Differential equation $E[X_t] = b + log(a) \int_{0}^{t} E[X_s]ds$

Question

How can I solve the following equation? $$E[X_t] = b + log(a) \int_{0}^{t} E[X_s]ds$$

I am not sure, how to start. Is there a special trick, how to solve ODE's like this?

Answer 1

Write $E[X_t]=f(t)$. Then your equation in differential form writes $$ \frac d{dt} f(t) = \log(a)f(t),\ t>0,\quad f(0)=b. $$ A solution is $f(t)=ba^t$. You can define the deterministic stochastic process $X_t=ba^t$ with probability $1$, then $f(t):=E[X_t]=ba^t$ solves your equation.