Stochastic Differential Equation for Exponential Decay
Define a stochastic process satisfying Ito stochastic differential equation

, describing exponential decay subject to Wiener noise:
Simulate the process for different values of variance parameter

:
Find the mean function of the process:
Its independence of the noise variance

suggests that it coincides with the deterministic solution:
Find the variance function of the process

:
Value of the process at time

is a random variable. Find the probability density function of the value of the process

: