I shall give an overview of reaction-diffusion fronts in
random flows, especially the variational formula of front speeds of
Kolmogorov-Petrovsky-Piskunov reactions. Large deviation of the random
flows is essential to the formula and the analysis of front
speed asymptotics.

The various concepts of volatility (realized, local, stochastic, implied), well defined or depending on a given model and/or statistical estimates, will be discussed. Backward and forward equations for call-option payoffs (Black-Scholes and Dupire equations) will be revisited. We will show that, besides the Black-Scholes model with constant volatility, fast mean reverting stochastic volatility models can reconcile local and implied volatilities. If time permits we will also look at the relation between volatility and correlation in the multidimensional case.
The talk is addressed to a general audience in Probability without any particular deep background in financial mathematics.

We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.

Let $B = (B_t: t\ge 0)$ be a real-valued Brownian motion and let
$L = (L_t: t\ge 0)$ denote its local time in state 0. We present a characterization of the measurable functions $H$ such that $M_t = H(B_t,L_t)$
is a continuous local martingale. It turns out that the class of such functions is considerably wider when one relaxes the smoothness conditions that would be needed for a facile application of It\^o's formula.

We collect together a number of examples of random walk
where the characteristic function of the first step has a
singularity at the point t=0. The function \log\varphi(t) has
two different expansions for positive and negative $t$ near the
origin; we call the coefficients of these expansions left and
right quasicumulants. Such examples include the trace of a
two dimensional random walk {(X_n,Y_n)} on the x-axis, and the
subordinated random walk (X_{\tau_n}) where (\tau_n) is an
appropriate sequence of random times. Using quasicumulants we derive an asymptotic expansion for the distribution of the sums of i.i.d. random variables, and assuming
further differentiability condition we are able to give sharp
estimate in the variable x of the remainder term.