Stochastic processes

Stochastic processes

This is a theoretical course.

Applied examples.

Zucco, Sacerdote, Kolokoltsov.

Exercises to do during the semester → no written part in June and lectures with exercises.

Schilling, Partzsch, Karlin and Taylor (towards applications).

Brownian motion

In 1888, a biologist observed the movement of pollen plant in the river.

The goal is to understand how they move.

They observed:

• Irregular movement, translation and rotation.
• Moves stop.
• Each particle moves independently from the others.
• Motion is more active the less viscous is the fluid.

In 1905, Einstein arrived to put together these observations → he understood that atoms are moving and bombarding macro-particles, they are moving accordingly to the movement of atoms.

We can have many different microscopic behavior that give rise to the same macroscopic behavior.

Microscopic ↔ macroscopic.

Stochastic processes

This is a theoretical course.

Applied examples.

Zucca, Sacerdote, Kolokoltsov.

Exercises to do during the semester no written part in June and lectures with exercises.

Schilling, Partzsch, Karlin and Taylor (Towards applications).

Brownian motion

In 1988, a biologist observed the movement of pollen plant in the river the goal is to understand how they move.

They observed:

• Irregular movement, translations and rotations.
• Never stop.
• Each particle moves independently from the others.
• Motion is more active the less viscous is the fluid.

In 1905, Einstein arrived to put together these observations he understood that atoms are moving and bombarding macroparticles; they are moving accordingly to the movement of atoms.

We can have many different microscopic behaviours that give rise to the same macroscopic behaviour.

Microscopic ↔ macroscopic.

White noise

White noise: when a signal has all possible frequencies.

Properties of Brownian motion

• The movement starts at x=0.
• Change position only at discrete times kΔt with k∈ℕ (fixed) and k∈ℕ.
• Maximum movement Δx (units fixed) to the left or to the right with probability 1/2.
• Δx does not depend on any previous position nor on the present position nor on time t=kΔt.
• As Δt →0 we also have Δx →0 given the fact that motion is continuous.

We consider x_t the position of the particle at time t, t∈[0,T], with T=NΔt and N=⌊t/Δt⌋ (the biggest integer part of t/Δt).

The left/right movement is independent and identically distributed.

N is the number of changing position.

Introducing i.i.d. Bernoulli random variables {ξ_k}_k≥1 such that P(ξ_k=1)=P(ξ_k=0)=1/2.

The number of movement (toward right is moving toward right).

S_N=∑_i=1^Nξ_i and the number of movement toward left is given by N-S_N.

So we get x_T=S_NΔx-(N-S_N)Δx= (2S_N-N)Δx=∑_i=1^N(2ξ_i-1)Δx.

Considering k=mΔt and T=NΔt we get:

X_T = (X_T - X_t) + (X_t - X₀)

where       ⇄   Δt   and   Δx   𝕁

We have  ε_i i.i.d. and this implies that:

X_T - X_t || X_t - X₀

This happens since:

X_T - X_t = ⁿ∑_k=m+1 (2ε_k - 1) Δx

while:

X_t - X₀ = ^m∑_k=1 (2ε_k - 1) Δx

Var (X_T) = Var (X_T - X_t) + Var (X_t - X₀) = σ² (T-t)+ σ²(t)

and Var X_t = σ²t.

So we have σ²(T) = σ² (T-t) + σ²(t), that is true for every t  ⇔ if and only if  σ² (t) is linear     σ² (T) = σ²T.

E ε² = P (ε_i = 0) 0 + P (ε_i = 1) 1 = 1/2, and Var ε² = E (ε - 1/2)² = [E ε² - 2 E ε + 1/4]= 1/2 - 1/2 + 1/4 = 1/4.

Var X_t = N [Δx]² = [    &nbs