SRJD Model

In this subpackage (ajdmom.mdl_srjd), we consider the following Square-Root Jump Diffusion:

\[dv(t) = k(\theta - v(t))dt + \sigma_v\sqrt{v(t)}dw^v(t) + dz(t),\]

which is the same as that in ajdmom.ito_cond2_mom, details can be referred to therein. Please note that we write \(v_t, z_t, v_0\) and \(v(t), z(t), v(0)\) interchangeably.

Conditional Moments - I

The solution to the SRJD process can be rewritten as:

\[e^{kt}(v_t-\theta) = (v_0-\theta) + \sigma_v I\!E_t + I\!E\!Z_t.\]

The first unconditional moment is calculated as \(\mathbb{E}[v] = \theta + \lambda\mu_v/k,\) since \(\mathbb{E}[I\!E\!Z_t] = \lambda\mu_v (e^{kt}-1)/k\), \(\mathbb{E}[I\!E_t] = 0\) and \(\mathbb{E}[v_t] = \mathbb{E}[v_0] = \mathbb{E}[v]\). This result allows us to rewrite solution to the SRJD in the following form:

\[e^{kt}(v_t-\mathbb{E}[v]) = \sigma_vI\!E_t + \overline{I\!E\!Z}_t + (v_0-\mathbb{E}[v]),\]

where \(\overline{I\!E\!Z}_t \mathrel{:=} I\!E\!Z_t - \mathbb{E}[I\!E\!Z_t]\) represents the centralized term. This centralized term can be decomposed similarly: \(\overline{I\!E\!Z}_t = \overline{I\!E\!Z}_{s} + \overline{I\!E\!Z}_{s,t}\) where \(\overline{I\!E\!Z}_{s,t} \mathrel{:=} I\!E\!Z_{s,t} - \mathbb{E}[I\!E\!Z_{s,t}]\). It is straightforward to verify that \(\mathbb{E}[\overline{I\!E\!Z}_{s,t}^m]\) can be expressed as a “polynomial”:

\[\mathbb{E}[\overline{I\!E\!Z}_{s,t}^m] = \sum_{\boldsymbol{j}} c_{\boldsymbol{j}} e^{j_1kt}e^{j_2ks}k^{-j_3}\lambda^{j_4}\mu_v^{j_5},\]

where, with a slight abuse of notation, \(\boldsymbol{j}\mathrel{:=} (j_1,\dots,j_5)\), and \(c_{\boldsymbol{j}}\) denotes the associated coefficient for the corresponding monomial. Therefore, the conditional joint moment \(\mathbb{E}[I\!E_t^{m_1}\overline{I\!E\!Z}_t^{m_2}|v_0]\) can be computed using the following recursive equation:

\[\mathbb{E}[I\!E_t^{m_1}\overline{I\!E\!Z}_t^{m_2}|v_0] = \sum_{i=0}^{m_2}\binom{m_2}{i}\sum_{\boldsymbol{j}}c_{\boldsymbol{j}} e^{j_1kt}k^{-j_3}\lambda^{j_4}\mu_v^{j_5}P(m_1,m_2),\]

where \(P(m_1,m_2) \mathrel{:=} [m_1(m_1-1)/2]\cdot(p_1 + p_2 + p_3 + p_4)\), and

\[\begin{split}\begin{align*} p_1 &\mathrel{:=} \int_0^t e^{(j_2+1)ks}\mathbb{E}[I\!E_s^{m_1-2}\overline{I\!E\!Z}_s^i|v_0] \mathrm{d} s \times (v_0-\mathbb{E}[v]),\\ p_2 &\mathrel{:=} \int_0^t e^{(j_2+2)ks}\mathbb{E}[I\!E_s^{m_1-2}\overline{I\!E\!Z}_s^i|v_0] \mathrm{d} s \times \mathbb{E}[v],\\ p_3 &\mathrel{:=} \int_0^t e^{(j_2+1)ks}\mathbb{E}[I\!E_s^{m_1-1}\overline{I\!E\!Z}_s^i|v_0] \mathrm{d} s \times \sigma_v,\\ p_4 &\mathrel{:=} \int_0^t e^{(j_2+1)ks}\mathbb{E}[I\!E_s^{m_1-2}\overline{I\!E\!Z}_s^{i+1}|v_0] \mathrm{d} s. \end{align*}\end{split}\]

For the special case \(m_1=1\), it is easy to find that \(\mathbb{E}[I\!E_t\overline{I\!E\!Z}_t^{m_2}|v_0] = 0\).

