# Vignette: Simulation of (weighted) trawl processes

#### Almut Veraart

#### 2022-08-18

# Definition of a (weighted) trawl process

The `ambit` package can be used to simulate univariate (weighted) trawl processes of the form \[ Y_t =\int_{(-\infty,t]\times \mathbb{R}} p(t-s)\mathbb{I}_{(0, a(t-s))}(x)L(dx,ds), \mbox{ for } t \ge 0. \] We refer to \(p\) as the weight/kernel function, \(a\) as the trawl function and \(L\) as the Lévy basis.

If the function \(p\) is given by the identity function, \(Y\) is a trawl process, otherwise we refer to \(Y\) as a weighted trawl process.

## Choice of the trawl function

This package only considers the case when the trawl function, denoted by \(a\), is strictly monotonically decreasing.

The following implementations are currently included in the function `sim_weighted_trawl`:

- Exponential trawl function (“Exp”): The trawl function is parametrised by one parameter \(\lambda>0\) and defined as

\[a(x)=\exp(-\lambda x), \qquad \mathrm{for \,} x \geq 0.\]

- supIG trawl function (“supIG”): The trawl function is parametrised by two parameters \(\delta\) and \(\gamma\), which are assumed to not be simultaneously equal to zero, and is defined as

\[a(x)=(1+2x\gamma^{-2})^{-1/2}\exp(\delta \gamma(1-(1+2x\gamma^{-2})^{1/2})), \qquad \mathrm{for \,} x \geq 0.\]

- supGamma trawl function (“LM”): The trawl function is parametrised by two parameters \(H> 1\) and \(\alpha > 0\), and is defined as

\[a(x) = (1+x/\alpha)^{-H}, \qquad \mathrm{for \,} x \geq 0.\]

Alternatively, the user can use the function `sim_weighted_trawl_gen` which requires specifying a monotonic trawl function \(a(\cdot)\).

## Choice of kernel/weight

The user can choose a suitable weight function \(p\). If no weight function is provided, then the function \(p(x)=1\) for all \(x\) is chosen. I.e. the resulting process is a trawl process rather than a weighted trawl process.

## Choice of Lévy bases

The driving noise of the process is given by a homogeneous Lévy basis denoted by \(L\) with corresponding Lévy seed \(L'\).

In the following, we denote by \(A\) a Borel set with finite Lebesgue measure.

The following infinitely divisible distributions are currently included in the implementation:

### Support on \((0, \infty)\)

- Gamma distribution (“Gamma”): \(L'\sim \Gamma(\alpha_g, \sigma_g)\), where \(\alpha_g>0\) is the shape parameter and \(\sigma_g>0\) the scale parameter. Then the corresponding density is given by \[ f(x)=\frac{1}{\sigma_g^{\alpha_g}\Gamma(\alpha_{g})}x^{\alpha_g-1}e^{-x/\sigma_g}, \] for \(x>0\). The characteristic function is given by \[ \psi(u)=(1-ui\sigma_g)^{\alpha_g}, \] for \(u\in \mathbb{R}\). We note that \(\mathbb{E}(L')=\alpha_g \sigma_g\), \(\mathrm{Var}(L')=\alpha_g \sigma_g^2\) and \(c_4(L')=6\alpha \sigma^4\). Here we have \[ L(A)\sim \Gamma(\mathrm{Leb}(A)\alpha_g, \sigma_g). \]

### Support on \(\mathbb{R}\)

- Gaussian case (“Gaussian”): \(L'\sim \mathrm{N}(\mu, \sigma^2)\). In this case, \[ L(A)\sim \mathrm{N}(\mathrm{Leb}(A)\mu, \mathrm{Leb}(A)\sigma^2). \] We note that \(\mathbb{E}(L')=\mu\), \(\mathrm{Var}(L')=\sigma^2\) and \(c_4(L')=0\).

- Cauchy distribution (“Cauchy”): \(L'\sim \mathrm{Cauchy}(l, s)\), where \(l\in \mathbb{R}\) is the location parameter and \(s>0\) the scale parameter. The corresponding density is given by \[ f(x)=\frac{1}{\pi s(1+(x-l)/s)^2}, \quad x \in \mathbb{R}, \] and the characteristic function is given by \[ \psi(u)=l i u-s|u|, \quad u \in \mathbb{R}. \] Here we have \[ L(A) \sim \mathrm{Cauchy}(l\mathrm{Leb}(A), s\mathrm{Leb}(A)). \]

- Normal inverse Gaussian case (“NIG”): \(L'\sim \mathrm{NIG}(\mu, \alpha, \beta, \delta)\), where \(\mu \in \mathbb{R}\) is the location parameter, \(\alpha \in \mathbb{R}\) the tail heaviness parameter, \(\beta \in \mathbb{R}\) the asymmetry parameter and \(\delta\in \mathbb{R}\) the scale parameter. We set \(\gamma=\sqrt{\alpha^2-\beta^2}\). The corresponding density is given by \[ f(x)=\frac{\alpha \delta K_1(\alpha\sqrt{\delta^2+(x-\mu)^2})}{\pi\sqrt{\delta^2+(x-\mu)^2}} \exp(\delta \gamma+\beta(x-\mu)), \quad x \in \mathbb{R}. \] Here \(K_1\) denotes the Bessel function of the third kind with index 1. The characteristic function is given by \[ \psi(u)=\exp(iu\mu+\delta(\gamma-\sqrt{\alpha^2-(\beta+iu)^2})), \quad u \in \mathbb{R}. \] In this case, we have \[ L(A)\sim \mathrm{NIG}(\mu \mathrm{Leb}(A), \alpha, \beta, \delta \mathrm{Leb}(A)). \] Also, \(\mathbb{E}(L')=\mu +\frac{\delta \beta}{\gamma}\), \(\mathrm{Var}(L')=\frac{\delta \alpha^2}{\gamma^3}\) and \(c_4(L')=\frac{3\alpha^2\delta(4\beta^2+\alpha^2)}{\gamma^7}\).

