ajdmom.itos_mom

Itô process moments under Superposition of Two Square-Root Diffusion Processes

See 2FSV Model for superposition of two square-root diffusion processes. I will demonstrate how to compute

(1)\[\mathbb{E}[m_4m_5m_6m_7m_8] \triangleq \mathbb{E}[I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8}|v_{1,n-1},v_{2,n-1}].\]

Result

The result is presented first. Function moment_IEI_IEII() is defined to compute equation (1) which returns a Poly with attribute

keyfor = ('((n_1m,n_2m,i_m),...,(n_11,n_21,i_1))', 'e^{(m_4*k1+m_6*k2)(n-1)h}','e^{(j_1*k1+j_2*k2)h}','h','v_{1,n-1}','theta1', 'sigma_v1','v_{2,n-1}','theta2','sigma_v2'),

i.e., with key components standing for

  • key[0]: \(((n_{1m},n_{2m},i_{m}),...,(n_{11},n_{21},i_{1}))\) for \((n_{1m}k_1+n_{2m}k_2)^{-i_m}\cdots (n_{11}k_1+n_{21}k_2)^{-i_1}\),

  • key[1]: \((m_4,m_6)\) for \(e^{(m_4k_1+m_6k_2)(n-1)h}\)

  • key[2]: \((j_1,j_2)\) for \(e^{(j_1k_1+j_2k_2)[t-(n-1)h]}\),

  • key[3]: \(i\) for \([t-(n-1)h]^i\),

  • key[4],key[5],key[6]: \(v_{1,n-1}, \theta_1, \sigma_{v1}\) raised to the respective power,

  • key[7],key[8],key[9]: \(v_{2,n-1}, \theta_2, \sigma_{v2}\) raised to the respective power.

Therefore, I write the result of equation (1) as

\[\begin{split}&\mathbb{E}[m_4m_5m_6m_7m_8]\\ &= \sum_{t0,(m_4,m_6),(i,i'),j,l,p,q,l',p',q'} b_{t0(m_4,m_6)(i,i')jlpql'p'q'} \cdot \\ &\quad (n_{1m}k_1+n_{2m}k_2)^{-i_m} \cdots (n_{11}k_1+n_{21}k_2)^{-i_1}\cdot e^{(m_4k_1+m_6k_2)(n-1)h}\cdot\\ &\quad e^{(ik_1+i'k_2)[t-(n-1)h]} [t-(n-1)h]^{j} v_{1,n-1}^{l}\theta_1^{p}\sigma_{v1}^{q} v_{2,n-1}^{l'}\theta_2^{p'}\sigma_{v2}^{q'}\end{split}\]

where \(t0 = ((n_{1m},n_{2m},i_{m}),...,(n_{11},n_{21},i_{1}))\).

Note that: \(\mathbb{E}[I\!E_{1,n}^{m_4} I_{1,n}^{m_5} I\!E_{2,n}^{m_6} I_{2,n}^{m_7} I_{n}^{*m_8}|v_{1,n-1},v_{2,n-1}] = \mathbb{E}[m_4m_5m_6m_7m_8|_{t=nh}]\).

I will show the deduction process in what follows.

Deduction

In order to compute equation (1), I expand it by taking derivative as the following equation shows

\[\begin{split}&d(I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8})\\ &\approx \frac{1}{2}m_8(m_8-1)I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8-2})v(t)dt\\ &\quad + c_1(t) I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8} dt\\ &\quad + c_2(t) I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I_{n-1,t}^{*m_8} dt\\ &\approx \frac{1}{2}m_8(m_8-1)I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8-2})v_1(t)dt\\ &\quad + \frac{1}{2}m_8(m_8-1)I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8-2})v_2(t)dt\\ &\quad + \frac{1}{2}m_4(m_4-1)e^{2k_1t}I\!E_{1,n-1,t}^{m_4-2}I_{1,n-1,t}^{m_5}I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8} v_1(t)dt\\ &\quad + \frac{1}{2}m_5(m_5-1)I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5-2} I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8} v_1(t)dt\\ &\quad + m_4m_5e^{k_1t}I\!E_{1,n-1,t}^{m_4-1}I_{1,n-1,t}^{m_5-1}I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7} I_{n-1,t}^{*m_8} v_1(t)dt\\ &\quad + \frac{1}{2}m_6(m_6-1)e^{2k_2t} I\!E_{2,n-1,t}^{m_6-2}I_{2,n-1,t}^{m_7}I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I_{n-1,t}^{*m_8} v_2(t)dt\\ &\quad + \frac{1}{2}m_7(m_7-1)I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7-2}I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I_{n-1,t}^{*m_8} v_2(t)dt\\ &\quad + m_6m_7e^{k_2t}I\!E_{2,n-1,t}^{m_6-1}I_{2,n-1,t}^{m_7-1}I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5} I_{n-1,t}^{*m_8} v_2(t)dt\end{split}\]

