We prove under appropriate hypotheses that the Hamilton-JacobiBellman dynamic programming equation with uniformly elliptic operators,
max
1
⩽
k
⩽
m
{
L
k
u
−
f
k
}
=
0
{\max _{1 \leqslant k \leqslant m}}\{{L^k}u - {f^k}\} = 0
, has a classical solution
u
∈
C
2
,
β
u \in {C^{2,\beta }}
, for some (small) Hölder exponent
β
>
0
\beta > 0
.