We study stochastic differential equations (SDEs) whose drift and diffusion
coefficients are path-dependent and controlled. We construct a value process on
the canonical path space, considered simultaneously under a family of singular
measures, rather than the usual family of processes indexed by the controls.
This value process is characterized by a second order backward SDE, which can
be seen as a non-Markovian analogue of the Hamilton-Jacobi-Bellman partial
differential equation. Moreover, our value process yields a generalization of
the G-expectation to the context of SDEs.

We consider dynamic sublinear expectations (i.e., time-consistent coherent
risk measures) whose scenario sets consist of singular measures corresponding
to a general form of volatility uncertainty. We derive a c\`adl\`ag nonlinear
martingale which is also the value process of a superhedging problem. The
superhedging strategy is obtained from a representation similar to the optional
decomposition. Furthermore, we prove an optional sampling theorem for the
nonlinear martingale and characterize it as the solution of a second order
backward SDE.

We study the leading term in the small-time asymptotics of at-the-money call
option prices when the stock price process $S$ follows a general martingale.
This is equivalent to studying the first centered absolute moment of $S$. We
show that if $S$ has a continuous part, the leading term is of order $\sqrt{T}$
in time $T$ and depends only on the initial value of the volatility.
Furthermore, the term is linear in $T$ if and only if $S$ is of finite
variation.

We study utility maximization for power utility random fields with and
without intermediate consumption in a general semimartingale model with closed
portfolio constraints. We show that any optimal strategy leads to a solution of
the corresponding Bellman equation. The optimal strategies are described
pointwise in terms of the opportunity process, which is characterized as the
minimal solution of the Bellman equation. We also give verification theorems
for this equation.