Foundations of Financial EconomicsIntroduction to stochastic processes

Paulo Brito

1pbrito@iseg.ulisboa.ptUniversity of Lisbon

May 15, 2020

Information set The information set is given by

(Ω,F ,P),F,P

where F is a filtration

F ≡ FtTt=0 = F0,F1, . . . ,FT

which is an ordered sequences of subsets of Ω such that: F0 = Ω, FT = F and Ft ⊂ Ft+1 meaning ”more information”

Then, we can consider a sequence of events up until time t

Wt = W0,W1, . . . ,Wt where Wt ∈ Ft

Filtration: exampleBinomial information tree: for T = 3 and Ω = ω1, . . . , ω8

ω1, ..., ω8

ω1, ..., ω4

ω1, ω2

ω1

ω2

ω3, ω4

ω3

ω4

ω5, ..., ω8

ω5, ω6

ω5

ω6

ω7, ω8

ω7

ω8

Observation: some events are being eliminated

Filtration: exampleSequence of events: W0,W1,W2,W3 where W1 = w1,1,w1,2

W0

w1,1

w2,1

w3,1

w3,2

w2,2

w3,3

w3,4

w1,2

w2,3

w3,5

w3,6

w2,4

w3,7

w3,8

Stochastic processesAdapted stochastic processes

Definition: the sequence of random variables Xt

Xt = X0, . . .Xt, t ∈ T

is an adapted stochastic process to the filtration F if

Xt = X(Wt),Wt ∈ Ft

if Xt is a random variable as regards the event Wt ∈ Ft intuition: the information as regards t has the same structure as

Ft, in the sense that some potencial sequences are beingeliminated across time.

Stochastic processesHistories

Let Nt = NtTt=0, N0 = 1 be the sequence of the number of

possible events (which are equal to the number of nodes for aninformation tree representing F)

We can represent an adapted stochastic process as a sequenceof possible realizations for every t ∈ 0, . . . ,T

Xt = X(Wt) =

xt,1. . .

xt,Nt

∈ RNt

where Nt is the number of possible realizations of the processat time t,

History: it is a particular realization of Xt = xt up until time twhere

xt = x0, . . . , xt, t ∈ T The set of all histories

Xt = X(Wt)

Probabilities Consider a particular history up until time t, Xt = xt

xt = x0, . . . , xt, t ∈ T

We call unconditional probability of history xt

P(xt) = P(Xt = xt,Xt−1,= xt−1, . . . ,X0 = x0),

Then, we have a sequence of unconditional probabilitydistributions

P0,P1, . . . ,Pt where Pt = Pt(Xt) where Xt are all histories until time t,

Pt =

πt,1. . .πt,Nt

Nt is the number of nodes of the information at t

then

P0 =

Nt∑s=1

πt,s = 1, for every t

A binomial stochastic process

x0

x1,1

x2,1x3,1

x3,2

x2,2

x3,3

x3,4

x1,2

x2,3x3,5

x3,6

x2,4

x3,7

x3,8

π1,1

π2,1

π3,1

π3,2

π2,2π3,3

π3,4

π1,2 π2,3

π3,5

π3,6

π2,4π3,7

π3,8

The process X0,X1,X2,X3 has 8 possible histories The sequence of uncontitional probability distributions is

1,P1,P2,P3 where∑2

s=1 π1,s =∑4

s=1 π2,s =∑8

s=1 π3,s = 1

Transition probabilities The conditional probability of xt+1 given a particular history xt is

P(xt+1|xt) = P(Xt+1 = xt+1|Xt = xt,Xt−1,= xt−1, . . . ,X0 = x0) =

P(Xt+1 = xt+1,Xt = xt,Xt−1,= xt−1, . . . ,X0 = x0)

P(Xt = xt,Xt−1,= xt−1, . . . ,X0 = x0)(1)

Definition we call transition probability for Xt+h = xt+hgiven the information history at t,

Pt(xt+h) = P(Xt+h = xt+h|Xt = xt)

we denote Pt+h|t = Pt(xt+h) where

Pt+h|t =

πt+h|t,1. . .

