lecture ii: ito’s formula and its uses in statistical ...jrojo/pasi/lectures/calderon lec2.pdf ·...
TRANSCRIPT
![Page 1: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/1.jpg)
Chris Calderon, PASI, Lecture 2
Lecture II: Ito’s Formula and Its Uses in Statistical Inference
Christopher P. CalderonRice University / Numerica Corporation
Research Scientist
![Page 2: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/2.jpg)
Chris Calderon, PASI, Lecture 2
OutlineI One View of ”Solving” SDEs and a Refresher on Ito’s Formula
II Extracting Information Implied by a Given SDE Even If Pathwise Solution Unavailable
III Using Techniques Above for Statistical Inference
IV Approximate Solutions and Applications e.g. Test if a Levy Process is Appropriate Given Observational Data Coming from a Complex System
![Page 3: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/3.jpg)
Chris Calderon, PASI, Lecture 2
ReferencesFor Items I - III Draw from:
For Item IV:
Protter, P. (2004) Stochastic Integration and Differential Equations, Springer-Verlag, Berlin.Bass, R. (1997) Diffusions and Elliptic Operators, Springer, NYC.Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.Romano, J. & Thombs, L. (1996) JASA 91, 590-600.
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
![Page 4: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/4.jpg)
Chris Calderon, PASI, Lecture 2
Suppose Given ODE:
Traditional View of “Solving” Would be To Find A Function Such That (e.g. Use Integrating Factor):
“Independent Variable/Index”
X(t) = f(t)
dX
dt= bo(X, t)
![Page 5: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/5.jpg)
Chris Calderon, PASI, Lecture 2
Suppose Given SDE:
dXt = b(Xt, t)dt+ σ(Xt, t)dBt
One View of “Solving” Would be To Find A Function Such That:
Xt = f(t, Bt)
“Independent Variable/Index”
A Process We Know Well
![Page 6: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/6.jpg)
Chris Calderon, PASI, Lecture 2
Or Suppose Given SDE:
Traditional View of “Solving”would be To Find A Function Such That:
“Independent Variable”
A Process We Know and Love
dYt = bo(Yt, t)dt+ σo(Yt, t)dXt
Yt = g(t,Xt)
![Page 7: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/7.jpg)
Chris Calderon, PASI, Lecture 2
More Concretely Given SDE:
Standard Example: Geometric Brownian Motion
“Independent Variable/Index”
dXt = αXtdt+ βXtdBt
Xt = exp (α− 12β
2)t+ βBt
A Process We Know Well
![Page 8: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/8.jpg)
Chris Calderon, PASI, Lecture 2
More Concretely Given SDE:
Found by Introducing:
“Independent Variable/Index”
dXt = αXtdt+ βXtdBt
Xt = exp (α− 12β
2)t+ βBt
A Process We Know Well
Y (X) = log(X)and Applying Ito’s Formula to Change Variables
![Page 9: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/9.jpg)
Chris Calderon, PASI, Lecture 2
Ito’s Formula• One of the Most Widely Known Results
Associated with SDEs:f(t,Xt)− f(t,Xo) =tR0
∂f(s,Xs)∂s ds+
tR0
∂f(s,Xs)∂X dXs +
12
tR0
∂2f(s,Xs)∂2X d[X,X]s
d[X,X]t = σ2(t,X)dt d[B,B]t = dt
For a General Diffusion: For Brownian Motion:
Something Unique to Stochastic Integration a la Ito
![Page 10: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/10.jpg)
Chris Calderon, PASI, Lecture 2
Ito’s Formula• One of the Most Widely Known Results
Associated with SDEs (For Time Homogeneous Functions):
f(Xt)− f(Xo) =tR0
∂f(Xs)∂X dXs +
12
tR0
∂2f(Xs)∂2X d[X,X]s
Something Unique to Stochastic Integration a la Ito
A More Fundamental Introduction On Quadratic Variation In The Next Lecture
![Page 11: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/11.jpg)
Chris Calderon, PASI, Lecture 2
Ito’s Formula• Result Also Applies to Jump Processes:
f(Xt)− f(Xo) =
P0<s≤t
f(Xs)− f(Xs−)− ∂f(Xs−)∂X
∆Xs − 12∂2f(Xs−)
∂2X(∆Xs)
2
tR0+
∂f(Xs−)∂X dXs +
12
tR0+
∂2f(Xs−)∂2X d[X,X]s+
![Page 12: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/12.jpg)
Chris Calderon, PASI, Lecture 2
Another Illustrative Application of Ito’s Formula
tR0
BsdBs =12 (B
2t − t)
dXt = 0dt+ 1dBt
Y = f(X) = X2/2
or Yt =tR0
XsdXs +12 t
“Brownian Variables/Coordinates”
Same Result in Terms of Newly Defined Variables
![Page 13: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/13.jpg)
Chris Calderon, PASI, Lecture 2
But Explicit Solutions Are Elusive
And Even For Cases That Are “Solvable” It May Not Be As Nice As We Think…..
