Solution of the Kolmogorov forward equation for a linear process

[dalves]

Consider the 1-D linear system governed by:

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

where "a" is a scalar and:
x(t) = system state
n(t) ~ N(0, sigma^2)

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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

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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.

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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.

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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.

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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.

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

 

[gtoguy97]

Your reasoning is incorrect when you assumed the solution of the KFE would be a Gaussian distribution with drift (beta*t). This solution does not satisfy the KFE since the drift term in the KFE has an x-dependence that is not present in the solution. Therefore, the solution you proposed does not satisfy the KFE and is not a valid solution.