  We obtain uniqueness and existence of a solution $u$ to the following
second-order stochastic partial differential equation (SPDE) : \begin{align}
  \label{abs eqn}
  du= \left( \bar a^{ij}(\omega,t)u_{x^ix^j}+ f \right)dt + g^k dw^k_t, \quad t
\in (0,T); \quad u(0,\cdot)=0, \end{align} where $T \in (0,\infty)$, $w^k$
$(k=1,2,\ldots)$ are independent Wiener processes, $(\bar a^{ij}(\omega,t))$ is
a (predictable) nonnegative symmetric matrix valued stochastic process such
that $$ \kappa |\xi|^2 \leq \bar a^{ij}(\omega,t) \xi^i \xi^j \leq K |\xi|^2
\qquad \forall (\omega,t,\xi) \in \Omega \times (0,T) \times {\mathbf{R}}^d $$
for some $\kappa, K \in (0,\infty)$, $$ f \in L_p\left( (0,T) \times
{\mathbf{R}}^d, dt \times dx ; L_r(\Omega, {\mathscr{F}} ,dP) \right), $$ and
$$ g, g_x \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(\Omega,
{\mathscr{F}} ,dP; l_2) \right) $$ with $2 \leq r \leq p < \infty$ and
appropriate measurable conditions.

Probability