An introduction of white noise theory
Recently, I have picked up some background of white noise theory after ready this paper
Starting from a classical results for a Browian montion, \(B_t\), the following holds
\[dB_t = (dt)^2\]This result leads to a new branch of calculus, stochastic calculus or Ito calculus. One of prominent lemmas is Ito’s lemma which is considered as a chain rule to compute the derivative of a function, \(f(t, B_t)\). A simple way to intepret Ito’s lemma is to use Talor approximation up to second order and then obtain
\[df(t, B_t)=\left(\frac{\partial f}{\partial t} +\frac{1}{2}\frac{\partial^2 f}{\partial x^2}\right)dt + \frac{\partial f}{\partial x}dB_t\]Now, what if the stochastic process here is not a Brownian motion? How do we do stochastic calculus for this case?
Let us start with a simple example of characteristic function of a univarite Normal random variable, \(X \sim \mathcal{N}(0, 1)\). The characteristic function is under the form
\[\varphi(t) = \mathbb{E}[\exp(itX)] = \exp\left(-\frac{1}{2}t^2\right)\]Extending to finite dimesion where we have to deal with isotropic multivariate Gaussian distribution, the characteristic function is
\[\varphi(t) = \mathbb{E}[\exp(i\langle t, X \rangle)] = \exp\left(-\frac{1}{2}\lVert t \rVert^2\right)\]where \(t = [t_1, \dots, t_n]^\top\) and \(\lVert t\rVert = \sqrt{t_1^2 +\dots, t_n^2}\). Here the inner product \(\langle x, y \rangle = x_1y_1 + \dots + x_ny_n\).
Obviously, this case is equivalent to Gaussian white noise with finite sample. However, stochastic process can be defined over a contiuous index set. It is natural to generalize even more.
Definition (White noise probability measure)
then the measure $\mu$ is called the white noise probability measure.
The Lévy process also is defined based on its characteristic function in Lévy-Khintchine formula.
We have the following properties:
We can see that Brownian motion can fit under the definition of white noise theory.
\[\chi_{[0, t]}(s)=\begin{cases} 1 & \text{ if } 0 \leq s \leq t\\ - 1 &\text{ if } t \leq s \leq 0 \\ 0 &\text{ otherwise} \end{cases}\]We can define \(\tilde{B}_t=\langle \omega, \chi_{[0, t]}(\cdot)\rangle\). Furthermore, we can say the \(\tilde{B}_t\) is a Gaussian process with zero mean and covariance
\[\mathbb{E}[\tilde{B}_t, \tilde{B}_s] = \min(t,s).\]This is exactly the definition of Brownian motion. We can verify this by examining the characteristic function
\[\begin{align*} \mathbb{E}\left[\exp(i \sum_{j=1}^n c_j\tilde{B}_{t_j} )\right] = & \mathbb{E}\left[\exp\left(i \langle \omega, \sum_{j=0}^n c_j\chi_{[0, t_j]} \rangle \right) \right] = \exp \left(-\frac{1}{2}\lVert \sum_j c_j\chi_{[0, t_j]}\rVert^2 \right) \\ =& \exp\left(-\frac{1}{2}\sum_{i,j}c_i c_j\int \chi_{[0, t_i]}(s)\chi_{[0, t_j]}(s)ds\right) \\ =& \exp\left(-\frac{1}{2}\sum_{i,j}c_i c_j \min(t_i, t_j)\right) \end{align*}\]This is the charateristic of a multivariate Gaussian distribution.
The covariance function of fBm is
\[k(s, t) = |t|^{2H} + |s|^{2H} - |t-s|^{2H}\]To derive the similar form like Brownian motion, we need an operator $M$
\[Mf(y) = |y|^{1/2 - H}\hat{f}(y), \quad y \in \mathbb{R}\] \[Mf(x) = C_H\int_\mathbb{R}\frac{f(x-t) - f(x)}{|t|^{3/2 - H}}dt\]We define \(M\chi_{[0, t]}(x) = M[0, t](x)\)
The fractional Brownian motion is
\[\tilde{B}^{(H)}(t) = \langle \omega, M[0,t](\cdot)\rangle\]We also can reobtain the covariance kernel for
\[\begin{aligned} \mathbb{E}[\tilde{B}^{(H)}(s)\tilde{B}^{(H)}(t)] & = \int_{\mathbb{R}} M[0,s](x) M[0, t](x) dx \\ & = \end{aligned}\]Although the mathematical formulation presented aboved is elegant, having applications in finance