With the preparations outlined above, the \(m\)-th conditional central moment of \(v_t\) can be calculated as:

\[e^{mkt}\mathbb{E}[(v_t-\mathbb{E}[v])^m|v_0] = \sum_{m_1+m_2+m_3=m}\binom{m}{m_1,m_2,m_3}\mathbb{E}[I\!E_t^{m_1} \overline{I\!E\!Z}_t^{m_2}|v_0]\sigma_v^{m_1}(v_0-\mathbb{E}[v])^{m_3}.\]

We note that the conditional central moment \(\mathbb{E}[(v_t-\mathbb{E}[v])^m|v_0]\) will be computed as a polynomial in \((v_0-\mathbb{E}[v])\):

\[\mathbb{E}[(v_t-\mathbb{E}[v])^m|v_0] = c_m(v_0-\mathbb{E}[v])^m + c_{m-1}(v_0-\mathbb{E}[v])^{m-1} + \cdots + c_1(v_0-\mathbb{E}[v]) + c_0,\]

where \(c_{m},\dots, c_0\) are coefficients, some of which may be zero and \(c_m = e^{-mkt}\). The reason is that the conditional joint moment \(\mathbb{E}[I\!E_t^{m_1}\overline{I\!E\!Z}_t^{m_2}|v_0]\) produces a polynomial in \((v_0 - \mathbb{E}[v])\) of order at most \(\lfloor m_1 / 2 \rfloor\).

Unconditional Moments

We further note that the unconditional central moment of \(v_t\) can be computed via:

\[\mathbb{E}[(v_t-\mathbb{E}[v])^m] = \mathbb{E}[\mathbb{E}[(v_t-\mathbb{E}[v])^m|v_0]].\]

Meanwhile, due to the assumption that \(v_t\) is strictly stationary, we have

\[\mathbb{E}[(v_t-\mathbb{E}[v])^m] = \mathbb{E}[(v_0-\mathbb{E}[v])^m].\]

Thus, the \(m\)-th unconditional central moment of \(v_t\) can be computed using the following recursive equation:

\[(1-e^{-mkt})\mathbb{E}[(v_0-\mathbb{E}[v])^m] = c_{m-1}\mathbb{E}[(v_0-\mathbb{E}[v])^{m-1}] + \cdots + c_1\mathbb{E}[(v_0-\mathbb{E}[v])] + c_0.\]

For example, the second central moment is computed as:

\[\mathbb{E}[(v_0-\mathbb{E}[v])^2] = \frac{\lambda\mu_v^2}{k} + \frac{\mathbb{E}[v]\sigma_v^2}{2k},\]

and the third central moment:

\[\mathbb{E}[(v_0-\mathbb{E}[v])^3] = \frac{2\lambda\mu_v^3}{k} + \frac{\sigma_v^2\lambda\mu_v^2}{k^2} + \frac{\mathbb{E}[v]\sigma_v^4}{2k^2}.\]

Using the above recursive equation, we can compute the fourth and any higher central moments recursively. Given the central moments, the corresponding non-central moments can be easily computed.

Conditional Moments - II

Given \(v_0\) and \(z_{s}, 0\le s \le t\),

\[e^{kt}v_t = \mu_{ev} + \sigma_v I\!E_t,\]

where \(\mu_{ev} \triangleq (v_0-\theta) + \theta e^{kt} + I\!E\!Z_t\). Thus, we have

\[\begin{split}\begin{align*} &\mathbb{E}[(e^{kt}v_t)^m|v_0, z_s, 0\le s \le t] \\ &\quad = \sum_{j=0}^mC_m^j \mu_{ev}^j \sigma_v^{m-j} \mathbb{E}[I\!E_t^{m-j}|v_0, z_s, 0\le s \le t], \end{align*}\end{split}\]

further,

\[\begin{split}\begin{align*} &\mathbb{E}[v_t^m|v_0, z_s, 0\le s \le t] \\ &= e^{-mkt} \sum_{j=0}^mC_m^j \mu_{ev}^j \sigma_v^{m-j} \mathbb{E}[I\!E_t^{m-j}|v_0, z_s, 0\le s \le t]. \end{align*}\end{split}\]

