Brownian motion

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Models of Financial Mathematics, MTMM.00.203 
 
Stochastic Differential Equation and Processes 
 
Stochastic differential equations (SDE) occur where a system described by differential 
equations is influenced by random noise. As one or more of the terms is a stochastic process, 
a solution of SDE is also a stochastic process. Stochastic differential equations are used in 
finance (interest rate, stock prices, …), biology (population, epidemics, …) and ohter 
applications. The evolution of the stochastic system, so called Itô Process is often modelled 
by an equation 
dX(t) = a(t,X(t))∙dt + b(t,X(t))∙dW(t). 
Here X(t) is a stochastic process at time t; W(t) denotes Brownian motion. 
A stochastic process X = { X(t) } is a time series of random variables. In other words, X(t) or 
Xt is a random variable for each time t and is usually called the state of the process at time t. 
A realization of X is called a sample path. Note that a sample path defines an ordinary 
function of t. 
If the times t form a countable set, X is called a discrete-time stochastic process or a time 
series. In this case, subscripts rather than parentheses are usually used, as in X= { Xn }. If the 
times form a continuum, X is called a continuous-time stochastic process. 
Random walks of various kinds are the foundations of discrete-time probabilistic models of 
asset prices. In fact, the binomial model of stock prices is a random walk. 
An important result is formula of Itô. Let f=f(S,t) be two times differentiable function which 
satisfies the SDE 
dtSdXSdS   . 
Then the next formula holds: 
dt
t
f
S
f
S
S
f
SdX
S
f
Sdf 











 )
2
1
(
2
2
22 . 
This formula of Itô is used to derive the Black – Scholes PDE for finding the option price V: 
0
2
1
2
2
22 








Vr
S
V
Sr
S
V
S
t
V  
peep.miidla@ut.ee 
 
 
 
Brownian motion is a stochastic process { X(t), t ≥ 0 } with the following properties: 
 
1. X(0) = 0, unless stated otherwise. 
2. For any 0 ≤ t0 < t1 < · · · < tn, the random variables X(tk)− X(tk−1) for 1 ≤ k≤ n are 
independent; so X(t)− X(s) is independent of X(r ) for r ≤ s < t. 
3. For 0 ≤ s < t, X(t)− X(s) is normally distributed with mean μ(t −s) and variance 
σ2(t −s), where μ and σ ≠ 0 are real numbers. 
 
Such a process is called a (μ, σ) Brownian motion with drift μ and variance σ2. Figure above 
plots a realization of a Brownian motion process. Although Brownian motion is a continuous 
function of t with probability one, it is almost nowhere differentiable. The (0, 1) Brownian 
motion is also called normalized Brownian motion or the Wiener process.