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  • Foundations of Financial Economics Introduction to stochastic processes

    Paulo Brito

    1pbrito@iseg.ulisboa.pt University of Lisbon

    May 15, 2020

  • Information set ▶ The information set is given by

    (Ω,F ,P),F,P

      ▶ where F is a filtration

    F ≡ {Ft}Tt=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: example Binomial 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: example Sequence 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 processes Adapted 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 being eliminated across time.

  • Stochastic processes Histories

    ▶ Let Nt = {Nt}Tt=0, N0 = 1 be the sequence of the number of possible events (which are equal to the number of nodes for an information tree representing F)

    ▶ We can represent an adapted stochastic process as a sequence of 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 process at time t,

    ▶ History: it is a particular realization of Xt = xt up until time t where

    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 probability distributions 

    {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

    x2,1 x3,1

    x3,2

    x2,2

    x3,3

    x3,4

    x1,2

    x2,3 x3,5

    x3,6

    x2,4

    x3,7

    x3,8

    π1 ,1

    π2,1

    π3,1

    π3,2

    π2,2 π 3,3

    π3,4

    π1,2 π2,3

    π3,5

    π3,6

    π2,4 π 3,7

    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+h given 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 the information node at xt,s;

    ▶ We have now Nt+h|t∑ s=1

    πt+h|t,s = 1, for every t  

  • A binomial stochastic process, after a t = 1 realization

    x0

    x1,1

    x2,1 x3,1

    x3,2

    x2,2

    x3,3

    x3,4

    π2,1 |1,1

    π3,1|1,1

    π3,2|1,1

    π2,2|1,1 π3,3|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 = 2 realizations

    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 the probability 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 is equal to the conditional probability conditional on the history until 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 the

    unconditional probability of Xt = xt is equal to the probability of the history xt, i.e.,

    Pt = P0(xt) = P(Xt = xt|X0 = x0) = P(xt) ▶ Then Markov processes verify the following relationship between

    conditional and unconditional probabilities Pt+1 = Pt+1|t ◦ Pt

     

  • A Markovian binomial process

    x0

    x1,1

    x2,1 x3,1

    x3,2

    x2,2

    x3,3

    x3,4

    x1,2

    x2,3 x3,5

    x3,6

    x2,4

    x3,7

    x3,8

    π

    π 2

    π 3

    π 2(1 − π)

    π(1 − π) π

    2 (1 − π)

    π(1 − π)2

    1− π

    (1 − π)π

    (1 − π)π 2

    (1 − π)2π

    (1 − π) 2 (1 − π)

    (1 − π)3

  • A Markovian binomial process after a t = 1 realization

    x0

    x1,1

    x2,1 x3,1

    x3,2

    x2,2

    x3,3

    x3,4

    π

    π 2

    π(1 − π)

    1 − π (1 − π)π

    (1 − π)2

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

    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 constant

    Et[A] = A

    ▶ if Xt = {xτ}tτ=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 operator should 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 = {Xτ}tτ=0 has the martingale property  if

    Et[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 × xt d × xt

    ) =

    ( u d

    ) 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 a martingale.

    ▶ Intuition: the martingale property is associated to the properties of the possible realisations of a stochastic process and of the probability 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