The main aim of this paper is to establish an existence theorem for the optimal weak solution \(\xi^{\ast}\) of the following problem: \[ \begin{split} E \int_{0}^{T} h(t,\xi^{\ast}_{t} ) \, dt & = \sup_{\xi} E \int_{0}^{T} h(t, \xi_{t} ) \, dt \\ \text{s.t.} \quad d\xi_{t} & \in F(t,\xi_{t} ) \, dt + G(t, \xi_{t} ) \, dW_{t} \\ P^{\xi_{0}}& = \mu , \end{split} \] where \(F\), \(G\) : \([0,T]\times \Re^{d} \to \text{Conv}(\Re^{d})\) are measurable, compact and convex valued multifunctions, \(W\) is an \(m\)-dimensional Wiener process on the filtered probability space \((\Omega, {\mathcal{F}}, \{ {\mathcal{F}}_{t} \}_{0 \leq t \leq T}, P)\) and \(\mu\) is a given probability measure on the space \((\Re^{d}, {\mathcal{B}}(\Re^{d}))\). Denote by \(SI ( F, G, \mu)\) the set of all weak solutions \(\xi\) of the differential inclusion, in the sense that \(\xi\) is a \(d\)-dimensional, continuous stochastic process defined on the probability space \((\Omega, {\mathcal F}, P)\) together with a Wiener process \((W^{t}, {\mathcal F}_{t}^{\xi} := \sigma \{ \xi_{s} : s \leq t\})\), and \(({\mathcal F}_{t}^{\xi})\)-adapted processes \(f_{t} \in F(t, \xi_{t})\), \(g_{t} \in G(t, \xi_{t})\, dt \times dP\) a.e. such that \[ \begin{split} \xi_{t} & = \xi_{0}+ \int_{0}^{t} f_{s} \, ds + \int_{0}^{t} g_{s} \, dW_{s}, \quad t \in [0,T] \\ P^{\xi_{0}} & = \mu. \end{split} \] Denote by \({\mathcal M}({\mathcal C})\) the set of all probability measures on \({\mathcal C}:=C([0,T], \Re^{d})\) endowed with the Borel \(\sigma\)-field and \(Q:=P^{\xi}\). An appropriate differential operator \(A_{t}\) is given so that one may show \(SI(F,G,\mu) \neq \emptyset\) if and only if \({\mathcal R}^{\text{loc}} (F,G,\mu) \neq \emptyset\), the latter set consisting of all probability measures \(Q \in {\mathcal M}({\mathcal C})\) which are solutions to the local martingale problem for \((F, Q, \mu)\): (i) \(Q^{\pi_{0}} = \mu\), (ii) there exists measurable mappings \(a:[0,T] \times {\mathcal C} \to \Re^{d}\) and \(b : [0,T] \times {\mathcal C} \to \Re^{d \times m}\) such that \(a(t,y) \in F(t, y(t))\), \(b(t,y(t)) \in G(t,y(t))\, dt \times dQ\) a.e., and for every \(f \in C^{2}_{b} (\Re^{d})\) the process \((M_{t}^{f})\) defined by \[ M_{t}^{f}:=f\circ \pi_{t} - f\circ \pi_{0} - \int_{0}^{t} (A_{s} f )\, ds,\quad t \in [0,T], \] is a local martingale. We may equip the space \({\mathcal R}^{\text{loc}} (F,G,\mu)\) with the topology of weak convergence of probability measures. Under fairly weak assumptions it is shown that \({\mathcal R}^{\text{loc}} (F,G,\mu)\) is nonempty and compact in \({\mathcal M}({\mathcal C})\). These results allow the existence of a solution to the extremal problem to be established in the following sense: there exists a measurable mapping \(Q^{\ast} :{\mathcal M}(\Re^{d}) \to {\mathcal M}({\mathcal C})\) such that \(Q^{\ast}_{\mu} \in {\mathcal R}^{\text{loc}} (F,G,\mu)\) for every \(\mu \in {\mathcal M}(\Re^{d})\) and \[ E_{Q^{\ast}_{\mu}}[\int_{0}^{T} h(t, \pi_{t} ) \, dt ] = \sup_{Q \in {\mathcal R}^{\text{loc}} (F,G,\mu)} E_{Q} [\int_{0}^{T} h(t, \pi_{t} ) \, dt ]. \]