Summarize some applications of stochastic calculus in engineering and physics.
In this blog, I will summarize several notable applications of stochastic calculus in engineering and physics. Although there are a lot more modern approaches, what will be discussed here is fundamental, still worth understanding in details. The content is based on Chapter 14 of this book.
As stochastic calculus mostly involves deriving integrations, in engineering applications, we may encounter some situations that we need to utilize such integrations.
What is filtering? Simply speaking, filtering is the process that we filter out noise out of a signal.
Consider a observed process \(Y(t)\) and its corresponding filtration \(\mathcal{F}_t^Y=\sigma(Y(s), s \leq t)\). Let’s say that there is a (linear) transformation from \(Y(t)\) to \(X\).
The goal here is to find an estimate of \(\pi_{X}(t)\) (probability distribution) such that
\[\mathbb{E}[(X(t) - Z(t))^2 \mid \mathcal{F}_t]\]where \(Z(t)\) varies over all \(\mathcal{F}_t\)-measurable processes. In general, we can define an adapted process \(h(t)\)
\[\pi_t(h) = \mathbb{E}[h(t)| \mathcal{F}_t].\]Here, we want to solve a general solution for a non-linear version of \(h(t)\).
\(\begin{align*} dh(t) = & H(t) dt + dM(t)\\ dY(t) = & A(t) dt + B(Y(t)) dW(t) \end{align*}\) where
The following theorem gives us a representation of an increment of the estimate \(\pi_t(h)\)
Theorem
\(d\pi_t(h)= \pi_t(H)dt + (\pi_t(D) + \frac{\pi_t(hA) - \pi_t(h)\pi_t(A)}{B(Y(t))} )d\overline{W}(t)\) where \(d\overline{W}(t) = \frac{dY(t) - \pi_t(A)}{B(Y(t))}\) is an \(\mathcal{F}^Y_t\)-Brownian motion and \(D(t) = \frac{d\langle M, W \rangle (t)}{dt}\)
Theorem Let \(A(t)\) be an \(\mathcal{F}_t\)-adapted process such that \(\int_0^T \mathbb{E}[\lvert A(t) \rvert] dt < \infty\) and \(V(t) = \int_0^t A(s) ds\). Then,
\[\pi_t(V) - \int_0^t \pi_s(A)ds = \mathbb{E}\left[\int_0^t A(s) ds \mid \mathcal{F}_t^Y\right] - \int_0^t\mathbb{E}\left[ A(s) \mid \mathcal{F}_t^Y\right] ds\]is a \(\mathcal{F}_t^Y\)-martingale.
Remark As $V(t)$ is a cumulative value of $A(s)$. This theorem is saying that if we exchange $\pi$ and $\int_0^t$, the difference become martingale which mean staying the same on average.
Theorem Let $M(t)$ be a $\mathcal{F}_t$ martingale. Then $\pi_t(M)$ is a $\mathcal{F}^Y_t$-martingale.
Corollary
\[\pi_t(h) = \pi_t(0) + \int_0^t \pi_s(H)ds + M_1(t) + M_2(t) + M_3(t)\]where $M_i(t)$ are martingales null at zero:
| $M_1 = \mathbb{E}[h(X(0)) | \mathcal{F}_t^Y] - h(0), $ |
| $M_2 = \mathbb{E}\left[\int_0^t H(s) ds | \mathcal{F}_t^Y \right] - \int_0^t \pi_s(H)ds, $ |
| $M_3 = \mathbb{E}[M(t) | \mathcal{F}_t^Y], $ |
Many real-world physical systems are governed by a differential equation
\[\ddot{x} + h(x, \dot{x}) = 0\]Note that the dot over a variable like \(\dot{x}\) indicates the derivative with respect to the time variable $t$.
However, the equation is not exactly equal zero but a white noise. Therefore, the stochastic version is under the form
\[\ddot{X} + h(x, \dot{X}) = \sum_i f_i(X, \dot{X})\dot{W}_i(t)\]where $W_i(t)$ are Brownian motions.
Now, all the results of Ito calculus become handy in this case. In fact, we need to reformulate the problem with $x_1 = x, x_2 = \dot{x}$ and use Ito’s lemma
\[\begin{align*} dX_1 =& X_2 dt\\ dX_2 =& \left(-h(X_1,X_2) + \pi \sum_{j,k}K_{ij}f_j(X_1,X_2)\frac{\partial f_k}{\partial x_2}(X_1, X_2)\right)dt + \sum_{i=1}^2 f_i(X_1, X_2)dW_i(t) \end{align*}\]This can be converted into a high-dimensional equation where \(\mathbf{X}(t) = [X_1(t), X_2(t)]^\top\)
The solution for these equations is \(\mathbf{X}(t) = \exp(Ft)\left(\mathbf{X}(0) + \int_0^t \exp(-Fs) [0, \sigma]^\top dB(s)\right)\)