I'm working with the following set of stochastic differential equations: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process $d\eta = \lambda \eta(t) dt + \alpha dW_t$
and:
$$\eta(t) = \alpha e^{-\lambda t} \int_0^{t}e^{\lambda s} dW_s$$
My goal is to show that $E\left[ \sum_{i=1}^n x_i(t)^p\right] \leq K < \infty$ for every $p$. I can prove this IF the following has bounded expectation:
$$ | \int_0^{t} e^s \eta(s)\sum_{i=1}^n x_i(s)^p ds |$$
For notation, I'll define $g_p(X_s) = \sum_{i=1}^n x_i(s)^p$.
So I have the following:
$$ \int_0^{t} e^s \eta(s)g_p(X_s) ds = \int_0^t e^sg_p(X_s) \left(\alpha e^{-\lambda s} \int_0^s e^{\lambda u} dW_u \right) ds$$
Naturally, I want to switch the order of integration, so I think I get:
$$= \alpha \int_0^t \left(\int_s^t g_p(X_u) e^{(1-\lambda)u}du \right) e^{\lambda s}dW_s$$
Now my two questions are:
- Did I switch the order of integration properly? I have concerns about that.
- The "du" integral is now inside the stochastic integral, so can I conclude the entire thing has mean $0$ which then solves the problem?