De los desafíos textuales

Example 1.3.12 (Bearings-only Tracking). Bearings-only tracking con- cerns online estimation of a target trajectory when the observations consist solely of the direction of arrivals (bearings) of a plane wavefront radiated by a target as seen from a known observer position (which can be fixed, but is, in most applications, moving). The measurements are blurred by noise, which ac- counts for the errors occurring when estimating the bearings. In this context, the range information (the distance between the object and the sensor) is not available. The target is usually assumed to be traveling in a two-dimensional space, the state of the target being its position and its velocity. Although the observations occur at regularly spaced instants, we describe the movement of the object in continuous time to be able to define the derivatives of the motion. The system model that we describe here is similar to that used in Gordon et al. (1993) and Chapter 6 of Ristic et al. (2004)—see also (Pitt and Shephard, 1999; Carpenter et al., 1999).

The state vector at time kT is Xk = (Px,k, ˙Px,k, Py,k, ˙Py,k)t, representing

the target’s position at time kT and its velocity, where T denotes the sampling period. One possible discretization of this model, based on a second order Taylor expansion, is given by (Gordon et al., 1993)

Xk+1= AXk+ RUk , (1.17) where A =     1 T 0 0 0 1 0 0 0 0 1 T 0 0 0 1     , R = σU     T2_{/2 0} T 0 0 T2_/2 0 T    

and {Uk}k≥0 is bivariate standard white Gaussian noise, Uk ∼ N(0, I2).

The scale σU characterizes the magnitude of the random fluctuations of

the acceleration between two sampling points. The initial position X0 is

multivariate Gaussian with mean (µx, ˙µx, µy, ˙µy) and covariance matrix

diag(σ_x2, ˙σ2_x, σ_y2, ˙σ2_y). The measurements {Yk}k≥0are modeled as

Yk= tan−1

 Py,k− Ry,k

Px,k− Rx,k



+ σVVk , (1.18)

where {Vk}k≥0is white Gaussian noise with zero mean and unit variance, and

(Rx,k, Ry,k) is the (known) observer position. It is moreover assumed that

{Uk} and {Vk} are independent. One important feature of this model is that

the amount of information about the range of the target that is present in the measurements is, in general, small. The only range information in the observa- tions arise due to the knowledge of the state equations, which are informative about the maneuvers that the target is likely to perform. Therefore, the ma- jority of range information contained in the model is that which is included

1.3 Examples 25 Target trajectory Target Observer trajectory 0 _x y θk P_{k= (Px,k, Py,k)} R_{k= (Rx,k, Ry,k)}

Fig. 1.9. Two-dimensional bearings-only target tracking geometry.

Example 1.3.13 (Stochastic Volatility). The distributional properties of speculative prices have important implications for several financial models. Let Sk be the price of a financial asset—such as a share price, stock index,

or foreign exchange rate—at time k. Instead of the prices, it is more cus- tomary to consider the relative returns (Sk − Sk−1)/Sk−1 or the log-returns

log(Sk/Sk−1), which both describe the relative change over time of the price

process. In what follows we often refer, for short, to returns instead of relative or log-returns (see Figure 1.10). The unit of the discrete time index k may be for example an hour, a day, or a month. The famous Black-Scholes model, which is a continuous-time model and postulates a geometric Brownian mo- tion for the price process, corresponds to log-returns that are i.i.d. and with a Gaussian N(µ, σ2_{) distribution, where σ is the volatility (the word volatility is}

the word used in econometrics for standard deviation). The Black and Scholes option pricing model provides the foundation for the modern theory of option valuation.

In actual applications, however, this model has certain well-documented deficiencies. Data from financial markets clearly indicate that the distribu- tion of returns usually have tails that are heavier than those of the normal distribution (see Figure 1.11). In addition, even though the returns are ap- proximately uncorrelated over times (as predicted by the Black and Scholes model), they are not independent. This can be readily verified by the fact that the sample autocorrelations of the absolute values (or squares) of the returns are non-zero for a large number of lags (see Figure 1.12). Whereas the former

26 1 Introduction 0 500 1000 1500 2000 2500 200 400 600 800 1000 1200 1400 1600

Time (in days)

S&P Index 0 500 1000 1500 2000 2500 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

Time (in days)

S&P Index

Fig. 1.10. Left: opening values of the Standard and Poors index 500 (S&P 500) over the period January 2, 1990–August 25, 2000. Right: log-returns of the opening values of the S&P 500, same period.

−0.10 −0.05 0 0.05 0.1 50 100 150 200 250 Return −4 −2 0 2 4 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08

Standard Normal Quantiles

Quantiles of Input Sample

Fig. 1.11. Left: histogram of S&P 500 log-returns. Right: quantile-quantile regres- sion plot of empirical quantiles of S&P 500 log-returns against quantiles of the standard normal distribution.

property indicates that the returns can be modeled by a white noise sequence (a stationary process with zero autocorrelation at all positive lags), the latter property indicates that the returns are dependent and that the dependence may even span a rather long period of time.