where

\[\begin{split}c_1(t) &\triangleq \bigg[ \frac{1}{2}m_4(m_4-1)I\!E_{1,n-1,t}^{m_4-2}I_{1,n-1,t}^{m_5}e^{2k_1t} + \frac{1}{2}m_5(m_5-1)I\!E_{1,n-1,t}^{m_4} I_{1,n-1,t}^{m_5-2}\\ &\qquad + m_4m_5I\!E_{1,n-1,t}^{m_4-1}I_{1,n-1,t}^{m_5-1}e^{k_1t} \bigg] v_1(t),\\ c_2(t) &\triangleq \bigg[ \frac{1}{2}m_6(m_6-1)I\!E_{2,n-1,t}^{m_6-2}I_{2,n-1,t}^{m_7}e^{2k_2t} + \frac{1}{2}m_7(m_7-1)I\!E_{2,n-1,t}^{m_6} I_{2,n-1,t}^{m_7-2}\\ &\qquad + m_6m_7I\!E_{2,n-1,t}^{m_6-1}I_{2,n-1,t}^{m_7-1}e^{k_2t} \bigg] v_2(t),\end{split}\]

and

\[\begin{split}v_{1}(t) &= e^{-k_1t}e^{k_1(n-1)h}(v_{1,n-1} - \theta_1) + \theta_1 + \sigma_{v1} e^{-k_1t}I\!E_{1,n-1,t},\\ v_{2}(t) &= e^{-k_2t}e^{k_2(n-1)h}(v_{2,n-1} - \theta_2) + \theta_2 + \sigma_{v2} e^{-k_2t}I\!E_{2,n-1,t}.\end{split}\]