### Support on \(\mathbb{N}_0=\{0, 1, \ldots\}\)

- Poisson case (“Poisson”): \(L' \sim \mathrm{Poi}(v)\) for \(v > 0\). In this case, \[ L(A)\sim \mathrm{Poi}(\mathrm{Leb}(A)v). \] We note that \(\mathbb{E}(L')=\lambda\), \(\mathrm{Var}(L')=\lambda\) and \(c_4(L')=\lambda\).

- Negative binomial case (“NegBin”): \(L'\sim \mathrm{NegBin}(m, \theta)\) for \(m>0, \theta \in (0, 1)\). I.e. the corresponding probability mass function is given by \(\mathrm{P}(L'=x)=\frac{1}{x!}\frac{\Gamma \left( m +x\right) }{\Gamma \left( m \right) }\left( 1-\theta \right)^{m }\theta^{x}\) for \(x \in \{0, 1, \ldots\}\). We note that \(\mathbb{E}(L')=m \theta/(1-\theta)\), \(\mathrm{Var}(L')=m \theta/(1-\theta)^2\) and \(c_4(L')=m\theta(\theta^2+4\theta+1)/(\theta-1)^4\). Then,

\[ L(A)\sim \mathrm{NegBin}(\mathrm{Leb}(A)m, \theta). \]

### Examples

We demonstrate the simulation of various trawl processes.

    library(ambit)
    library(ggplot2)

We start off with a trawl with standard normal marginal distribution and exponential trawl function.

    #Set the number of observations
    n <-2000
    #Set the width of the grid
    Delta<-0.1

    #Determine the trawl function
    trawlfct="Exp"
    trawlfct_par <-0.5

    #Choose the marginal distribution
    distr<-"Gauss"
    #mean 0, std 1
    distr_par<-c(0,1)

    #Simulate the path
    set.seed(233)
    path <- sim_weighted_trawl(n, Delta, trawlfct, trawlfct_par, distr, distr_par)$path

    #Plot the path
    df <- data.frame(time = seq(0,n,1), value=path)
    p <- ggplot(df, aes(x=time, y=path))+
        geom_line()+
        xlab("l")+
        ylab("Trawl process")
    p

    #Plot the empirical acf and superimpose the theoretical one

    #Plot the acf
    my_acf <- acf(path, plot = FALSE)
    my_acfdf <- with(my_acf, data.frame(lag, acf))
    #Confidence limits
    alpha <- 0.95
    conf.lims <- c(-1,1)*qnorm((1 + alpha)/2)/sqrt(n)
    q <- ggplot(data = my_acfdf, mapping = aes(x = lag, y = acf)) +
           geom_hline(aes(yintercept = 0)) +
           geom_segment(mapping = aes(xend = lag, yend = 0))+
            geom_hline(yintercept=conf.lims, lty=2, col='blue') +
            geom_function(fun = function(x) acf_Exp(x*Delta,trawlfct_par), colour="red", size=1.2)+
            xlab("Lag")+
            ylab("Autocorrelation")
    q

The same trawl process can be obtained using the `sim_weighted_trawl_gen` instead as follows:

    #Set the number of observations
    n <-2000
    #Set the width of the grid
    Delta<-0.1

    #Determine the trawl function
    trawlfct_par <-0.5
    a <- function(x){exp(-trawlfct_par*x)}

    #Choose the marginal distribution
    distr<-"Gauss"
    #mean 0, std 1
    distr_par<-c(0,1)

    #Simulate the path
    set.seed(233)
    path <- sim_weighted_trawl_gen(n, Delta, trawlfct_gen=a, distr, distr_par)$path

    #Plot the path
    df <- data.frame(time = seq(0,n,1), value=path)
    p <- ggplot(df, aes(x=time, y=path))+
        geom_line()+
        xlab("l")+
        ylab("Trawl process")
    p

    #Plot the empirical acf and superimpose the theoretical one

    #Plot the acf
    my_acf <- acf(path, plot = FALSE)
    my_acfdf <- with(my_acf, data.frame(lag, acf))
    #Confidence limits
    alpha <- 0.95
    conf.lims <- c(-1,1)*qnorm((1 + alpha)/2)/sqrt(n)
    q <- ggplot(data = my_acfdf, mapping = aes(x = lag, y = acf)) +
           geom_hline(aes(yintercept = 0)) +
           geom_segment(mapping = aes(xend = lag, yend = 0))+
            geom_hline(yintercept=conf.lims, lty=2, col='blue') +
            geom_function(fun = function(x) acf_Exp(x*Delta,trawlfct_par), colour="red", size=1.2)+
            xlab("Lag")+
            ylab("Autocorrelation")
    q