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 mathematical formalism of Brownian motion, let’s consider a simple example to illustrate the concept. Consider a time series data \(W_t\) where \(t\) represents a time index, such that:

1. Initial Condition: \(W_{0} = 0\).
2. 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 \(Z \sim N(0, 1)\).

To simulate this process, given a time step \(\Delta t\) and a total time horizon, we can generate a path of Brownian motion by iteratively applying the above formula starting from the initial condition. Consider the following interactive demonstration that simulates the above-mentioned time series for a given time step \(\Delta t\) and total time horizon of 1 unit of time.

As we can see from the demonstration, as \(\Delta t \rightarrow 0\), the times series data approaches a very interesting object, a function that is continuous but nowhere differentiable, which is the hallmark of Brownian motion. A Brownian motion is precisely defined as the limit of the above time series as \(\Delta t \to 0\), and it is denoted by \(W_t\).

Formally, a Brownian motion is defined as a stochastic process \(\{W_t : t \geq 0\}\) that satisfies the following properties:

1. Initial Condition: \(W_{0} = 0\) almost surely.
2. Independent Increments: For any \(0 \leq t_1 < t_2 < \ldots < t_n\), the increments \(W_{t_2} - W_{t_1}, W_{t_3} - W_{t_2}, \ldots, W_{t_n} - W_{t_{n-1}}\) are independent random variables.
3. Stationary Increments: The distribution of the increment \(W_{t+s} - W_{t}\) depends only on \(s\) and not on \(t\). Specifically, \(W_{t+s} - W_{t} \sim N(0, s)\) for all \(t, s \geq 0\).
4. Continuity: The sample paths of \(W_t\) are continuous functions of time.

The third property implies that the increments of Brownian motion are normally distributed with mean 0 and variance equal to the time increment. This is a crucial property and implies that at any time \(t\), the value of \(W_t\) is normally distributed with mean 0 and variance \(t\), i.e., \(W_t \sim N(0, t)\). To visualize this property, we can simulate multiple paths of Brownian motion and plot the distribution of the values at a fixed time \(t\).

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.

Brownian motion is the building block for many other stochastic processes, with many applications in various fields. As common in standard calculus, time processes are defined via differential equations, and the same is true for stochastic processes. However, for stochastic processes, we need to use stochastic differential equations (SDEs) since Brownian motion is not differentiable and does not abide by the rules of calculus. A general SDE may be expressed as,

\[\mathrm{d}X_t = \mathrm{d}f(t, W_t)\]

Where \(X_t\) is the stochastic process we want to model, and \(f(t, W_t)\) is a function that depends on time and the Brownian motion. To solve such an SDE, we need to compute the derivative of \(f\) with respect to time and Brownian motion, which is done using the Itô calculus. If we expand the function \(f\) using Taylor’s expansion, we can express the differential of \(f\) as:

\[\mathrm{d}f(t, W_t) = \frac{\partial f}{\partial t} \mathrm{d}t + \frac{\partial f}{\partial w} \mathrm{d}W_t + \frac{1}{2} \frac{\partial^2 f}{\partial w^2} (\mathrm{d}W_t)^2\]

Noticed that we went to the second order derivative of \(f\) with respect to \(w\), which is a consequence of the fact that Brownian motion is not differentiable and has infinite variation. Indeed \((\mathrm{d}W_t)^2\) is not zero, but rather equal to \(dt\), which is a key property of Brownian motion. This comes from the fact that the increment \(\mathrm{d} W_t\) is normally distributed with mean 0 and variance \(dt\) (a consequence of the stationary increments property), and thus the expected value of \((\mathrm{d}W_t)^2\), i.e. the variance, is equal to \(dt\). Substituting this into the above expression gives us the Itô formula:

\[\mathrm{d}f(t, W_t) = \left(\frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial w^2} \right)\mathrm{d}t + \frac{\partial f}{\partial w} \mathrm{d}W_t\]

