Brownian motion and Stochastic calculus
Brownian motion is a fundamental stochastic process that describes the random movement of particles suspended in a fluid. It is named after the botanist Robert Brown, who first observed this phenomenon in 1827. Brownian motion has been extensively studied in various fields, including physics, finance, and mathematics, due to its wide range of applications.
Before diving into the math, let’s first simulate a Brownian motion path using Wolfram Cloud. Consider the variable \(t\) representing time, and let \(W_{t}\) denote the Brownian motion at time \(t\). To simulate a Brownian motion path, we can use the following properties:
- Initial Condition: \(W_{0} = 0\).
- Increments: The value of \(W_{t}\) at any time \(t + \Delta t\) is given by \(W_{t + \Delta t} = W_{t} + \sqrt{\Delta t} \cdot Z\), where \(Z\) is a standard normal random variable \(Z \sim N(0, 1)\).
Play with the following interactive demonstration to see that by reducing the time step \(\Delta t\), the path of the Brownian motion starts to approach a very particular object. Indeed as \(\Delta t \to 0\), the path of the Brownian motion converges to a continuous function that is nowhere differentiable (i.e. very rough).
We now turn to the mathematical properties of Brownian motion. One of the key properties is that Brownian motion has independent and normally distributed increments. This means that for any two time intervals, the increments of Brownian motion are independent of each other and follow a normal distribution. Specifically, for any \(0 \leq s < t\), the increment \(W_{t} - W_{s}\) is normally distributed with mean 0 and variance \(t - s\), i.e., \(W_{t} - W_{s} \sim N(0, t - s)\).
Another important property of Brownian motion is that it has continuous paths. This means that the function \(t \mapsto W_{t}\) is continuous with probability 1. However, despite being continuous, Brownian motion paths are almost surely nowhere differentiable, which makes them very rough and irregular. This property can be visualized in the interactive demonstration above, where as we decrease the time step \(\Delta t\), the path becomes more and more jagged, illustrating the rough nature of Brownian motion.
To view all these properties in action, consider the following interactive demonstration that simulates multiple paths of Brownian motion and allows you to visualize the distribution of their values at different time points.
As you can see from the demonstration, as time progresses, the distribution of the Brownian motion values becomes wider, reflecting the increasing variance over time. This illustrates the property that the variance of Brownian motion at time \(t\) is equal to \(t\), which is a direct consequence of the normal distribution of increments.
To formalize this property, we must first enter the real of Stochastic calculus, which is a branch of mathematics that deals with integration and differentiation of stochastic processes. In particular, we can define the Itô integral, which allows us to integrate functions with respect to Brownian motion. This is crucial for understanding how Brownian motion can be used to model various phenomena in finance and physics, such as stock prices and particle movement. Since a Brownian motion is not differentiable, we cannot use the standard rules of calculus. We denote as d\(W_{t}\) the infinitesimal increment of Brownian motion at time \(t\). The increment d\(W_{t}\) is normally distributed with mean 0 and variance dt, i.e., d\(W_{t} \sim N(0, dt)\). This means that the expected value of d\(W_{t}\) is zero, and the variance of d\(W_{t}\) is equal to the time increment dt.
We can use this property to explain the Itô integral, which is defined as the limit of Riemann sums of the form:
\[\int_{0}^{t} f(s) dW_{s} = \lim_{n \to \infty} \sum_{i=0}^{n-1} f(t_i) (W_{t_{i+1}} - W_{t_i})\]where \(f(s)\) is a deterministic function and \(t_i\) are partition points of the interval [0, t]. In this case, the Itô integral is quite simple and is equal to a normal random variable with mean 0 and variance equal to the integral of the square of the function \(f(s)\) over the interval [0, t], i.e.,
\[\int_{0}^{t} f(s) dW_{s} \sim N\left(0, \int_{0}^{t} f(s)^2 ds\right)\]As an example consider the Wiener process Stochastic differential equation (SDE) given by:
\[dX_{t} = \mu dt + \sigma dW_{t}\]where \(\mu\) is the drift term and \(\sigma\) is the volatility term. The solution to this SDE can be expressed as:
\[X_{t} = X_{0} + \mu t + \sigma \int_{0}^{t} dW_{s}\]Since the integral of d\(W_{s}\) is a normal random variable with mean 0 and variance t, we can conclude that \(X_{t}\) is normally distributed with mean \(X_{0} + \mu t\) and variance \(\sigma^2 t\), i.e.,
\[X_{t} \sim N(X_{0} + \mu t, \sigma^2 t)\]If we want to integrate with respect to a function of Brownian motion, we can use the Itô formula, which is a generalization of the chain rule for stochastic processes. The Itô formula states that if \(f(t, W_{t})\) is a twice continuously differentiable function, then the differential of \(f\) can be expressed as:
\[df(t, W_{t}) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial w} dW_{t} + \frac{1}{2} \frac{\partial^2 f}{\partial w^2} (dW_{t})^2\]Since (d\(W_{t})^2\) is equal to \(dt\), we can rewrite the Itô formula as:
\[df(t, W_{t}) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial w} dW_{t} + \frac{1}{2} \frac{\partial^2 f}{\partial w^2} dt\]As a concluding example, consider the Ornstein-Uhlenbeck (OU) process, which is a mean-reverting stochastic process defined by the following SDE:
\[dX_{t} = \theta (\mu - X_{t}) dt + \sigma dW_{t}\]where \(\theta\) is the rate of mean reversion, \(\mu\) is the long-term mean, \(\sigma\) is the volatility. To solve this SDE, we first divide both sides by \(e^{\theta t}\) to get:
\[e^{\theta t} dX_{t} = \theta \mu e^{\theta t} dt - \theta X_{t} e^{\theta t} dt + \sigma e^{\theta t} dW_{t}\]This can be rewritten as: \(d(e^{\theta t} X_{t}) = \theta \mu e^{\theta t} dt + \sigma e^{\theta t} dW_{t}\)
Integrating both sides from 0 to t gives:
\[e^{\theta t} X_{t} - X_{0} = \theta \mu \int_{0}^{t} e^{\theta s} ds + \sigma \int_{0}^{t} e^{\theta s} dW_{s}\]Solving the integral on the right-hand side yields: \(e^{\theta t} X_{t} = X_{0} + \mu (e^{\theta t} - 1) + \sigma \int_{0}^{t} e^{\theta s} dW_{s}\)
Finally, we can express the solution for \(X_{t}\) as:
\[X_{t} = \mu + (X_{0} - \mu) e^{-\theta t} + \sigma e^{-\theta t} \int_{0}^{t} e^{\theta s} dW_{s}\]The last integral is a normal random variable with mean 0 and variance given by:
\[\int_{0}^{t} e^{2\theta s} ds = \frac{e^{2\theta t} - 1}{2\theta}\]You can now play with the following interactive demonstration to see how the OU process behaves for different values of the parameters \(\theta\), \(\mu\), and \(\sigma\).
This concludes our brief introduction to Brownian motion and Stochastic calculus. We have seen how Brownian motion can be simulated, its key properties, and how it can be used to model various phenomena through stochastic differential equations. The interactive demonstrations provided allow you to explore these concepts further and gain a deeper understanding of the behavior of Brownian motion and related processes.