Suppose $X^x$ is solution of $$d X_t = X^3_t dW_t, \quad X_0 = x>0.$$ In the above, $W$ is a Brownian motion in a given filtered probability space. Such an equation has unique strong solution, since $x^3$ is locally lipschitz. In fact, it is a strict local martingale starting from initial $x$.
[Q.] Anybody could either prove or disprove $$\lim_{(y,t) \to (x,0)} \mathbb E [X_t^y] = x ?$$
limit in t. $X_t$ is a positive local martingale, hence a supermartingale so the limit as $t \rightarrow 0$ exists and is $\le x$. On the other hand, by path continuity $X_t \approx x$ for small t, so $\mathbb E^x X_t > x - \epsilon $ for those same t.
Continuity in y. I think must follow from monotonicity in y and continuity in t