If we now try to integrate the RHS we would be left with an integral with respect to \(\mathrm{d}W_t\). This is a not a standard Riemann integral, but rather an Itô integral. To compute an Itô integral of a deterministic function \(f(s)\) we shall use the definition of a Brownian motion as the limit of the time series data we defined at the beginning. Specifically, we can express the Itô integral as:

\[\int_{0}^{t} f(s) \mathrm{d}W_{s} = \lim_{\Delta t \to 0} \sum_{i=0}^{n-1} f(t_i) (W_{t_{i+1}} - W_{t_i}) = \lim_{\Delta t \to 0} \sum_{i=0}^{n-1} f(t_i) \sqrt{\Delta t} Z_i\]

where \(Z_i \sim N(0,1)\). This integral is a random variable and is expressed as a linear combination of normal random variables. Since any linear combination of normal random variables is also normally distributed, we can conclude that the Itô integral of a deterministic function with respect to Brownian motion is normally distributed. Specifically its mean is given by the expected value of the integral, which can be computed as:

\[\mathbb{E}\left[\int_{0}^{t} f(s) \mathrm{d}W_{s}\right] = \lim_{\Delta t \to 0} \sum_{i=0}^{n-1} f(t_i) \sqrt{\Delta t} \mathbb{E}[Z_i] = 0\]

Its variance can be computed as:

\[\mathbb{V}\left[\int_{0}^{t} f(s) \mathrm{d}W_{s}\right] = \lim_{\Delta t \to 0} \sum_{i=0}^{n-1} f(t_i)^2 \Delta t \mathbb{V}[Z_i] = \int_{0}^{t} f(s)^2 ds\]

This shows that the Itô integral of a deterministic function with respect to Brownian motion is normally distributed with mean 0 and variance equal to the integral of the square of the function over the interval [0, t].

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.,

\[\int_{0}^{t} f(s) dW_{s} \sim N\left(0, \int_{0}^{t} f(s)^2 ds\right)\]

Consider, for example, the integral of \(f(s) = s\) with respect to Brownian motion over the interval [0, t]. In this case, the Itô integral simplifies to:

\[\int_{0}^{t} s dW_{s} \sim N\left(0, \int_{0}^{t} s^2 ds\right) = N\left(0, \frac{t^3}{3}\right)\]

Consider now the integral of a Brownian motion itself \(W_s\) with respect to Brownian motion over the interval [0, t]. To compute this integral, we can use the Itô formula with \(f(s, W_s) = \frac{1}{2} W_s^2\). Applying the Itô formula gives us:

\[\mathrm{d}\left(\frac{1}{2} W_s^2\right) = W_s dW_s + \frac{1}{2} \mathrm{d}s\]

Integrating both sides from 0 to t gives us: \(\frac{1}{2} W_t^2 - \frac{1}{2} W_0^2 = \int_{0}^{t} W_s dW_s + \frac{1}{2} t\)

Since \(W_0 = 0\), we can simplify this to: \(\int_{0}^{t} W_s \mathrm{d}W_s = \frac{1}{2} W_t^2 - \frac{1}{2} t\)

The above formula can be generalized to compute the integral of any function of Brownian motion with respect to Brownian motion itself. For example, if we want to compute the integral of \(f(W_s)\) with respect to Brownian motion, we can use the Itô formula with \(f(s, W_s) = F(W_s)\), where \(F\) is an antiderivative of \(f\). Applying the Itô formula gives us:

\[\mathrm{d}F(W_s) = f(W_s) \mathrm{d}W_s + \frac{1}{2} f'(W_s) \mathrm{d}s\]

Integrating both sides from 0 to t gives us: \(F(W_t) - F(W_0) = \int_{0}^{t} f(W_s) \mathrm{d}W_s + \frac{1}{2} \int_{0}^{t} f'(W_s) \mathrm{d}s\)

Hence we can express the integral of \(f(W_s)\) with respect to Brownian motion as: \(\int_{0}^{t} f(W_s) \mathrm{d}W_s = F(W_t) - F(W_0) - \frac{1}{2} \int_{0}^{t} f'(W_s) \mathrm{d}s\)

Now we have all the tools we need to solve problems in SDEs. For example, consider the process defined as the integral of a Brownian motion

\[X_t = \int_{0}^{t} W_s \mathrm{d}s\]

We can compute the mean and variance of \(X_t\) as follows:

\[\mathbb{E}[X_t] = \mathbb{E}\left[\int_{0}^{t} W_s \mathrm{d}s\right] = \int_{0}^{t} \mathbb{E}[W_s] \mathrm{d}s = 0\] \[\mathbb{V}[X_t] = \mathbb{V}\left[\int_{0}^{t} W_s \mathrm{d}s\right] = \int_{0}^{t} \int_{0}^{t} \mathbb{E}[W_s W_u] \mathrm{d}s \mathrm{d}u = \int_{0}^{t} \int_{0}^{t} \min(s, u) \mathrm{d}s \mathrm{d}u = \frac{t^3}{3}\]

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.