This paper does not suppose a priori that the evolution of the price of a
financial asset is a semimartingale. Since possible strategies of investors are
self-financing, previous prices are forced to be finite quadratic variation
processes. The non-arbitrage property is not excluded if the class
$\mathcal{A}$ of admissible strategies is restricted. The classical notion of
martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus
related to $\mathcal{A}$-martingales with some examples is developed.

For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m(m\geq 1)$,
we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$
denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and
only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension
$n-3$ and H(1) is the Heisenberg algebra of dimension 3.

For a large class of vanilla contingent claims, we establish an explicit
F\"ollmer-Schweizer decomposition when the underlying is a process with
independent increments (PII) and an exponential of a PII process. This allows
to provide an efficient algorithm for solving the mean variance hedging
problem. Applications to models derived from the electricity market are
performed.

Quadratic entry locus manifold of type $\delta$ $X\subset\mathbb P^N$ of
dimension $n\geq 1$ are smooth projective varieties such that the locus
described on $X$ by the points spanning secant lines passing through a general
point of the secant variety $SX\subseteq\mathbb P^N$ is a smooth quadric
hypersurface of dimension $\delta=2n+1-\dim(SX)$ equal to the secant defect of
$X$.

Small codimensional embedded manifolds defined by equations of small degree
are Fano and covered by lines. They are complete intersections exactly when the
variety of lines through a general point is so and has the right codimension.
This allows us to prove the Hartshorne Conjecture for manifolds defined by
quadratic equations and to obtain the list of such Hartshorne manifolds. Using
the geometry of the variety of lines through a general point, we characterize
scrolls among dual defective manifolds. This leads to an optimal bound for the
dual defect, which improves results due to Ein.