Recursive Equation

Thus, we have the following recursive equation

(2)\[\begin{split}&\mathbb{E}[m_4m_5m_6m_7m_8]&\\ &= \frac{m_4(m_4-1)}{2}e^{k_1(n-1)h}(v_{1,n-1} - \theta_1) &\int_{(n-1)h}^t e^{k_1s}\mathbb{E}[(m_4-2)m_5m_6m_7m_8]ds\\ &\quad + \frac{m_4(m_4-1)}{2}\theta_1 &\int_{(n-1)h}^t e^{2k_1s}\mathbb{E}[(m_4-2)m_5m_6m_7m_8]ds\\ &\quad + \frac{m_4(m_4-1)}{2}\sigma_{v1} &\int_{(n-1)h}^t e^{k_1s}\mathbb{E}[(m_4-1)m_5m_6m_7m_8]ds\\ &\quad + \frac{m_5(m_5-1)}{2}e^{k_1(n-1)h}(v_{1,n-1} - \theta_1) &\color{blue}\int_{(n-1)h}^t e^{-k_1s}\mathbb{E}[m_4(m_5-2)m_6m_7m_8]ds\\ &\quad + \frac{m_5(m_5-1)}{2}\theta_1 &\color{blue}\int_{(n-1)h}^t \mathbb{E}[m_4(m_5-2)m_6m_7m_8]ds\\ &\quad + \frac{m_5(m_5-1)}{2}\sigma_{v1} &\color{blue}\int_{(n-1)h}^t e^{-k_1s}\mathbb{E}[(m_4+1)(m_5-2)m_6m_7m_8]ds\\ &\quad + m_4m_5e^{k_1(n-1)h}(v_{1,n-1} - \theta_1) &\int_{(n-1)h}^t \mathbb{E}[(m_4-1)(m_5-1)m_6m_7m_8]ds\\ &\quad + m_4m_5\theta_1 &\int_{(n-1)h}^t e^{k_1s}\mathbb{E}[(m_4-1)(m_5-1)m_6m_7m_8]ds\\ &\quad + m_4m_5\sigma_{v1} &\int_{(n-1)h}^t \mathbb{E}[m_4(m_5-1)m_6m_7m_8]ds\\ &\quad + \frac{m_6(m_6-1)}{2}e^{k_2(n-1)h}(v_{2,n-1} - \theta_2) &\color{blue}\int_{(n-1)h}^t e^{k_2s}\mathbb{E}[m_4m_5(m_6-2)m_7m_8]ds\\ &\quad + \frac{m_6(m_6-1)}{2}\theta_2 &\color{blue}\int_{(n-1)h}^t e^{2k_2s}\mathbb{E}[m_4m_5(m_6-2)m_7m_8]ds\\ &\quad + \frac{m_6(m_6-1)}{2}\sigma_{v2} &\color{blue}\int_{(n-1)h}^t e^{k_2s}\mathbb{E}[m_4m_5(m_6-1)m_7m_8]ds\\ &\quad + \frac{m_7(m_7-1)}{2}e^{k_2(n-1)h}(v_{2,n-1} - \theta_2) &\int_{(n-1)h}^t e^{-k_2s}\mathbb{E}[m_4m_5m_6(m_7-2)m_8]ds\\ &\quad + \frac{m_7(m_7-1)}{2}\theta_2 &\int_{(n-1)h}^t \mathbb{E}[m_4m_5m_6(m_7-2)m_8]ds\\ &\quad + \frac{m_7(m_7-1)}{2}\sigma_{v2} &\int_{(n-1)h}^t e^{-k_2s}\mathbb{E}[m_4m_5(m_6+1)(m_7-2)m_8]ds\\ &\quad + m_6m_7e^{k_2(n-1)h}(v_{2,n-1} - \theta_2) &\color{blue}\int_{(n-1)h}^t \mathbb{E}[m_4m_5(m_6-1)(m_7-1)m_8]ds\\ &\quad + m_6m_7\theta_2 &\color{blue}\int_{(n-1)h}^t e^{k_2s}\mathbb{E}[m_4m_5(m_6-1)(m_7-1)m_8]ds\\ &\quad + m_6m_7\sigma_{v2} &\color{blue}\int_{(n-1)h}^t \mathbb{E}[m_4m_5m_6(m_7-1)m_8]ds\\ &\quad + \frac{m_8(m_8-1)}{2}e^{k_1(n-1)h}(v_{1,n-1} - \theta_1) &\int_{(n-1)h}^t e^{-k_1s}\mathbb{E}[m_4m_5m_6m_7(m_8-2)]ds\\ &\quad + \frac{m_8(m_8-1)}{2}\theta_1 &\int_{(n-1)h}^t \mathbb{E}[m_4m_5m_6m_7(m_8-2)]ds\\ &\quad + \frac{m_8(m_8-1)}{2}\sigma_{v1} &\int_{(n-1)h}^t e^{-k_1s}\mathbb{E}[(m_4+1)m_5m_6m_7(m_8-2)]ds\\ &\quad + \frac{m_8(m_8-1)}{2}e^{k_2(n-1)h}(v_{2,n-1} - \theta_2) &\color{blue}\int_{(n-1)h}^t e^{-k_2s}\mathbb{E}[m_4m_5m_6m_7(m_8-2)]ds\\ &\quad + \frac{m_8(m_8-1)}{2}\theta_2 &\color{blue}\int_{(n-1)h}^t \mathbb{E}[m_4m_5m_6m_7(m_8-2)]ds\\ &\quad + \frac{m_8(m_8-1)}{2}\sigma_{v2} &\color{blue}\int_{(n-1)h}^t e^{-k_2s}\mathbb{E}[m_4m_5(m_6+1)m_7(m_8-2)]ds.\end{split}\]

Initial Moments

For order 0, i.e., \(m_4+\cdots+m_8=0\), \(\mathbb{E}[m_4m_5m_6m_7m_8] = 1\). And for order 1, \(m_4+\cdots+m_8=1\), \(\mathbb{E}[m_4m_5m_6m_7m_8] = 0\).

For order 2, i.e., \(m_4+\cdots+m_8=2\), \(\mathbb{E}[m_4m_5m_6m_7m_8] = 0\), except for

  • \(m_4+m_5=2\):

    \[\begin{split}&\mathbb{E}[I\!E_{1,n-1,t}^2|v_{1,n-1}]\\ &= e^{2k_1t}\frac{1}{2k_1}\theta_1 + e^{k_1t + k_1(n-1)h}\frac{1}{k_1} (v_{1,n-1}-\theta_1) - e^{2k_1(n-1)h}\frac{1}{2k_1}\left(2v_{1,n-1} - \theta_1 \right),\\ % &\mathbb{E}[I\!E_{1,n-1,t}I_{1,n-1,t}|v_{1,n-1}]\\ &=e^{k_1t}\frac{1}{k_1}\theta_1 + e^{k_1(n-1)h}(v_{1,n-1}-\theta_1)[t-(n-1)h] - e^{k_1(n-1)h}\frac{1}{k_1}\theta_1,\\ % &\mathbb{E}[I_{1,n-1,t}^2|v_{1,n-1}]\\ &= -e^{-k_1t + k_1(n-1)h}\frac{1}{k_1}(v_{1,n-1}-\theta_1) + \theta_1[t-(n-1)h] + (v_{1,n-1}-\theta_1)\frac{1}{k_1};\end{split}\]