Complicated Functions of B.M. Not Always Easy to Express in Terms of Normal Density or Distribution (Think Dickey Fuller Tables)
![Page 14: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/14.jpg)
Chris Calderon, PASI, Lecture 2
But Explicit Solutions Are Elusive
However….Ito’s Formula Can Be Used to Extract Useful Information, e.g. Moments or a Transition Density
(Different Than Solving in Terms of “Paths”)
Ito’s Formula is Very Useful In Statistical Modeling Because it Does Allow Us to Quantify Some Properties Implied by an Assumed SDE
![Page 15: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/15.jpg)
Chris Calderon, PASI, Lecture 2
Cox Ingersoll Ross (CIR) Process dXt = κ(α−Xt)dt+ σ
√XtdBt
Rewrite Above Using New Constants
Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.
dXt = (a+ bXt)dt+ σ√XtdBt
XT −Xt =TRt
(a+ bXs)ds+TRt
σ√XsdBs
Then Integrate from t to T (Assume Known Deterministically)Xt
![Page 16: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/16.jpg)
Chris Calderon, PASI, Lecture 2
Cox Ingersoll Ross (CIR) Process
Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.
E[XT −Xt|Ft] = E[TRt
(a+ bXs)ds+TRt
σ√XsdBs|Ft]
E[XT −Xt|Ft] = E[TRt
(a+ bXs)ds|Ft] + 0
XT −Xt =TRt
(a+ bXs)ds+TRt
σ√XsdBs
Then Take Conditional Expectation
Brownian Increments are Mean Zero Martingales
![Page 17: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/17.jpg)
Chris Calderon, PASI, Lecture 2
Cox Ingersoll Ross (CIR) Process
Y (T ) = E[XT |Ft]
Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.
dXt = (a+ bXt)dt+ σ√XtdBt
dY (T )dT = a+ bY (T )
Y (T ) = Y (t) exp(b(T − t)) + ab
¡exp(b(T − t))− 1
¢
E[XT −Xt|Ft] = E[TRt
(a+ bXs)ds|Ft] + 0
Define: Then Differentiate Deterministic Function Above
![Page 18: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/18.jpg)
Chris Calderon, PASI, Lecture 2
Cox Ingersoll Ross (CIR) Process
Y (T ) = E[XT |Ft]
Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.
dXt = (a+ bXt)dt+ σ√XtdBt
dY (T )dT = a+ bY (T )
Y (T ) = Y (t) exp(b(T − t)) + ab
¡exp(b(T − t))− 1
¢
Conditional Independence of B.M and Ordinary Differential Equation Results (Integrating Factor) Provides Conditional Mean
![Page 19: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/19.jpg)
Chris Calderon, PASI, Lecture 2
Cox Ingersoll Ross (CIR) Process
Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.
dXt = (a+ bXt)dt+ σ√XtdBt
Conditional Independence of B.M and Ordinary Differential Equation Results (Integrating Factor) Provides Conditional Mean
Conditional Variance Can Be Readily Found by Ito Formula and Well-Known Statistical Identity
Z(T ) = E[X2T |Ft] Y (T ) = E[XT |Ft]
V AR[XT |Ft] = Z(T )−¡Y (T )
¢2
![Page 20: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/20.jpg)
Chris Calderon, PASI, Lecture 2
Cox Ingersoll Ross (CIR) Process dX2
t = 2XtdXt + σ2Xtdt
Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.
Rewrite in Integrated Form, Taking Conditional Expectations and Then Differentiating Again Yields:
dZ(T )dT = (2a+ σ2)Y (T ) + 2bZ(T )
dX2t = 2(aXt + bX
2t )dt+ 2σX
3/2t dBt + σ2Xtdt
[X,X]t = σ2Xtf(X) = X2
![Page 21: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/21.jpg)
Chris Calderon, PASI, Lecture 2
Useful for Method of Moments or Martingale Estimating Equations
Bibby, B. & Sorensen, M. (1995) Bernoulli 1, 17-39.
