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 :