  • \(m_6+m_7=2\):

    \[\begin{split}&\mathbb{E}[I\!E_{2,n-1,t}^2|v_{2,n-1}]\\ &= e^{2k_2t}\frac{1}{2k_2}\theta_2 + e^{k_2t + k_2(n-1)h}\frac{1}{k_2} (v_{2,n-1}-\theta_2) - e^{2k_2(n-1)h}\frac{1}{2k_2}\left(2v_{2,n-1} - \theta_2 \right),\\ % &\mathbb{E}[I\!E_{2,n-1,t}I_{2,n-1,t}|v_{2,n-1}]\\ &=e^{k_2t}\frac{1}{k_2}\theta_2 + e^{k_2(n-1)h}(v_{2,n-1}-\theta_2)[t-(n-1)h] - e^{k_2(n-1)h}\frac{1}{k_2}\theta_2,\\ % &\mathbb{E}[I_{2,n-1,t}^2|v_{2,n-1}]\\ &= -e^{-k_2t + k_2(n-1)h}\frac{1}{k_2}(v_{2,n-1}-\theta_2) + \theta_2[t-(n-1)h] + (v_{2,n-1}-\theta_2)\frac{1}{k_2};\end{split}\]

  • \(m_8=2\):

    \[\begin{split}&\mathbb{E}[I_{n-1,t}^{*2}|v_{1,n-1},v_{2,n-1}]\\ &= -e^{k_1(n-1)h}(v_{1,n-1}-\theta_1)\frac{1}{k_1}(e^{-k_1t} - e^{-k_1(n-1)h}) +\theta_1 [t-(n-1)h]\\ &\quad -e^{k_2(n-1)h}(v_{2,n-1}-\theta_2)\frac{1}{k_2}(e^{-k_2t} - e^{-k_2(n-1)h}) +\theta_2 [t-(n-1)h].\end{split}\]

Implementation

We have [1],

\[\begin{split}\int e^{(n_1k_1+n_2k_2)t} t^m dt = \begin{cases} \sum_{i=0}^m c_{n_1n_2mi} e^{(n_1k_1+n_2k_2)t} t^{m-i} & \text{if } n_1k_1+n_2k_2\neq 0, m \neq 0,\\ \frac{1}{n_1k_1+n_2k_2}e^{(n_1k_1+n_2k_2)t}t^0 & \text{if } n_1k_1+n_2k_2\neq 0, m = 0,\\ \frac{1}{m+1}e^{0kt}t^{m+1} & \text{if } n_1k_1+n_2k_2 = 0, m \neq 0,\\ e^{0kt}t^1 & \text{if } n_1k_1+n_2k_2 =0 , m=0, \end{cases}\end{split}\]

where \(c_{n_1n_2m0} \triangleq \frac{1}{n_1k_1+n_2k_2}\) and

(3)\[c_{n_1n_2mi} \triangleq \frac{(-1)^{i}}{(n_1k_1+n_2k_2)^{i+1}} \prod_{j=m-i+1}^{m} j,~~ 1\le i \le m.\]

The coefficient \(c_{n_1n_2mi}\) is implemented in function c().

For the definite integral

\[\int_{(n-1)h}^t e^{(n_1k_1+n_2k_2)[s-(n-1)h]}[s-(n-1)h]^mds = F(t-(n-1)h) - F(0)\]

which is defined in int_et(), where \(F(t) = \int e^{(n_1k_1+n_2k_2)t} t^m dt\).

In summary, I defined

  1. int_et() which uses c().

  2. recursive_IEI_IEII() which uses int_mIEI_IEII() and coef_poly().

  3. moment_IEI_IEII().

Functions

c(n1, n2, m, i)

Constant \(c_{n_1n_2mi}\) in (3)

coef_poly(coef, poly, tp)

Multiply poly with different type coefficients

int_et(n1, n2, m)

\(\int_{(n-1)h}^t e^{(n_1k_1+n_2k_2)[s-(n-1)h]}[s-(n-1)h]^mds\)

int_mIEI_IEII(i, m, n4, n5, n6, n7, n8, IEI_IEII)

\(\int_{(n-1)h}^te^{mk_is}\mathbb{E}[n_4n_5n_6n_7n_8]ds\)

moment_IEI_IEII(n4, n5, n6, n7, n8[, return_all])

Moment of \(\mathbb{E}[m_4m_5m_6m_7m_8]\)

recursive_IEI_IEII(n4, n5, n6, n7, n8, IEI_IEII)

Recursive equation (2)

t_mul_t0(t, t0)

multiply quant t0 with quant t