Approach Allows Estimation and Useful Approximations (Can Handle Cases Where MLE Is Problematic [Hyperbollic Diffusion])
Though Asymptotic Variance Maybe Inefficient
Bigger Problem Lies with Assessing Various Model Assumptions via Goodness-of-Fit (Return to This Point Later….)
bt=κXt√1+Xt
![Page 22: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/22.jpg)
Chris Calderon, PASI, Lecture 2
For Likelihood Based Inference One Requires Transition Density
Maximum Likelihood is Useful for Asymptotically Efficient Inference
However Assessing the Validity of “Independent Increments” Assumption Benefits from Using Information Implied by SDE
Transition Density Can Also Be Obtained by Ito’s Formula (Using Backward and Forward Kolmogorov Equations)
![Page 23: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/23.jpg)
Chris Calderon, PASI, Lecture 2
Ito’s Formula and PDEs
An Associated Cauchy PDE:
A[f(X)] = b(X)∂f(X)∂X + σ2(X) 12∂2f(X)∂2X
∂u(x,t)dt
= A[u(x, t)] ∀t > 0u(x, 0) = f(x)
Can Solve Deterministic Problem By Taking Expectation Over SDE Paths (See Board)
dXt = b(Xt)dt+ σ(Xt)dBt
![Page 24: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/24.jpg)
Chris Calderon, PASI, Lecture 2
PDEs Help Characterize SDEs
An Associated Cauchy PDE:
A[f(X)] = b(X)∂f(X)∂X + σ2(X) 12∂2f(X)∂2X
∂u(x,t)dt
= A[u(x, t)] ∀t > 0u(x, 0) = f(x)
If PDE Solvable, Can Compute “Pay-Off” Functions. Smooth Adjoint Solutions Yield Fokker Planck Equation [Useful in Likelihood Based Inference].
dXt = b(Xt)dt+ σ(Xt)dBt
p(XT |Xt)
![Page 25: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/25.jpg)
Chris Calderon, PASI, Lecture 2
Though Most PDEs Rarely Admit Closed-Form Solutions
An Associated Cauchy PDE:
A[f(X)] = b(X)∂f(X)∂X + σ2(X) 12∂2f(X)∂2X
∂u(x,t)dt
= A[u(x, t)] ∀t > 0u(x, 0) = f(x)
However the Generator Above Can Be Used To Help Approximate PDE Solutions [More Later (Maybe)…..]
dXt = b(Xt)dt+ σ(Xt)dBt
![Page 26: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/26.jpg)
Chris Calderon, PASI, Lecture 2
dXt = κ(α−Xt)dt+ σdBt
Assume Ornstein Uhlenbeck Process Fits Data
Data Generating Process One (Discrete Version of Assumed Model)
Xi+1 = FXi + σoZiData Generating Process Two (Discrete Misspecified Model)
Xi+1 = FXi + σoZiZi−1Zi iid Normal RandomVariables
Testing Validity of An Implicit Levy Process Assumption
![Page 27: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/27.jpg)
Chris Calderon, PASI, Lecture 2
Testing Validity of An Implicit Levy Process Assumption
dXt = κ(α−Xt)dt+ σdBt
Assume Ornstein Uhlenbeck Process Fits Data
Data Generating Process Two (Discrete Misspecified Model)
Xi+1 = FXi + σoZiZi−1
Zi iid Normal RandomVariables.Romano, J. & Thombs, L. (1996) JASA 91, 590-600.
“Noise”Increments Uncorrelated but Not Serially Independent
![Page 28: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/28.jpg)
Chris Calderon, PASI, Lecture 2
Testing Validity of An Implicit Levy Process Assumption
dXt = κ(α−Xt)dt+ σdBt
Assume Ornstein Uhlenbeck Process Fits Data
Data Generating Process Two (Discrete Misspecified Model)
Xi+1 = FXi + σoZiZi−1
“Noise”Increments Uncorrelated but Not Serially Independent
φi := ZiZi−1
E[φji+1φki ]
![Page 29: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/29.jpg)
Chris Calderon, PASI, Lecture 2
Forcing Both the Square and Circular Peg Into The Circular Hole
Assume Ornstein Uhlenbeck Process Fits Data
Data Generating Process One (Discrete Version of Assumed Model)
Xi+1 = FXi + σoZiData Generating Process Two (Discrete Misspecified Model)
Xi+1 = FXi + σoZiZi−1 {Xi}Ni=1
{Xi}Ni=1
θ ≡ (α,κ,σ)
θ̂ ≡ maxθ p(X0; θ)p(X1|X0; θ) . . . p(XN |XN−1; θ)θ̂ ≡ maxθ p(X0; θ)p(X1|X0; θ) . . . p(XN |XN−1; θ)
![Page 30: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/30.jpg)
Chris Calderon, PASI, Lecture 2
Forcing Both the Square and Circular Peg Into The Circular Hole
Assume Ornstein Uhlenbeck Process Fits Data
{Xi}Ni=1
{Xi}Ni=1
θ ≡ (α,κ,σ)
θ̂ ≡ maxθ p(X0; θ)p(X1|X0; θ) . . . p(XN |XN−1; θ)θ̂ ≡ maxθ p(X0; θ)p(X1|X0; θ) . . . p(XN |XN−1; θ)
