3.7. Derive from first principles the Fokker–Planck equation

∂p

∂t

=

∂2

∂x2

(12 x2p)

for the PDF p(t,x) of solutions X(t) of the SDE dX = XdW

(this SDE is the case of the financial SDE dX = αXdt+βXdW for no stock drift,

α = 0 , and unit stock volatility, β = 1).

• First, assume small time steps of size Δt = h and approximate theWiener process by the up/down binomial steps ΔW = ±

√

h . Show that for the stochastic system to reach (t+h,x) it must have come from (t,ξ) for ξ = x/(1±

√

h) .

• Second, use Taylor series about p(t,x) to deduce p(t+h,x) = p+hp+2xh

∂p

∂x

+ 12 x2h

∂2p

∂x2

+· · · , where the right-hand side is evaluated at (t,x). Youmayuse that (1±

√

h)−1=

1∓

√

h+h+· · · .

• Lastly, rearrange and take the limit as the time step h→0 to derive the corresponding Fokker–Planck equation.