Lecture 3 Financial Engineering II (E4707)

16/06/2024, 03:31 Lecture 3




Lecture 3: Martingales and Quadratic Variation Processes



1 Martingales
Martingales play a fundamental role in stochastic calculus. The basic idea of a
martingale is that of a fair game.
Definition 1.1. Let (Ω, F, P) be a probability space. A process {Xt }t≥0 is a martin-
gale (respectively, submartingale, supermartingale) with respect to a filtration {Ft }t≥0
if it is adapted to that filtration, E|Xt | < ∞ for all t and

E[Xt Fs ] = (respectively, ≥, ≤)Xs , ∀s ≤ t.
Example: Prove that a Brownian motion {Wt }t≥0 is a martingale with respect
to Ft := σ (Wu ; 0 ≤ u ≤ t):
  
E WtFs = E [Wt |Fs ]
= E [Wt − Ws |Fs ] + E [Ws |Fs ]
= E [Wt − Ws ] + Ws
= Ws , s ≤ t.
Hence, the best estimate of the future value of the Brownian motion is its present
value.

Example: Prove that {Wt2}t≥0 is a submartingale with respect to Ft = σ (Wu ; 0 ≤ u ≤ t),
where {Wt }t≥0 is a Brownian motion:
      2
E Wt2Fs ≥ E WtFs
= Ws2
where we have used Jensen’s inequality since f (x) = x2 is a convex function.

Example: Prove that {Wt2−t}t≥0 is a martingale with respect to Ft = σ (Wu ; 0 ≤ u ≤ t),
where {Wt }t≥0 is a Brownian motion.
  
Hint: Compute E Wt2 − Ws2Fs and use independent increment and normality.

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