
























We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) \begin{align} \label{abs eqn} du=(a^{ij}(ω,t)u_{x^ix^j}+f)dt + (σ^{ik}(ω,t)u_{x^i}+g^k)dw^k_t, \quad u(0,x)=u_0, \end{align} where $\{w^k_t:k=1,2,\cdots\}$ is a sequence of independent Brownian motions. The coefficients are merely measurable in $(ω,t)$ and can be unbounded and fully degenerate, that is, coefficients $a^{ij}$, $σ^{ik}$ merely satisfy \begin{align} \label{abs only} \left(α^{ij}(ω,t)\right)_{d\times d}:= \left(a^{ij}(ω,t)-\frac{1}{2}\sum_{k=1}^{\infty} σ^{ik}(ω,t)σ^{jk}(ω,t)\right) \geq 0. \end{align} In this article, we prove that there exists a unique solution $u$ to \eqref{abs eqn}, and \begin{align} \notag \|u_{xx}\|_{\mathbb{H}^γ_p(τ,δ)} &\leq N(d,p) \bigg( \|u_0\|_{\mathbb{B}_p^{γ+2 \left(1-1/ p \right)}} + \| f\|_{\mathbb{H}^γ_p( τ,δ^{1-p} )} \label{abs est} &\qquad \qquad+\|g_x\|^p_{\mathbb{H}^γ_p( τ, |σ|^p δ^{1-p},l_2)}+ \| g_x\|_{\mathbb{H}^γ_p( τ,δ^{1-p/2},l_2)} \bigg), \end{align} where $p\geq 2$, $γ\in \mathbf{R}$, $τ$ is an arbitrary stopping time, $δ(ω, t)$ is the smallest eigenvalue of $α^{ij}(ω, t)$, $\mathbb{H}_p^γ(τ, δ)$ is a weighted stochastic Sobolev space, and $\mathbb{B}_p^{γ+2 \left(1-1/ p \right)}$ is a stochastic Besov space.
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