The variance of returns tends to change over time: the large and small values in the sample occur in clusters. Large changes tend to be followed by

1.3 Examples 27 0 50 100 150 200 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Lag Correlation Coefficient 0 50 100 150 200 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Lag Correlation Coefficient

Fig. 1.12. Left: correlation coefficients of S&P 500 log-returns over the period January 2, 1990–August 25, 2000. The dashed lines are 95% confidence bands

(±1.96/√n) corresponding to the autocorrelation function of i.i.d. white Gaussian

noise. Right: correlation coefficients of absolute values of log-returns, same period.

large changes—of either sign—and small changes tend to be followed by small changes, a phenomenon often referred to as volatility clustering.

Most models for return data that are used in practice are of a multiplicative form,

Yk= σkVk, (1.19)

where {Vk}k≥0 is an i.i.d. sequence and the volatility process {σk}k≥0 is a

non-negative stochastic process such that σk and Vk are independent for all

k. Mostly, {σk} is assumed to be strict sense stationary. It is often assumed

that Vk is symmetric or, at least, has zero mean. The rationale for using

these models is quite simple. First of all, the direction of the price changes is modeled by the sign of Vk only, independently of the order of magnitude of

this change, which is directed by the volatility. Because σk and Vk are inde-

pendent and Vk is assumed to have unit variance, σk2 is then the conditional

variance of Xk given σk. Most models assume that σk is a function of past

values. The simplest model assumes that σk is a function of the squares of the

previous observations. This leads to the celebrated autoregressive conditional heteroscedasticity (ARCH) model developed by Engle (1982),

Yk = p XkVk, Xk = α0+ p X i=1 αiYk−i2 , (1.20)

28 1 Introduction

where α0, . . . , αpare non-negative constants. In the Engle (1982) model, {Vk}

is normal; hence the conditional error distribution is normal, but with con- ditional variance equal to a linear function of the p past squared observa- tions. ARCH models are thus able to reproduce the tendency for extreme values to be followed by other extreme values, but of unpredictable sign. The autoregressive structure can be seen by the following argument. Writ- ing νk = Yk2− Xk = Xk(Vk2− 1), one obtains

Y_k2−

p

X

i=1

αiYk−i2 = α0+ νk. (1.21)

Because {Vk} is an i.i.d. sequence with zero mean and unit variance, {νk}k≥0

is an uncorrelated sequence. Because ARCH(p) processes do not fit log-returns very well unless the order p is quite large, various people have thought about improvements. As (1.21) bears some resemblance to an AR structure, a possi- ble generalization is to introduce an ARMA structure. This construction leads to the so-called GARCH(p, q) process (Bollerslev et al., 1994). This model dis- plays some striking similarities to autoregressive models with Markov regime; this will be discussed in more detail below.

An alternative to the ARCH/GARCH framework is a model in which the variance is specified to follow some latent stochastic process. Such models, referred to as stochastic volatility (SV) models, appear in the theoretical lit- erature on option pricing and exchange rate modeling. In contrast to GARCH- type processes, there is no direct feedback from past returns to the volatility process, which has been questioned as unnatural by some authors. Empiri- cal versions of the SV model are typically formulated in discrete time, which makes inference problems easier to deal with. The canonical model in SV for discrete-time data is (Hull and White, 1987; Jacquier et al., 1994),

Xk+1= φXk+ σUk , Uk ∼ N(0, 1) ,

Yk= β exp(Xk/2)Vk , Vk∼ N(0, 1) , (1.22)

where the observations {Yk}k≥0 are the log-returns, {Xk}k≥0 is the log-

volatility, which is assumed to follow a stationary autoregression of order 1, and {Uk}k≥0 and {Vk}k≥0 are independent i.i.d. sequences. The parameter

β plays the role of the constant scaling factor, φ is the persistence (mem- ory) in the volatility, and σ is the volatility of the log-volatility. Despite a very parsimonious representation, this model is capable of exhibiting a wide range of behaviors. Like ARCH/GARCH models, the model can give rise to a high persistence in volatility (“volatility clustering”). Even with φ = 0, the model is a Gaussian scale mixture that will give rise to excess kurtosis in the marginal distribution of the data. In ARCH/GARCH models with normal er- rors, the degree of kurtosis is tied to the roots of the volatility equation; as the volatility becomes more correlated, the degree of kurtosis also increases. In the stochastic volatility model, the parameter σ governs the degree of mixing independently of the degree of smoothness in the variance evolution.

1.3 Examples 29

It is interesting to note that stochastic volatility models are related to conditionally Gaussian linear state-space models. By taking logarithms of the squared returns, one obtains,

Xk = φXk−1+ σUk−1,

log Y_k2= log β2+ Xk+ Zk, where Zk = log Vk2.

If Vk is standard normal, Zk follows the log χ21distribution. This distribution

may be approximated with arbitrary accuracy by a finite mixture of Gaussian distributions, and then the SV model becomes a conditionally Gaussian linear state-space model (Sandmann and Koopman, 1998; Durbin and Koopman, 2000). This time, the latent variable Ck is the mixture component and the

model writes

Wk+1= φWk+ Uk, Uk ∼ N(0, 1) ,

Yk = Wk+ (µ(Ck) + σV(Ck)Vk) , Vk∼ N(0, 1) .

This representation of the stochastic volatility model may prove useful when deriving numerical algorithms to filter the hidden state or estimate the model

parameters.