Consider the 1-D linear system governed by:(adsbygoogle = window.adsbygoogle || []).push({});

"dx/dt = a*x(t) + n(t)"

where "a" is a scalar and:

x(t) = system state

n(t) ~ N(0, sigma^2)

****************************

We can write Ito's stochastic differential equation of the previous process as:

"dx = a*x*dt + 1/2*sigma^2*dW_t"

where:

x = system state

a*x = drift term

1/2*sigma(t,x)^2 = diffusion term

W_t = Wiener process

****************************

The time evolution of the system state's probability density function " pdf(x(t)) = p(x,t) " is governed by the Kolmogorov forward equation (aka Fokker-Planck equation):

"dp(x,t)/dt = -d/dx[a*x*p(x,t)] + 1/2*sigma^2*d^2p(x,t)/dx^2"

which after expanding gives:

"dp(x,t)/dt = -a*p(x,t) - a*x*dp(x,t)/dt + 1/2*sigma^2*d^2p(x,t)/dx^2"

where "d/dt", "d/dx" and "d^2/dx^2" are partial derivatives.

****************************

Because the process is linear and "n(t) ~ N(0, sigma^2)", if we further assume that "x(t=0) ~ N(x0, sigma_x^2)", then "pdf(x) = p(x,t)" is Gaussian for t >= 0. Which is basically why the Kalman filter works.

****************************

Furthermore, in the case of "a = 0", i.e. pure diffusion, the solution of the KFE is:

"p(x,t) = ((2*pi*sigma*t)^(-1/2))*exp(-((x-x0)^2)/(2*sigma*t))"

which is indeed a Gaussian distribution thus supporting the previous statement.

****************************

I therefore expected that a solution of the type (gaussian drift + diffusion):

"p(x,t) = ((2*pi*sigma*t)^(-1/2))*exp(-((x-beta*t-x0)^2)/(2*sigma*t))"

would solve the KFE. However, it is easy to verify that this is not a solution of the KFE above.

****************************

Undoubtedly my reasoning is wrong and, most likely, in more than one place. Where?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Solution of the Kolmogorov forward equation for a linear process

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**