What is this conditional probability?

Question

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$":

$$\mathbb{P} [C(t_{i+1})<y|\mathcal{F}_{t_i}]=\mathbb{P} [C(t_{i+1})<y|C(t_i)]$$

I have no problem understanding the second probability. The C is a portfolio value by the way.The problem is the $\mathcal{F}_{t_i}$, I want to know what it stands for. I know this is probably a stupid question, but I don't know where to look exactly.

Answer 1

$\mathcal{F}_{t_i}$ is a member of a filtration which is a set of $\sigma$ algebras indexed by time. To be measurable with respect to $\mathcal{F}_{t_i}$ means to only depend on the information available up to and including time $t_i$. Conditioning probability on it means given the information available up to and including $t_i$, what is the probability.

Answer 2

$\mathscr{F}_{t_i}$ represents a sigma algebra, which is a fancy term for "available information". In this case your only available information for computing the probability is the portfolio value $C(t_i)$.