ajdmom.mdl_svcj.ieziziz_mom

Joint Moments of CPPs \(I\!E\!Z_{s,t}, I\!Z_{s,t}, I\!Z_{s,t}^{*}\)

In this module, we present the computation of the joint moments of (\(I\!E\!Z_{s,t}, I\!Z_{s,t}, I\!Z_{s,t}^*\)). This is accomplished by deriving the joint moment-generating function (MGF) of these processes.

The joint MGF of (\(I\!E\!Z_{s,t}, I\!Z_{s,t}, I\!Z_{s,t}^*\)) can be derived as follows:

\[\begin{split}\begin{align*} M_{I\!E\!Z_{s,t},I\!Z_{s,t},I\!Z_{s,t}^*}(\boldsymbol{a}) &\mathrel{:=} \mathbb{E}[e^{a_1I\!E\!Z_{s,t} + a_2I\!Z_{s,t} + a_3I\!Z_{s,t}^*}]\\ %&= \mathbb{E}[\mathbb{E}[e^{a_1 \sum_{i=1}^{N(t-s)}e^{ks_i}J_i + a_2\sum_{i=1}^{N(t-s)}J_i + a_3\sum_{i=1}^{N(t-s)}J_i^*}|N(t-s) = n]]\\ &= \mathbb{E}[\mathbb{E}[e^{\sum_{i=1}^n [(a_1 e^{ks_i} + a_2)J_i + a_3J_i^*]} |N(t-s) = n]]\\ &= \mathbb{E}[\mathbb{E}[\left(\mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)]\cdot M_{J_i^*}(a_3)\right)^n |N(t-s) = n]]\\ &= \sum_{n=0}^{\infty} \left(\mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)] \cdot M_{J_i^*}(a_3)\right)^n \frac{[\lambda (t-s)]^n e^{-\lambda (t-s)}}{n!}\\ &= e^{-\lambda (t-s)} \sum_{n=0}^{\infty} \left[\lambda (t-s)\mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)]\cdot M_{J_i^*}(a_3)\right]^n/n!\\ %&= e^{-\lambda (t-s)} e^{\lambda (t-s)\mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)]\cdot M_{J_i^*}(a_3)}\\ &= e^{\lambda (t-s)\left(\mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)]\cdot M_{J_i^*}(a_3) - 1\right)}, \end{align*}\end{split}\]

where \(\boldsymbol{a}\) is a vector of real numbers, i.e., \(\boldsymbol{a} \mathrel{:=} (a_1, a_2, a_3)\), \(N(\cdot)\) is the shared counting process for (\(I\!E\!Z_{s,t}, I\!Z_{s,t}, I\!Z_{s,t}^*\)), and \(M_{J_i}(\cdot)\) and \(M_{J_i^*}(\cdot)\) are the MGFs of \(J_i^*\) and \(J_i\), respectively. Note that, conditional on \(N(t-s) = n\), the unsorted arrival times are uniformly distributed over the interval \((s,t]\). For notational simplicity, we use \(\{s_1,\dots, s_n\}\) to denote these unsorted arrival times and omit the conditional notation \(|N(t-s) = n\) in the expectation. In what follows, we will demonstrate that \(\mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)]\) admits a closed-form expression.

First, we substitute the MGF of \(J_i\) (conditioned on \(s_i\)) with its known formula:

\[\begin{align*} \mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)] = \mathbb{E}[\mathbb{E}[e^{(a_1e^{ks_i} + a_2)J_i}|s_i]] = \mathbb{E}\left[\frac{1}{1-(a_1e^{ks_i} + a_2 )\mu_v} \right]. \end{align*}\]

By introducing two new variables \(a \mathrel{:=} - a_1\mu_v, b \mathrel{:=} 1 - a_2\mu_v\), the above expectation simplifies to:

\[\begin{align*} \mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)] = \mathbb{E}\left[\frac{1}{ae^{ks_i} + b}\right] = \frac{1}{t-s}\int_{s}^t\frac{1}{ae^{ks_i}+b}\mathrm{d} s_i. \end{align*}\]

