Studies in Chaos and Finance: Chapter II

In the previous chapter we proposed the Ornstein-Uhlenbeck process as the foundation for our stochastic model:

$dx^{i,j}_t = \alpha x^{i,j}_tdt + \tilde{W}(z_i, z_j)$ $(18)$

The results derived in (16) follow a special case of state variables $x$ and $y$ expressed in terms of standard normal variables $z_1$ and $z_2$ . Using the Cholesky decomposition of a covariance matrix $\Sigma$ :

$\Sigma = \textbf{L} \textbf{L}^{T}$ $(19)$

we can show that

$\Sigma = \begin{bmatrix} \sigma_{1} & 0 \\ 0 & \sigma_{2} \end{bmatrix} \begin{bmatrix} 1 & \rho \\ \rho & 1 \end{bmatrix} \begin{bmatrix} \sigma_{1} & 0 \\ 0 & \sigma_{2} \end{bmatrix} \\ = \begin{bmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{bmatrix} \begin{bmatrix} l_{11} & l_{12} \\ 0 & l_{22} \end{bmatrix}.$ $(20)$

Solving for $L$ we get:

$l_{11} = 1, \indent l_{21} = \rho, \indent l_{22} = \sqrt{1 - \rho^2}$ $(21)$

Now we can express these variables in terms of $L$ and the standard normal variables:

$\begin{aligned} x &= \sigma_1 l_{11} z_1 = \sigma_1 z_1 \\ y &= \sigma_1 (l_{21} z_1 + l_{22} z_2) = \sigma_2 (\rho z_1 + \sqrt{1 - \rho^2} z_2) \end{aligned}$ $(22)$

matching the stated forms in (12) and (13). The result in (22) will be applied in its general form when solving the n-state extension of our SDE in (18). It will help to rearrange the expression into its proper differential form with an additional term, $\boldsymbol{\mu}$ , to account for the rate at which our stochastic process changes with time

$\dot{\textbf{x}}(t) + \textbf{A} (\textbf{x}(t) - \boldsymbol{\mu}) = \tilde{W}(\mathbf{z})$ $(23)$

where,

$\frac{\partial {\textbf{x}}(t)}{\partial t} = \begin{bmatrix} \frac{\partial x_1(t)}{\partial t} \\ \frac{\partial x_2(t)}{\partial t} \\ \vdots \\ \frac{\partial x_i(t)}{\partial t} \end{bmatrix}, \indent \textbf{A} = \begin{bmatrix} \alpha_{11} & \alpha_{12} & \cdots & \alpha_{1j} \\ \alpha_{21} & \alpha_{22} & \cdots & \alpha_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{i1} & \alpha_{i2} & \cdots & \alpha_{ij} \\ \end{bmatrix}, \indent \boldsymbol{\mu} = \begin{bmatrix} \mu_{11} \\ \mu_{21} \\ \vdots \\ \mu_{i1} \end{bmatrix}.$ $(24)$

Note that $\textbf{A}$ is an upper triangle matrix:

$\textbf{A}_{ij} = \begin{cases} a_{ij}, & \text{if } i \leq j \\ 0, & \text{if } i > j \end{cases}$ $(25)$

This is what is known as a Linear First Order Inhomogeneous Stochastic Differential Equation with function coefficients written as

$\frac {\partial x(t)}{\partial t} + P(t) x(t) = Q(t)$ $(26)$

and can be solved by its general solution

$e^{-\int_ {\:}^{t} P(t) \, dt} \left[ {\int_ {\:}^{t} e^{\int_ {\:}^{\lambda} P(\epsilon) \, d\epsilon} Q(\lambda)\, d\lambda} + t_0 \right]$ $(27)$

where $e^{\int_ {\:}^{t} P(t) \, dt}$ is the integrating factor. Plugging (18) into our solution we have

$\begin{aligned} \textbf{x}(t) = \boldsymbol{x_0} e^{-\textbf{A} t} + \boldsymbol{\mu} (\textbf{I} - e^{-\textbf{A} t}) - e^{-\textbf{A} t} \int_0^t \boldsymbol{W}_s^T (e^{\textbf{A} s} \circ \Sigma^{-1}) \boldsymbol{W}_s \, ds \\ \end{aligned}$ $(28)$

given $\boldsymbol{x_0} \in \mathbb{R}^{n \times 1}, A \in \mathbb{R}^{n \times n}, \boldsymbol{\mu} \in \mathbb{R}^{n \times 1}$ and identity matrix $I \in \mathbb{R}^{n \times n}$ . The deterministic part of the (28) is rather trivial. In order to address the non-deterministic integrands, however, requires working knowledge of Itô Calculus. In the next chapter we will lay out the logic to make sense of these integrals.