= Solution
Use the same <Brownian motion> for both solutions: this is a <synchronous coupling>. For $D_t=X_t-Y_t$, the <Itô formula> gives
$$
dD_t^2=\left[2D_t(b(X_t)-b(Y_t))+(\sigma(X_t)-\sigma(Y_t))^2\right]\,dt+2D_t(\sigma(X_t)-\sigma(Y_t))\,dW_t.
$$
On any fixed finite horizon the <stochastic integral> has mean zero. Indeed, boundedness of $\sigma$ and the assumed second-moment bounds make the expectation of its squared integrand integrable in time. With $h(t)=\mathbb ED_t^2$, the contraction condition yields, for every $s\leq t$,
$$
h(t)\leq h(s)-k\int_s^th(r)\,dr.
$$
The allowed <Gronwall inequality> gives the <mean-square contraction of synchronously coupled diffusions>
$$
\boxed{\mathbb E(X_t-Y_t)^2\leq e^{-kt}\mathbb E(X_0-Y_0)^2.}
$$
The identical estimate can also be obtained by localizing the nonnegative local supermartingale $e^{kt}D_t^2$ and using Fatou's lemma, a formulation useful when the drift has linear growth.
There is a genuine compatibility issue in the printed global assumptions. If $|b|\leq B$, then for $d=|x-y|$,
$$
2(x-y)(b(x)-b(y))+(\sigma(x)-\sigma(y))^2\geq-4Bd.
$$
It cannot be at most $-kd^2$ for all $d>4B/k$. Thus \b[a globally bounded drift cannot satisfy the stated strict contraction on all of the real line]. The stochastic estimate above is the requested conditional calculation. For a non-vacuous application, global boundedness of the drift must be relaxed while retaining suitable existence and moment hypotheses; it is not silently changed here.
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