The problem Univariate Analysis Multivariable Analysis Conclusion How mathematicians predict the future? Mattia Zanella Group 5 Costanza Catalano, Angela Ciliberti, Goncalo S. Matos, Allan S. Nielsen, Olga Polikarpova, Mattia Zanella Instructor: Dr in ˙ z . Agnieszka Wylomańska (Hugo Steinhaus Center) December 22, 2011 How mathematicians predict the future? ECMI European Consortium for Mathematics in Industry
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The problem Univariate Analysis Multivariable Analysis Conclusion
How mathematicians predict the future?
Mattia Zanella
Group 5Costanza Catalano, Angela Ciliberti, Goncalo S. Matos, Allan S. Nielsen,
Olga Polikarpova, Mattia Zanella
Instructor: Dr inz. Agnieszka Wyłomańska (Hugo Steinhaus Center)
December 22, 2011
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Introduction and definitions
Introduction
SPOT RATEINFLATION RATENOMINAL RATEREAL RATE
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Introduction and definitions
Datas
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Detecting Trends
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Ornstein-Uhlenbeck Process
Definition
Let (Ω,F ,P) a probability space and F = (Ft)t≥0 a filtrationsatisfying the usual hypotheses. A stochastic process Xt is anOrnstein-Uhlenbeck process if it satisfies the following stochasticdifferential equation
dXt = λ (µ− Xt) dt + σdWt
X0 = x0
where λ ≥ 0, µ and σ ≥ 0 are parameters, (Wt)t≥0 is a Wienerprocess and X0 is deterministic.
If (St)t≥0 is the process implied/real/nominal inflation we will inour model consider St = expXt ∀t ≥ 0.
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationMaximum Likelihood Estimation
Let (Xt0 , ...,Xtn) n + 1− observations, the Likelihood Function ofXti |Xti−1 is
n∏i=1
fi(Xti ;λ, µ, σ|Xti−1
)
The Log-Likelihood function is defined as
L(X , λ, µ, σ) =n∑
i=1
log f (Xti ;λ, µ, σ|Xti−1).
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationMaximum Likelihood Estimation
Let (Xt0 , ...,Xtn) n + 1− observations, the Likelihood Function ofXti |Xti−1 is
n∏i=1
fi(Xti ;λ, µ, σ|Xti−1
)The Log-Likelihood function is defined as
L(X , λ, µ, σ) =n∑
i=1
log f (Xti ;λ, µ, σ|Xti−1).
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationMaximum Likelihood Estimation
Now we have to find
arg maxλ∈R,µ∈R,σ∈R+
L(X , λ, µ, σ)
putting conditions of the first and second order.
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationResults
λ=9.9241 µ = 2.8656 σ = 2.2687
λ=5.8952 µ = 4.4358 σ = 3.1919
λ=4.5916 µ = 1.5487 σ = 2.3572
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationResults
λ=9.9241 µ = 2.8656 σ = 2.2687
λ=5.8952 µ = 4.4358 σ = 3.1919
λ=4.5916 µ = 1.5487 σ = 2.3572
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationResults
λ=9.9241 µ = 2.8656 σ = 2.2687
λ=5.8952 µ = 4.4358 σ = 3.1919
λ=4.5916 µ = 1.5487 σ = 2.3572
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical Approximations
Consider a general SDE
dXt = a(Xt)dt + b (Xt) dWt , t ∈ [0,T ]
and a partition of the time interval [0,T ] into n equal subintervalsof width δ = T
n0 = t0 < t1 < ... < tn = T
How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry
The problem Univariate Analysis Multivariable Analysis Conclusion