Question

Let \(\\{X(t), t \geqslant 0\\}\) be a Brownian motion process with drift coefficient \(\mu\) and variance parameter \(\sigma^{2}\). What is the joint density function of \(X(s)\) and \(X(t), s

Short answer

The joint density function of \(X(s)\) and \(X(t)\) for a Brownian motion process with drift coefficient \(\mu\) and variance parameter \(\sigma^2\) is: \[ f_{X(s),X(t)}(x_s,x_t) = \frac{1}{2\pi\sigma^2\sqrt{st(t-s)}}\exp\left\{-\frac{(x_t-x_s-\mu(t-s))^2}{2\sigma^2(t-s)}-\frac{(x_s-\mu s)^2}{2\sigma^2 s}\right\} \]

Step-by-step solution

Define Brownian Motion with drift

A Brownian motion process with drift coefficient \(\mu\) and variance parameter \(\sigma^2\) can be represented as a stochastic differential equation: \[dX(t)=\mu dt+\sigma dW(t)\] where \(W(t)\) is a standard Brownian motion.

Write the process in integral form

Let's write the stochastic differential equation in integral form: \[X(t)=X(0)+\mu\int_{0}^{t}ds+\sigma\int_{0}^{t}dW(s)\]

Conditional Increment

The increment between \(X(t)\) and \(X(s)\) for \(s<t\) is given by: \[ X(t) - X(s) = \mu\int_{s}^{t}du+\sigma\int_{s}^{t}dW(u) \]

Normal Distribution of Increment

Since \(W(t)\) is a standard Brownian motion, the increment between \(X(t)\) and \(X(s)\) has a normal distribution: \[ X(t) - X(s) \sim \mathcal{N}\left(\mu(t-s),\sigma^2(t-s)\right) \] So, the conditional density function of \(X(t)\) given \(X(s)\) is: \[ f_{X(t)|X(s)}(x_t|x_s) = \frac{1}{\sqrt{2\pi\sigma^2(t-s)}}\exp\left\{-\frac{(x_t-x_s-\mu(t-s))^2}{2\sigma^2(t-s)}\right\} \]

Marginal Density

To find the joint density function, we need to find the marginal density function of \(X(s)\). The distribution of \(X(s)\) is also a normal distribution: \[ X(s) \sim \mathcal{N}\left(\mu s, \sigma^2 s\right) \] So, the marginal density function of \(X(s)\) is: \[f_{X(s)}(x_s) = \frac{1}{\sqrt{2\pi\sigma^2 s}}\exp\left\{-\frac{(x_s-\mu s)^2}{2\sigma^2 s}\right\} \]

Joint Density Function

Now we can find the joint density function of \(X(s)\) and \(X(t)\) by multiplying the conditional density function of \(X(t)\) given \(X(s)\) with the marginal density function of \(X(s)\): \[ f_{X(s),X(t)}(x_s,x_t) = f_{X(t)|X(s)}(x_t|x_s) f_{X(s)}(x_s) \] \[ f_{X(s),X(t)}(x_s,x_t) = \frac{1}{2\pi\sigma^2\sqrt{st(t-s)}}\exp\left\{-\frac{(x_t-x_s-\mu(t-s))^2}{2\sigma^2(t-s)}-\frac{(x_s-\mu s)^2}{2\sigma^2 s}\right\} \] The joint density function of \(X(s)\) and \(X(t)\) for a Brownian motion process with drift coefficient \(\mu\) and variance parameter \(\sigma^2\) is: \[ f_{X(s),X(t)}(x_s,x_t) = \frac{1}{2\pi\sigma^2\sqrt{st(t-s)}}\exp\left\{-\frac{(x_t-x_s-\mu(t-s))^2}{2\sigma^2(t-s)}-\frac{(x_s-\mu s)^2}{2\sigma^2 s}\right\} \]

Key concepts

Stochastic Differential Equation

When trying to understand the Brownian motion process, particularly one with drift, it's essential to begin with the concept of a stochastic differential equation (SDE). An SDE is a type of equation used to describe the evolution of random or stochastic processes over time. It is different from traditional deterministic differential equations because it incorporates random variables that introduce chance into the system.

As seen in the textbook solution, the SDE for Brownian motion with drift includes a term \(\mu dt\), representing the predictable drift or trend of the process, and a term \(\sigma dW(t)\), where \(dW(t)\) signifies the unpredictable influence of Brownian motion, also known as the 'Wiener process'. It's crucial to note that the term \(\sigma dW(t)\) captures the essence of randomness within the process and is the component that gives the Brownian motion its characteristic unpredictable movement.

To fully understand this, think of \(\mu\) as the force pushing the particle steadily in one direction, while \(\sigma\) governs the speed at which the particle zigzags erratically. In our case, these terms allow for the modeling and prediction of an asset or particle's behavior within the often chaotic financial markets or physical systems.

Normal Distribution Increment

A hallmark of Brownian motion, especially in financial contexts, is the assumption that the increments of the movement are normally distributed. This concept is crucial because it implies that the future state of the process, within a certain interval \(t-s\), can be predicted using a probability distribution that most people are familiar with — the normal distribution.

In the given solution for the increment \(X(t) - X(s)\), the distribution is expressed as normal with mean \(\mu(t-s)\) and variance \(\sigma^2(t-s)\). This tells us that over a short time, the average change we expect in our process is proportional to the time passed, scaled by the drift \(\mu\), while the variability or uncertainty of the change scales with both the time passed and the variance parameter \(\sigma^2\).

Therefore, when dealing with Brownian motion, the normal distribution increment aspect is vital as it allows us to consider the probability of the process reaching certain values at a given time, which has far-reaching implications in fields like risk management and option pricing in finance, or diffusion and particle modeling in physics.

Joint Density Function

The joint density function is an important concept that extends into the realm of multivariate analysis. This function helps us understand the likelihood of two connected random variables occurring simultaneously. For a Brownian motion process with drift \(\mu\) and variance \(\sigma^2\), the joint density function allows us to gauge the probability of the process being at a specific point \(X(s)\) at time \(s\) and another point \(X(t)\) at a later time \(t\), with \(s
To find this joint density function, as we have in the textbook solution, you multiply the marginal density function of \(X(s)\), which captures the likelihood of the process being at \(X(s)\) regardless of \(X(t)\), by the conditional density function of \(X(t)\) given \(X(s)\), which characterizes how likely we are to find the process at \(X(t)\) given that it was at \(X(s)\).

Understanding this multi-step calculation is crucial for anticipating the behaviors of variables over time in stochastic processes. It's not just about knowing where you are or where you're going; it's about understanding the relationship between those two states and how the probability of being at one point affects the probability of being at another later on. This nuanced comprehension is highly significant in complex systems where multiple factors and outcomes are interrelated.