Chapter 6
Brownian Motion

Consider a stochastic process \(\left \{X(t), \, t\geq 0\right \}\) whose state space is an interval of real line such a process has stationary increments in distribution of \(X\left (t + s\right ) - X(t)\) does not depend on \(t\). This process has independent increments if for \(0 < t_1 < t_2 < \cdots < t_n\) the random variables \[X(t_n) - X(t_{n - 1}),\, X(t_{n - 1}) - X(t_{n - 2}), \, \cdots \, , X(t_2) - X(t_1), \, X(t_1) - X(0)\] are independent.

A stochastic process \(\left \{X(t),\, t\geq 0\right \}\) with a real state space is called Brownian motion if the following are true:

(i)

\(X(0) = 0\)

(ii)

the process has stationary and independent increments

(iii)

for every \(t>0, \, X(t)\) has a normal distribution with mean zero and variance \(t\sigma ^2\).

If \(\sigma ^2 =1\), the Brownian motion is called standard Brownian motion.

Consider a standard Brownian motion \(\left \{X(t), \, \geq 0\right \}\). Let \(0 < t_1 < t_2 < \, \cdots \, < t_n\). The set of equations \[X(t_n) = x_n, \, \, X(t_{n - 1}) = x_{n - 1}, \, \cdots \, \, , \, X(t_1) = x_1\] is equivalent to \begin {align*} X(t_n) - X(t_{n - 1}) & = x_n - x_{n - 1}\\ X(t_{n - 1}) - X(t_{n - 2}) & = x_{n - 1} - x_{n - 2}\\ \vdots & \\ X(t_2) - X(t_1) &= x_2, - x_1\\ X(t_1) & = x_1 \end {align*}

where random variables \[X(t_n) - X(t_{n - 1}), \, \cdots \, , \, X(t_1)\] are independent and satisfy the condition of stationarity.

Hence the pdf of \(X(t)\) is \[f_t(x) = \frac {1}{\sqrt {2\pi t}}\, e^{-\frac {x^2}{2t}}\] and the distribution of \(X(t_i) - X(t_{i -1})\) is normal with mean 0 and variance \((t_i - t_{i -1})\) (the variance is the length of the interval.)

The joint density of \(X(t_1), \, X(t_2), \, \cdots \, ,\, X(t_n)\) is equal to the joint density of \[X(t_n) - X(t_{n - 1}), \, \cdots \, , \, X(t_1).\] Hence \(f(x_1, \, \cdots \, ,\, x_n) = f_{t_1}(x_1)\, f_{t_2-t_1}(x_2 - x_1)\, \cdots \, f_{t_n - t_{n -1}}(x_n - x_{n - 1}).\)

Let \(s < t\).
Finding conditional distribution of \(X(s)\) given \(X(t) = B\) \begin {align*} f_{s/t}(x,B) & = \frac {f_{s,t}(x, B)}{f_t(B)} = \frac {f_s(x)\, f_{t-s}(B - x)}{f_t(B)}\\\\ & = \frac {\frac {1}{\sqrt {2\pi s}}e^{-\frac {x^2}{2s}}\, \cdot \, \frac {1}{\sqrt {2\pi (t - s)}}e^{-\frac {(B - x)^2}{2(t - s)}}}{f_t(B)}\\\\ & = K\, \exp \left [\frac {x^2}{2s} - \frac {(B - x)^2}{2(t - s)}\right ]\\\\ & = K_1\, \exp \left [\frac {-\left (x - \dfrac {Bs}{t}\right )}{\dfrac {2s(t-s)}{t}}\right ]. \end {align*}

For \(s <t \) \[E\left (X(s)| X(t) = B\right ) = \frac {Bs}{t}\]

\[var\left (X(s)\, |\, X(t) = B\right ) = \frac {s(t - s)}{t}\]

A.

Rationale and Objects
This course links the probability theory to its applications in the physical and social sciences and in operations research.

At the end of the course, students will have understood and be able to use Markov chains, Poisson processes, continuous-time Markov chains, renewal theory, queuing theory, reliability theory, Brownian motion and stationary processes.

B.

Topic Outline
Markov Chain: Markov chains having two states, transition functions and initial distribution, transient and recurrent states, decomposition of state space (absorption probabilities and Martingales), birth and death chains. Stationary distributions of a Markov chain: elementary properties of stationary distributions, average number of visits to recurrent state, null recurrent and positive recurrent states, queuing chain, convergence to the stationary distribution. Markov pure jump processes: construction of jump processes, birth and death processes, properties of a Markov pure jump process. Second order processes: mean and covariance functions, Gaussian processes, Wiener processes. Continuity, integration and differentiation of second order processes: continuity assumptions, integration, differentiation, white noise.

C.

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