Stochastic Differential Equation for Exponential Decay
Define a stochastic process satisfying Ito stochastic differential equation
![](Files/StochasticDifferentialEquationForExponentialDecay.en/1.png)
, describing exponential decay subject to Wiener noise:
Simulate the process for different values of variance parameter
![](Files/StochasticDifferentialEquationForExponentialDecay.en/2.png)
:
Find the mean function of the process:
Its independence of the noise variance
![](Files/StochasticDifferentialEquationForExponentialDecay.en/3.png)
suggests that it coincides with the deterministic solution:
Find the variance function of the process
![](Files/StochasticDifferentialEquationForExponentialDecay.en/4.png)
:
Value of the process at time
![](Files/StochasticDifferentialEquationForExponentialDecay.en/5.png)
is a random variable. Find the probability density function of the value of the process
![](Files/StochasticDifferentialEquationForExponentialDecay.en/6.png)
: