{H} (\Delta_{mj} )} } \right) = \sum\nolimits_{k = 1}^{n} {\sum\nolimits_{j = 1}^{m} {\varphi_{nk} \psi_{mj} \frac{1}{{h_{n} h_{m} }}E} } \left[ {B^{H} (\Delta_{nk} )B^{H} (\Delta_{mj} )} \right] \to $$

$\frac{{\sigma^{2} }}{2\pi }V(H)\int_{a}^{b} {\int_{c}^{d} {\phi (r)\psi (s)\left| {r - s} \right|^{2H - 2} } } drds$, as $n,m \to \infty$. Pipiras and Taqqu (2002) obtain similar results (p. 191 Eq. 6.11) via fractional calculus.

1.2 Proof of Theorem 1

Under Assumption 1 the solution of (1) in the interval ((t-1)h, th) satisfies

$$ r_{th} = e^{ah} r_{{\left( {t - 1} \right)h}} + v + \xi_{th} ,\;t = 1,\,2, \ldots ,N $$

(28)

where $r_{th} = r(th)$,

$$ v = c\int_{0}^{h} {e^{as} ds} $$

(29)

and

$$ \xi_{th} = \int_{{\left( {t - 1} \right)h}}^{th} {e^{{a\left( {th - s} \right)}} dB^{H} (s)ds} = \int_{0}^{h} {e^{as} dB^{H} \left( {th - ds} \right)} $$

(30)

See for example Eq. (8) in the exact discretization of Phillips (1991) and Theorem 1 in Bergstrom (1984). To obtain the exact discrete model for the observed h-period log-price returns $\int_{{\left( {t - 1} \right)h}}^{th} {r(s)ds} =$ $\log P(th) - \log P(th - h)$, t = 1, 2, … N, we integrate (28) over ((t-1)h, th) to obtain

$$ \int_{{\left( {t - 1} \right)h}}^{th} {r(s)ds} = e^{ah} \int_{{\left( {t - 2} \right)h}}^{{\left( {t - 1} \right)h}} {r(s)ds} + vh + \kappa_{th} ,\;t = 2,\,3, \ldots N $$

(31)

where $\kappa_{th}$ is stated in (12) and

$$ K_{1} (r) = \int_{0}^{r} {e^{as} ds} , $$

(32)

$$ K_{2} (r) = \int_{r}^{h} {e^{as} ds} . $$

(33)

The stated formulae (15)–(17) are deduced by (32), (33). In view of Eq. (12), Assumption 2 and the definition

$$ \begin{aligned} \gamma_{k}^{F} (\tau h) & = E\left[ {\int_{0}^{h} {K_{1} \left( r \right)} dB^{H} \left( {th - r} \right) + \int_{0}^{h} {K_{2} \left( r \right)} dB^{H} \left( {th - h - r} \right)} \right] \\ & \times \left[ {\int_{0}^{h} {K_{1} \left( s \right)} dB^{H} \left( {th + \tau h - s} \right) + \int_{0}^{h} {K_{2} \left( s \right)} dB^{H} \left( {th + \tau h - h - s} \right)} \right] \\ \end{aligned} $$

we obtain the auto-covariance function (15).

1.3 Derivation of the Whittle likelihood function

The Whittle approximation of the Gaussian likelihood has been noted by Hannan and Deistler (1988), p. 224) for the estimation of linear systems and by Beran (1994, p. 109) and Giraitis and Robinson (2002, p. 234) for the estimation of long memory models. In the current context under regularity conditions we can write

$$ \frac{1}{N}\log \left| V \right| \to \frac{1}{2\pi }\int_{ - \pi }^{\pi } {\log f_{{\bar{x}}} } (\omega )d\omega \;{\text{as}}\;N \to \infty $$

(34)

where $V$ and $f_{{\bar{x}}} (\omega )$, the covariance matrix and spectral density function respectively of the DGM, are stated in (16) and (20). Moreover, $\frac{1}{N}\bar{x}'V^{ - 1} \bar{x}\,\left( { = \frac{1}{N}tr\left[ {V^{ - 1} \bar{x}\bar{x}'} \right]} \right)$ can be approximated by the quadratic form $\frac{1}{N}\bar{x}'S\bar{x}\,$ where $S\,$ is a $N \times N$ matrix with the $\left( {m,n} \right)$ element $S\left( {m - n} \right)$:

$$ S(j) = \frac{1}{2\pi }\int_{ - \pi }^{\pi } {\frac{{e^{ij\omega } }}{{f_{{\bar{x}}} (\omega )}}} d\omega . $$

(35)

Thus, we can easily deduce that

$$ \frac{1}{N}\bar{x}'S\bar{x}\, = \frac{1}{2\pi }\int_{ - \pi }^{\pi } {\frac{I(\omega )}{{f_{{\bar{x}}} (\omega )}}} d\omega $$

(36)

where $I(\omega )$ is the periodogram of the data. The matrix $S$ can be viewed as an approximation of the inverse of the covariance matrix $V$. From the above discussion, we easily conclude that the Gaussian likelihood (21) possesses the approximation:

$$ \frac{1}{2\pi }\int_{ - \pi }^{\pi } {\left[ {\log f_{{\bar{x}}} (\omega ) + \frac{I(\omega )}{{f_{{\bar{x}}} (\omega )}}} \right]} d\omega . $$

(36)

Further approximating the above integral with summation, we finally obtain the objective function (27).