The integral can be computed explicitly using the variable substitution method. Let us introduce \(x \mathrel{:=} e^{ks_i}\). Then, \(s_i = \frac{1}{k}\log x\), \(\mathrm{d} s_i = \frac{1}{k}\frac{1}{x}\mathrm{d} x\). The integral thus becomes:

\[\begin{align*} \int_{s}^t\frac{1}{ae^{ks_i}+b}\mathrm{d} s_i = \int_{e^{ks}}^{e^{kt}} \frac{1}{ax + b} \frac{1}{k} \frac{1}{x} \mathrm{d} x = \frac{1}{k}\int_{e^{ks}}^{e^{kt}} \frac{1}{ax + b} \frac{1}{x} \mathrm{d} x. \end{align*}\]

If \(a = 0\), i.e., \(a_1 = 0\), the integral simplifies to:

\[\begin{equation*} \int_{s}^t\frac{1}{ae^{ks_i}+b}\mathrm{d} s_i = \frac{1}{b}(t-s). \end{equation*}\]

Otherwise, for \(a_1\neq 0\), in any neighborhood of the origin of the vector \((a_1,a_2)\) such that \(ax + b \approx 1\), we have:

\[\begin{split}\begin{align*} \int_{e^{ks}}^{e^{kt}} \frac{1}{ax + b} \frac{1}{x} \mathrm{d} x %&= \int_{e^{ks}}^{e^{kt}} \left(\frac{-a/b}{ax + b} + \frac{1/b}{x}\right) dx\\ %&= -\frac{1}{b}\int_{e^{ks}}^{e^{kt}} \frac{1}{ax+b}\mathrm{d}(ax+b) + \frac{1}{b}\int_{e^{ks}}^{e^{kt}}\frac{1}{x}\mathrm{d} x\\ &= -\frac{1}{b}\left[\log(ae^{kt} + b) - \log(ae^{ks} + b)\right] + \frac{1}{b}k(t-s). \end{align*}\end{split}\]

Therefore, the integral evaluates to:

\[\begin{align*} \int_{s}^t\frac{1}{ae^{ks_i}+b}\mathrm{d} s_i &= -\frac{1}{kb}\left[\log(ae^{kt} + b) - \log(ae^{ks} + b)\right] + \frac{1}{b}(t-s), \end{align*}\]

provided that \((a_1,a_2)\) lies within a sufficiently small neighborhood of the origin \((0,0)\). Finally, we obtain the closed-form expression for \(\mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)]\) as:

\[\begin{equation} \mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)] = \frac{1}{b}\left[1 -\frac{1}{k(t-s)}\left(\log(ae^{kt} + b) - \log(ae^{ks} + b)\right)\right], \end{equation}\]

where \(a = -a_1\mu_v\), \(b = 1 - a_2\mu_v\), and \((a_1,a_2)\) is restricted to a small neighborhood around the origin \((0,0)\).

To simplify the notation, we define a new function \(M_{E\!J,J,J^*}(\cdot)\) as the following product:

\[\begin{equation*} M_{E\!J,J,J^*}(\boldsymbol{a}) \mathrel{:=} \mathbb{E}[M_{J_i}(a_1e^{ks_i} + a_2)]\cdot M_{J_i^*}(a_3). \end{equation*}\]

We know that the MGF of \(J_i^*\) has the following expression:

\[\begin{equation} M_{J_i^*}(a_3) = e^{\mu_s a_3 + \sigma_s^2a_3^2/2} \end{equation}\]

since \(J_I^*\) is normally distributed with mean \(\mu_s\) and variance \(\sigma_s^2\). Combining these results, we obtain the following closed-form expression for the joint MGF of \((I\!E\!Z_{s,t}, I\!Z_{s,t}, I\!Z_{s,t}^*)\):

\[\begin{equation} M_{I\!E\!Z_{s,t},I\!Z_{s,t},I\!Z_{s,t}^*}(\boldsymbol{a}) = e^{\lambda (t-s) (M_{E\!J,J,J^*}(\boldsymbol{a})-1)}, \end{equation}\]

where \(\boldsymbol{a} = (a_1,a_2,a_3)\), \(a = -a_1\mu_v\), \(b = 1-a_2\mu_v\), and

\[\begin{equation*} M_{E\!J,J,J^*}(\boldsymbol{a}) = \frac{1}{b} \left[ 1 - \frac{1}{k(t-s)}\left(\log(ae^{kt} + b) - \log(ae^{ks} + b)\right) \right] e^{\mu_s a_3 + \sigma_s^2 a_3^2/2}. \end{equation*}\]

Given the joint MGF of \((I\!E\!Z_{s,t}, I\!Z_{s,t}, I\!Z_{s,t}^*)\), we can compute the joint moment of \((I\!E\!Z_{s,t}, I\!Z_{s,t},I\!Z_{s,t}^*)\) of any order.

