I am trying to learn some stochastic calculus, and I have gotten stuck trying to answer what I thought would be a simple question. I think I am misunderstanding something fundamental about stochastic calculus, and I hope that answering this question will help me figure out what that is!
Let $Y:\Omega \times [0,\infty)$ be a diffusion process with $$dY_t = \mu(Y_t)dt + \sigma(Y_t)dW_t$$ where $W_t$ is a Wiener process. Suppose $Y_t$ represents a stochastic "flow" value, such as the amount of water flowing in or out of a tub. Then, for any $\omega \in \Omega$, the "stock" resulting from this flow (e.g. the random amount of water in the tub) is $$X(t,\omega) \equiv \int_0^t Y(s,\omega) ds$$
Is $X$ also a diffusion process? I keep trying to apply Ito's lemma, but $X(t,\omega)$ is a function of $(Y(s,\omega))_{s\leq t}$ rather than of $Y(t,\omega)$. Am I doing something stupid?
Ito diffusions $dY_t=b(Y_t)dt+\sigma(Y_t)dW_t$ have SDE coefficients which satisfy the Lipschitz condition $$|b(x)-b(y)|+|\sigma(x)-\sigma(y)|\leq D|x-y|$$ for all $x,y\in \mathbb{R}$ and are all Markov processes (ref: Oksendal, Th.7.1.2. p. 111). The simple example $dY_t=dW_t$ yields $X_t=\int_0^tY_sds=\int_0^tW_sds$ which is not Markov, so it can't be a Ito diffusion.