= Solution
A sufficient condition is that $x\mapsto\sqrt{2\theta(x)}$ is <globally Lipschitz>. A more familiar, stronger condition implying this is that $\theta$ is <globally Lipschitz> and $\theta\geq\varepsilon>0$. Both conditions repair the existence issue in part (i).
To connect this to the suggested <stochastic differential equation>, apply the coordinate and coordinate-product cutoff tests to any solution of the <martingale problem>. As before they give
$$
[X^i,X^j]_t=2\delta_{ij}\int_0^t\theta(X_s)\,ds.
$$
Define, using <stochastic integrals> against the coordinates of $X$,
$$
W_t^i=\int_0^t\frac{1}{\sqrt{2\theta(X_s)}}\,dX_s^i.
$$
These integrals exist even without a global positive lower bound: their <quadratic variations> are exactly $t$. Indeed,
$$
[W^i,W^j]_t
=\int_0^t\frac{1}{2\theta(X_s)}\,d[X^i,X^j]_s
=\delta_{ij}t.
$$
The <Lévy characterization of multidimensional Brownian motion> identifies $W$ as standard $d$-dimensional <Brownian motion>. By <associativity of stochastic integration>,
$$
\boxed{dX_t=\sqrt{2\theta(X_t)}\,dW_t,\qquad X_0=0.}
$$
The diffusion matrix $\sqrt{2\theta(x)}I$ is <globally Lipschitz> under the stated condition, and boundedness of $\theta$ gives bounded growth. The <global existence theorem for stochastic differential equations with Lipschitz coefficients> gives a unique <strong solution of a stochastic differential equation> and <uniqueness in law>. Every solution to the <martingale problem> has just been represented as a <weak solution of a stochastic differential equation> of this form, so they all have the same law. Conversely, the <Itô formula> applied to any $f\in C_c^\infty$ gives a true <martingale>, since its stochastic integrand is bounded. Thus the <martingale problem> is well posed under this sufficient condition.
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