πt+h|t,Nt+h|t

where Nt+h|t is the number of nodes, at t + h, of theinformation node at xt,s;

We have nowNt+h|t∑s=1

πt+h|t,s = 1, for every t

A binomial stochastic process, after a t = 1 realization

x0

x1,1

x2,1x3,1

x3,2

x2,2

x3,3

x3,4

π2,1|1,1

π3,1|1,1

π3,2|1,1

π2,2|1,1π3,3|1,1

π3,4|1,1

Conditional probabilities satisfy:∑2

s=1 π2,s|1,1 =∑4

s=1 π3,s|1,1 = 1

A binomial stochastic process, after t = 1 and t = 2realizations

x0

x1,1

x2,2

x3,3

x3,4

π3,3|2,21,1

π3,4|2,21,1

Conditional probabilities satisfy:∑2

s=1 π2,s|2,21,1= 1

Markovian processes Definition: a stochastic process has the Markov property if

the probability conditional to a history is the same as theprobability conditional on the last realization

P(Xt+h = xt+h|Xt = xt) = P(Xt+h = xt+h|Xt = xt)

In other words: the transition probability from Xt = xt isequal to the conditional probability conditional on the historyuntil time t

Pt+h|t = Pt(xt+h) ≡ P(Xt+h = xt+h|Xt = xt)

Observe that a general property of adapted processes is that theunconditional probability of Xt = xt is equal to the probability ofthe history xt, i.e.,

Pt = P0(xt) = P(Xt = xt|X0 = x0) = P(xt)

Then Markov processes verify the following relationship betweenconditional and unconditional probabilities

Pt+1 = Pt+1|t Pt

A Markovian binomial process

x0

x1,1

x2,1x3,1

x3,2

x2,2

x3,3

x3,4

x1,2

x2,3x3,5

x3,6

x2,4

x3,7

x3,8

π

π2

π3

π 2(1 − π)

π(1 − π)π

2 (1 − π)

π(1 − π)2

1−π

(1 −π)π

(1 − π)π2

(1 − π)2π

(1 − π) 2(1 − π)

2π

(1 − π)3

A Markovian binomial process after a t = 1 realization

x0

x1,1

x2,1x3,1

x3,2

x2,2

x3,3

x3,4

π

π2

π(1 − π)

1 − π (1 − π)π

(1 − π)2

A Markovian binomial process after a t = 1 and t = 2realization

x0

x1,1

x2,2

x3,3

x3,4

π

1 − π

Mathematical expectation for stochastic processes Unconditional mathematical expectation of Xt is a number

E0[xt] = E[xt|x0] =

Nt∑s=1

P0(xt,s)xt,s =

Nt∑s=1

πt,sxt,s

Unconditional variance of Xt is

V0[xt] = V[xt|x0] = E0[(xt − E0(xt))2] =

Nt∑s=1

πt,s(xt,s − E0(xt))2.

The conditional mathematical expectation

Eτ [xt] = E[xt | xτ ]is an adapted stochastic process because

Eτ [xt] = (Eτ,1(xt), . . .Eτ,Nτ (xt))

where

Eτ,i[xt] =

Nt|τ,i∑j=1

P(Xt = xt,j|xτ )xt,i =

Nt|τ,i∑j=1

πt|τ,jxt,j, i = 1, . . . ,Nτ

where∑

i Nt|τ,i = Nτ

Properties of conditional mathematical expectation: Et

if A is a constantEt[A] = A

if Xt = xτtτ=0 is an adapted process

Et[xt] = xt

law of the iterated expectations:

Et−s[Et[xt+r]] = Et−s[xt+r], s > 0, r > 0

this is a very important property: the expected value operatorshould be taken from the time in which we have the least information

if Yt is a predictable process (i.e., Ft−1-adapted)

Et[yt+1] = yt+1

Martingales

Definition: a process Xt = Xτtτ=0 has the martingale

property ifEt[xt+r] = xt, r > 0

Definition: super-martingale if

Et[xt+r] ≤ xt, r > 0

Definition: sub-martingale if

Et[xt+r] ≥ xt, r > 0

Example Let

xt+1 =

(u × xtd × xt

)=

(ud

)xt

d and u are known constants such that 0 < d < u and assume that

Pt+1|t =

(P(xt+1 = u × xt|xt)P(xt+1 = d × xt|xt)

)=

(p

1 − p

) for 0 < p < 1

Then the conditional mathematical expectation is

Et[xt+1] = (pu + (1 − p)d)xt.

If pu + (1 − p)d = 1 then Et[xt+1] = xt, that is Xt is amartingale.

Intuition: the martingale property is associated to the propertiesof the possible realisations of a stochastic process and of theprobability sequence.

Wiener process (or Standard Brownian Motion)

The process Xt = Xt, t ∈ [0,T) is a Wiener process if:

x0 = 0, E0[Xt] = 0, V0[Xt − Xτ ] = t − τ

for any pair t, τ ∈ [0,T). in particular: V0[Xt − Xt−1] = 1 observe that the process has asymptotically infinite unconditional

variance limt→∞ V0[Xt − Xτ ] = ∞ for a finite τ ≥ 0 The variation of the process then follows a stationary standard

normal distribution

∆Xt = Xt+1 − Xt ∼ N(0, 1)

Wiener process

10 replications of a Wiener process