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, traslazione and rotazione
• moves stops
• 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 arise 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 stops
• 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 raise to the same macroscopic behaviour

microscopic ↔ macroscopic

★ White noise: when a signal have all possible frequencies

⚠ Properties of Brownian motion

• the movement start 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 give 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 imply 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