The \(n\)-th (\(n\ge1\)) partial derivative of \(M_{E\!J,J,J^*}(\boldsymbol{a})\) with respect to \(a_1\) is given by

\[\begin{equation*} \frac{\partial^nM_{E\!J,J,J^*}}{\partial a_1^{n}} = \frac{1}{b}\left[\frac{e^{nkt}}{(ae^{kt}+b)^{n}} - \frac{e^{nks}}{(ae^{ks}+b)^{n}} \right] \frac{(n-1)! \mu_v^n}{k(t-s)} M_{J_i^*}(a_3). \end{equation*}\]

Consequently, the \(n\)-th moment of \(E\!J\) can be expressed as

\[\begin{equation} \mathbb{E}[(e^{ks_i}J_i)^n] = (e^{nkt} - e^{nks})\frac{1}{k(t-s)}(n-1)!\mu_v^n. \end{equation}\]

For \(n_1\ge 1\) and \(n_2\ge 1\), the following formula holds:

\[\begin{align} \frac{\partial^{n_1+n_2}M_{E\!J,J,J^*}}{\partial a_1^{n_1}\partial a_2^{n_2}} &= \sum_{i=0}^{n_2} \frac{c(n_1,n_2,i)}{b^{n_2-i+1}}\left[\frac{e^{n_1kt}}{(ae^{kt}+b)^{n_1+i}} - \frac{e^{n_1ks}}{(ae^{ks}+b)^{n_1+i}} \right] \frac{\mu_v^{n_1+n_2}}{k(t-s)} M_{J_i^*}(a_3), \end{align}\]

where \(c(n_1,n_2,i) \mathrel{:=} n_2!(n_1 - 1 + i)!/(i!)\). An alternative approach involves directly computing the joint moment directly, yielding:

\[\begin{equation} \mathbb{E}[(e^{ks_i}J_i)^{n_1}J_i^{n_2}] = \mathbb{E}[e^{n_1ks_i}]\mathbb{E}[J_i^{n_1+n_2}] = \frac{1}{n_1k(t-s)}(e^{n_1kt} - e^{n_1ks})(n_1+n_2)!\mu_v^{n_1+n_2}. \end{equation}\]

For \(n\ge 1\), we have:

\[\begin{split}\begin{align*} \frac{\partial^{n}M_{E\!J,J,J^*}}{\partial a_2^{n}} &= \frac{n!}{b^{n+1}} \left[ 1 - \frac{1}{k(t-s)}\left(\log(ae^{kt} + b) - \log(ae^{ks} + b)\right) \right]\mu_v^n M_{J_i^*}(a_3)\\ &\quad + \sum_{i=1}^n\binom{n}{i}\frac{(n-i)!(i-1)!}{b^{n-i+1}} \left[\frac{1}{(ae^{kt}+b)^i} - \frac{1}{(ae^{ks} + b)^i}\right]\frac{1}{k(t-s)} \mu_v^n M_{J_I^*}(a_3)\\ &= \frac{n!}{b^{n+1}} \left[ 1 - \frac{1}{k(t-s)}\left(\log(ae^{kt} + b) - \log(ae^{ks} + b)\right) \right]\mu_v^n M_{J_i^*}(a_3)\\ &\quad + \sum_{i=1}^n\frac{n!/i}{b^{n-i+1}} \left[\frac{1}{(ae^{kt}+b)^i} - \frac{1}{(ae^{ks} + b)^i}\right]\frac{1}{k(t-s)} \mu_v^n M_{J_I^*}(a_3). \end{align*}\end{split}\]

Thus, the \(n\)-th moment of \(J\) is given by

\[\begin{equation} \mathbb{E}[J_i^n] = n!\mu_v^n. \end{equation}\]

Functions

d1_times_key(key, wrt)
decode(poly)
dmgf_ieziziz(poly, wrt)
dterm(key, coef, wrt)
m_ieziziz(order, par)
moment_ejjj(n1, n2, n3)
moment_ieziziz(n1, n2, n3)	joint moment of \(E[IEZ_{s,t}^{n_1} IZ_{s,t}^{n_2} IZ_{s,t}^{*n_3}]\)
poly2num(poly, par)