We have, \(\forall m \ge 2\),

(1)\[\begin{split}\begin{align*} &\mathbb{E}[I\!E_t^m|v_0, z(s), 0\le s \le t]\\ &= \frac{1}{2}m(m-1)(v_0-\theta)\int_0^te^{ks}\mathbb{E}[I\!E_s^{m-2} |v_0, z(s), 0\le s \le t]ds\\ &\quad + \frac{1}{2}m(m-1)\theta\quad~~~ \int_0^te^{2ks}\mathbb{E} [I\!E_s^{m-2}|v_0, z(s), 0\le s \le t]ds\\ &\quad + \frac{1}{2}m(m-1)\quad~~~ \int_0^t(e^{ks}I\!E\!Z_s)\mathbb{E} [I\!E_s^{m-2}|v_0, z(s), 0\le s \le t]ds\\ &\quad + \frac{1}{2}m(m-1)\sigma_v\quad \int_0^te^{ks}\mathbb{E} [I\!E_s^{m-1}|v_0, z(s), 0\le s \le t]ds, \end{align*}\end{split}\]

where \(\mathbb{E}[I\!E_t^0|v_0, z(s), 0\le s \le t] = 1\) and \(\mathbb{E}[I\!E_t|v_0, z(s), 0\le s \le t] = 0\).

We decode \(\mathbb{E}[I\!E_t^m|v_0, z(s), 0\le s \le t]\) as a Poly object of the following form:

\[\begin{split}\begin{align*} &\mathbb{E}[I\!E_t^m|v_0, z(s), 0\le s \le t]\\ &\equiv \sum_{\boldsymbol{j}, \boldsymbol{l}} c_{\boldsymbol{j}, \boldsymbol{l}} v_0^{j_1} k^{-j_2} \theta^{j_3} \sigma_v^{j_4} e^{j_5kt} f_{Z_t}(\boldsymbol{l}), \end{align*}\end{split}\]

where \(\boldsymbol{j}\equiv (j_1,\dots,j_5)\), and

\[\begin{split}\boldsymbol{l}\equiv \begin{cases} (l_1,\dots,l_n) & \text{if } n \ge 1,\\ () & \text{otherwise}, \end{cases}\end{split}\]

noting that \(n\) denotes the number of compound Poisson processes being multiplied together. The function \(f_{Z_t}(\cdot)\) is defined as

(2)\[\begin{split} f_{Z_t}(\boldsymbol{l}) \equiv \begin{cases} \sum_{i_1=1}^{N(t)}\cdots\sum_{i_n=1}^{N(t)} e^{ks_{i_1} + \cdots + ks_{i_n}} J_{i_1}\cdots J_{i_n} \cdot e^{l_1 k s_{i_1} + \cdots+ l_{n} k(s_{i_1}\vee\cdots\vee s_{i_n})} & \text{if } n \ge 1,\\ 1 & \text{otherwise}. \end{cases}\end{split}\]

The conditional moments of \(I\!E_t\) and \(v(t)\) are implemented in moment_IE() and moment_v(), respectively, in the ajdmom.mdl_srjd subpackage.

For the conditional central moments, by defining \(\overline{v}(t)\triangleq v(t)-\mathbb{E}[v(t)|v_0, z(s), 0\le s\le t]\), thus \(\overline{v}(t) \equiv e^{-kt}\sigma_vI\!E_t\).

\[\begin{split}\begin{align*} &\mathbb{E}[\overline{v}^m(t)|v_0, z(s), 0\le s\le t]\\ &= e^{-mkt}\sigma_v^m \mathbb{E}[I\!E_t^m|v_0, z(s), 0\le s\le t]. \end{align*}\end{split}\]

The conditional central moments are implemented in cmoment_v() in the ajdmom.mdl_srjd subpackage.

API

ajdmom.mdl_srjd.mom	Moments of the SRJD model
ajdmom.mdl_srjd.cmom	Central Moments of the SRJD model
ajdmom.mdl_srjd.cond_mom	Conditional Moments (I)
ajdmom.mdl_srjd.cond_cmom	Conditional Central Moments (I)
ajdmom.mdl_srjd.cond2_mom	Conditional moments (II)
ajdmom.mdl_srjd.cond2_cmom	Conditional central moments (II)