Simulate 100 Time Series Batches Using Common Noise Sequence and Find MLEs
![Page 31: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/31.jpg)
Chris Calderon, PASI, Lecture 2
Forcing Both the Square and Circular Peg Into The Circular Hole
Correlation (A Low Order Moment) In “Residuals” Seems Similar
1 2 3 4 5 6 7 8 9 10−0.2
0
0.2
0.4
0.6
0.8
1
(0.02,.23,.30)
Lag
AC
Z
1 2 3 4 5 6 7 8 9 10−0.2
0
0.2
0.4
0.6
0.8
1
(−0.03,.24,.30)
Lag
AC
Z
Xi+1 − Eθ̂[Xi+1|Xi]
![Page 32: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/32.jpg)
Chris Calderon, PASI, Lecture 2
Correlation (A Low Order Moment) In “Residuals” Seems Similar
Xi+1 − Eθ̂[Xi+1|Xi]
However if Transition Density (All Moments) Used Along with Tools From Lecture 4, the Misspecification Becomes Apparent
0 0.2 0.4 0.6 0.8 10
50
100
150
200
250
300
350
Z
freq
(Z)
(0.02,.23,.30)
0 0.2 0.4 0.6 0.8 10
50
100
150
200
250
300
350
Z
freq
(Z)
(−0.03,.24,.30)
![Page 33: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/33.jpg)
Chris Calderon, PASI, Lecture 2
The Problem: Transition Density Rarely Available in Closed-form
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
Many Approaches to Dealing with This Problem (Assess Performance Using Simulated Data of Known Data Generating Process)
![Page 34: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/34.jpg)
Chris Calderon, PASI, Lecture 2
The Problem: Transition Density Rarely Available in Closed-form
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
The Ait-Sahalia Method. Use Generator of Transformed Process to Approximate Transition Density
dXt = b(Xt)dt+ σ(Xt)dBt
dYt = μ(Yt)dt+ 1dBt
Use Ito to Get a new SDE
(assume well behaved and invertible)Y :=
XZ1
σ(u)du
![Page 35: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/35.jpg)
Chris Calderon, PASI, Lecture 2
The Problem: Transition Density Rarely Available in Closed-form
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
The Ait-Sahalia Method. Use Generator of TranformedProcess to Approximate Transition Density
dYt = μ(Yt)dt+ 1dBt
AY [f(Y )] = μ(Y )∂f(X)∂y + 12∂2f(Y )∂2Y
![Page 36: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/36.jpg)
Chris Calderon, PASI, Lecture 2
The Problem: Transition Density Rarely Available in Closed-form
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
The Ait-Sahalia Method. Use Generator of TranformedProcess to Approximate Transition Density
limδt→0
δ−(J+1)nE[f(Yt+δ)|Yt = yo]−
JPj=1
AjY [f(yo)] δj
j!
o=
AJY [f(yo)](J+1)!
Various Methods for Exploiting This and Other Related Expansions for Inference
![Page 37: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/37.jpg)
Chris Calderon, PASI, Lecture 2
The Problem: Transition Density Rarely Available in Closed-form
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
pZ(δt, z|yo; θ) ≈
φ(z)
KXj=0
η(j)Z (δt, yo; θ)Hj(z)
The Ait-Sahalia Method. Use Generator of Transformed Process to Approximate Transition Density
![Page 38: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/38.jpg)
Chris Calderon, PASI, Lecture 2
The Problem: Transition Density Rarely Available in Closed-form
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
The Ait-Sahalia Method. Use Generator of Transformed Process to Approximate Transition Density
η(j)Z (δt, yo; θ) ≡
1
j!
∞Z−∞
Hj(z)pZ(δt, z|yo; θ)dz :=
1
j!E[Hj
³δt−
12 (Yt+δt − yo)
´|Yt = yo; θ].
![Page 39: Lecture II: Ito’s Formula and Its Uses in Statistical ...jrojo/PASI/lectures/CALDERON Lec2.pdf · Chris Calderon, PASI, Lecture 2 Useful for Method of Moments or Martingale Estimating](https://reader035.vdocuments.us/reader035/viewer/2022071000/5fbcb92e0ebcd9064e05c34a/html5/thumbnails/39.jpg)
Chris Calderon, PASI, Lecture 2
The Problem: Transition Density Rarely Available in Closed-form
Prakasa Rao, (1999) Statistical Inference for Diffusion Type Processes, Arnold, London.
Highly Recommended to Test (Both Estimation and Inference Results)All Approximations Using Controlled Data.
Can Give Negative Densities and/or Gradients of Log Likelihood Can Act Funny In Points
Calderon (2007). SIAM Mult. Mod. & Sim. 6.