  Spike variation technique plays a crucial role in deriving Pontryagin's type
maximum principle of optimal controls for differential equations of several
types, including ordinary differential equations (ODEs), partial differential
equations (PDEs), and stochastic differentia equations (SDEs), when the control
domains are not assumed to be convex. This technique also applies to
(deterministic forward) Volterra intrgral equations (FVIEs). It is natural to
expect that such a technique could be extended to the case of (forward)
stochastic Volterra integral equations (FSVIEs). However, by mimicking the case
of SDEs, one encounters an essential difficulty of handling an involved
quadratic term. To overcome the difficulty, we introduce an auxiliary process
for which one can use It\^o's formula, and adopt a trick used in
linear-quadratic stochastic optimal control problems. Then a suitable
representation of the above-mentioned quadratic form is obtained, and the
second order adjoint equations are derived. Consequently, the maximum principle
of Pontryagin type is established. Some relevant extensions are investigated as
well.
