Studies in Chaos and Finance: Chapter IV

In this chapter we will use the conditional representations in (32) and (42) to calculate the first statistical moment, the mean. The diffusion integral in (29) can be re-written as:

$\sum_ {i = 1}^{n} \sum_ {j = 1}^{n} \Sigma^{-1}_{ij} \begin{cases} W_t^{\{i\}, 2} \frac {e^{\alpha_{ii} t}}{\alpha_{ii}} - \frac {2}{\alpha_{ii}} \int_0^t W_s^{\{i\}} e^{\alpha_{ii} s} \, dW_s^{\{i\}} + \frac {1 - e^{\alpha_{ii} t}}{\alpha_{ii}^2}, & \text{if } i = j \\ \frac{1}{\alpha_{ij}} e^{\alpha_{ij} t} W_t^{\{i\}} W_t^{\{j\}} - \frac{1}{\alpha_{ij}} \int_0^t e^{\alpha_{ij} s} \left[ W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} \right] - \delta_{ij} \frac{e^{\alpha_{ij} t} - 1}{\alpha_{ij}^2}, & \text{if } i \ne j \end{cases}$ $(43)$

The expectation under $i = j$ can be derived as follows:

$\begin{aligned} & = \langle W_t^{\{i\}, 2} \frac {e^{\alpha_{ii} t}}{\alpha_{ii}} \rangle - \langle \frac {2}{\alpha_{ii}} \int_0^t W_s^{\{i\}} e^{\alpha_{ii} s} \, dW_s^{\{i\}} \rangle + \langle \frac {1 - e^{\alpha_{ii} t}}{\alpha_{ii}^2} \rangle \\ & = \langle W_t^{\{i\}, 2} \rangle \frac {e^{\alpha_{ii} t}}{\alpha_{ii}} - \frac {2}{\alpha_{ii}} \int_0^t \langle W_s^{\{i\}} e^{\alpha_{ii} s} \rangle \, dW_s^{\{i\}} + \langle \frac {1 - e^{\alpha_{ii}t}}{\alpha_{ii}^2} \rangle \\ & = \langle W_t^{\{i\}, 2} \rangle \frac {e^{\alpha_{ii} t}}{\alpha_{ii}} - \frac {2}{\alpha_{ii}} \int_0^t \underbrace {\langle W_s^{\{i\}} \rangle }_{= \, 0} e^{\alpha_{ii} s} \, dW_s^{\{i\}} + \langle \frac {1 - e^{\alpha_{ii} t}}{\alpha_{ii}^2} \rangle \\ & = \frac{t e^{\alpha_{ii} t}}{\alpha_{ii}} + \frac{1 - e^{\alpha_{ii} t}}{\alpha_{ii}^2} \end{aligned}$ $(44)$

where we can interchange expectation and integration in the second step by Fubini’s theorem. The expectation under $i \neq j$ is a little more nuanced and will require more wrangling. Recall the definition of covariace between two non-overlapping processes:

$Cov(W_t^{i}, W_t^{j}) = \langle W_t^{i} W_t^{j} \rangle - \langle W_t^{i} \rangle \langle W_t^{j} \rangle$ $(45)$

We know that $\langle W_t^{i} \rangle \langle W_t^{j} \rangle = 0$ and that using the Quadratic Covariation we can show:

$Cov(W_t^{i}, W_t^{j}) = \langle W_t^{i} W_t^{j} \rangle = \rho_{ij} \sqrt{Var(W_t^{i}) Var(W_t^{j})} = \rho_{ij} \sqrt{t^2}$ $(46)$

The expectation under $i \neq j$ in (49) can be expressed as:

$\frac{e^{\alpha_{ij} t}}{\alpha_{ij}} \langle W_t^{\{i\}} W_t^{\{j\}} \rangle - \frac{1}{\alpha_{ij}} \int_0^t e^{\alpha_{ij} s} \left[ \langle W_s^{\{i\}} \rangle\, dW_s^{\{j\}} + \langle W_s^{\{j\}} \rangle\, dW_s^{\{i\}} \right] - \langle \delta_{ij} \frac{e^{\alpha_{ij} t} - 1}{\alpha_{ij}^2} \rangle$ $(47)$

Following the results where $\langle W_s^{\{i\}} \rangle = \langle W_s^{\{j\}} \rangle = 0$ and $\langle \delta_{ij} \rangle = \rho_{ij}$ our expression in (47) simplifies to:

$\rho_{ij} \left( \frac{t e^{\alpha_{ij} t}}{\alpha_{ij}} + \frac{1 - e^{\alpha_{ij} t}}{\alpha_{ij}^2} \right)$ $(48)$

Interestingly, the only noted difference between $i = j$ and $i \neq j$ is the correlation term $\rho_{ij}$ . Therefore, the expectation of the diffusion integral in (29) is given by:

$\langle \int_0^t \boldsymbol{W}_s^T (e^{\textbf{A} s} \circ \Sigma^{-1}) \boldsymbol{W}_s \, ds \, \rangle = \sum_ {i = 1}^{n} \sum_ {j = 1}^{n} \Sigma^{-1}_{ij} \left(\frac{t e^{\alpha_{ij} t}}{\alpha_{ij}} + \frac{1 - e^{\alpha_{ij} t}}{\alpha_{ij}^2} \right) \times \begin{cases} 1, & \text{if } i = j \\ \rho_{ij}, & \text{if } i \ne j \end{cases}$ $(49)$