foundations of financial economics introduction to ... · stochastic processes adapted stochastic...
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Foundations of Financial EconomicsIntroduction to stochastic processes
Paulo Brito
[email protected] of Lisbon
May 15, 2020
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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
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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
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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
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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.
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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A Markovian binomial process after a t = 1 and t = 2realization
x0
x1,1
x2,2
x3,3
x3,4
π
1 − π
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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
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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
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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
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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.
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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)
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Wiener process
10 replications of a Wiener process