# How is the time integral of a stochastic process distributed?

Consider a continuous-time stochastic process $$\{X_{t\in [a,b]}\}$$ which is sufficiently well-behaved so that it can be integrated in some well-defined sense over $$[a,b]$$ (continuous, finite-valued, etc.). This integral would plausibly be (slightly abusing notation): $$W_t = \int_a^b X_t dt$$ How can one determine in general the distribution of $$W_t$$, the integral of this stochastic process with respect to time?

COMMENT: The answer to this question proposes a solution in the context of Brownian motion by suggesting to find the moments of $$W_t$$ to obtain the distribution in their example; are there no "integration rules" similar to those in traditional or Itô calculus that could be applied here that avoid this process?

• I think it's impossible to answer the question in this generality. Typically, you would try to determine the distribution of sums of the form $$\sum_{j=1}^n \alpha_j X_{t_j}$$ for real numbers $\alpha_j$, and then you can try to study the limiting behaviour of this object. For Brownian motion this works very well (because the sum is Gaussian and the limit of Gaussians is again Gaussian) but in general it's very hard. – saz Oct 2 '18 at 9:35
• Thank you for the insight! At face value it’s surprising that such a “small” tweak (randomness) makes something as “simple” as integration so hard, but it looks like that’s the case. – aghostinthefigures Oct 2 '18 at 15:30