... the function (8, x,y) J (8, x,y;ZJi,,Tl(.)) is continuous. Thus, by the definition of V, it is necessary that V(s, x, y) is upper semi-continuous. On the other hand, by (2.6) and (3.2), taking ... (i) ~,~(s, ~, y) and V ~,~(s, x, y) are continuous in (x, y) 9 ~" • ~m, uniformly in s C [0, T] and 6, E >_ O; For fixed 6 > 0 and e > O, ~5,~ (s, x, y) and Va'~(s,x,y) ... similar to that of Proposition 3.1 and the proof of (ii) and (iii) are by now standard, which we omit here for simplicity of presentation (see Yong-Zhou [1] and Fleming-Soner [1], for details)....