Stochastic Calculus: Basic Concepts

All concepts in financial mathematics are defined within a certain probability space (Ω,F,Q).

Ω denotes the total space,

Ft denotes the σ-algebra of all the information that is known at time t

Q denotes the risk neutral probability measure, which governs the probabilities of events occurring in this space.

The domain of Q is F.

 

A random variable (RV) is a function that assigns values to outcomes of a probabilistic experiment. It’s future value is uncertain. If the value of a particular RV, Xt, is known at time t it is said to be Ft-measurable (Xt ∈ Ft). For any time t2 after t, the value of the RV cannot be determined at time t.

 

The collection, X, of Ft-measurable random variables, {Xt : 0 ≤ t ≤ T}, is a stochastic process. If a stochastic process Y behaves such that every realisation Yt is Ft-measurable, then it is said that it is adapted to the filtration {Ft}0 ≤ t ≤ T . Adapted processes are also known as non-anticipating processes, since their values do not depend on future events.

 

Conditional expectation is the expected value of a random variable given, conditional on, a certain amount of information. Let G be a σ-algebra contained in F. Then the conditional expectation of X given the information contained in G is denoted by   

E[X | G]

It then follows that when X is adapted to the filtration {Ft}0≤t≤T

E[Xt | Ft] = Xt

 

If X ∈/ G, then its value is unknown at time t and the expected value, under the probability measure Q, is an objective prediction of the future value. Formally this means

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