Mathematical Modeling in Finance with Stochastic Processes

160 CHAPTER 5. BROWNIAN MOTION

In the limit ask→∞andn→∞,vk,nwill be the sampling of the function
v(t,x) at time intervalsr, so thatkr=t, and space intervals so thatn·δ=
x. That is, the function v(t,x) should be an approximate solution of the
difference equation:

v(t+r−^1 ,x) =pv(t,x−δ) +qv(t,x+δ)

We assumev(t,x) is a smooth function so that we can expandv(t,x) in a
Taylor series at any point. Using the first order approximation in the time
variable on the left, and the second-order approximation on the right in the
space variable, we get (after canceling the leading termsv(t,x) )

∂v(t,x)

∂t

= (q−p)·δr

∂v(t,x)

∂x

+ (1/2)δ^2 r

∂^2 v(t,x)

∂x^2

In our passage to the limit, the omitted terms of higher order tend to zero,
so may be neglected. The remaining coefficients are already accounted for in
our limits and so the equation becomes:

∂v(t,x)

∂t

=−c

∂v(t,x)

∂x

+ (1/2)D

∂^2 v(t,x)

∂x^2

This is a specialdiffusion equation, more specifically, a diffusion equation
with convective or drift terms, also known as the Fokker-Planck equation for
diffusion. It is a standard problem to solve the differential equation forv(t,x)
and therefore, we can find the probability of being at a certain position at a
certain time. One can verify that

v(t,x) = (1/(

√

2 πDt)) exp(−[x−ct]^2 /(2Dt))

is a solution of the diffusion equation, so we reach the same probability
distribution forv(t,x).
The diffusion equation can be immediately generalized by permitting the
coefficients c and Dto depend on x, and t. Furthermore, the equation
possesses obvious analogues in higher dimensions and all these generalization
can be derived from general probabilistic postulates. We will ultimately
describe stochastic processes related to these equations asdiffusions.

Sources

This section is adapted from W. Feller, inIntroduction to Probability Theory
and Applications, Volume I, Chapter XIV, page 354.