In this thesis we investigate the conditions which lead to explosion of Stochastic Differential Delay Equations (SDDE) without the drift term. SDDE’s are stochastic differential equations with delay in the argument. In particular, we will be studying explosion of these equations in finite time. We will show that, with positive probability, the following SDDE:

\[d X ( t ) = \left ( \int_{0}^{1} { | X ( t - s ) |^{q} \ ds } \right ) d W ( t ), \ q > 1 , \ t \in{ ( 0, 1 ) } ,\]

with the past condition:

\[X ( t ) = X_{-} ( t ), \ \ t \in [ - 1 , 0 ] ,\]

explodes in finite time. Here \(X_{-} ( t ) , \ t \in [ - 1 , 0 ]\) is a given stochastic process with continuous paths and \(W(t), \ -\infty < t < \infty\), is the 1-dimensional Wiener process with \(W (0) = 0\).

Furthermore, we will consider some generalizations of above equation, and show that, with positive probability, blow up occurs in finite time.