could somebody explain to me why the drift coefficient of a stochastic differentuial equation (the one that multiplies dt) is usually assumed to be absolutely integrable? Is not enough to assume that this coefficient is (only) integrable (meaning that its integral with respect to time is a.s. finite)? Many thanks!
coefficient of a stochastic differential equation
0
$\begingroup$
stochastic-processes
stochastic-calculus
1 Answers
1
Note that in Lebesgue sense, absolutely integrable and integrable are the same. They both mean $\int |f(x)|dx<\infty$.
Generally, it is not enough to only know $\int f(t)dt<\infty$, which only implies $EX<+\infty$. But quite often, we also want to guarantee $EX>-\infty$. Here $X$ is the solution to your stochastic differential equation and $f$ is the drift.