Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 An Introduction T The Ito Calculus And
o
Stochastic Differential Equations
Recall the earlier discussion about Ito and Stratonovich
integrals that arose in discussing the Langevin equation.
The key problem was when we had internal, ws.
external, noise t
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 An Introduction T Stochastic Processes And
o
Related Subjects
1.1 A Road Map to T
opics
Stochastic Processes
Markov
Processes
Martingales
Brownian Motion
Diffusion Processes
Ito Processes
Markov
Chains
Branching
Processes
Renewal
Processes
Stochastic
D
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 An Introduction T Stochastic Calculus
o
You are used to :
A) Deterministic differential equations:
xt = f (xt, t)
 we have discussed their solutions, their qualitative
properties.
B) Deterministic maps
xt+1 = f (xt, t)
and their qualitative properties
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 Nonlinear Time Series, Stochastic Processes
And Dynamical Systems: Some Important
Relationships Explored
Basic Outline:
Nonlinear differential equations
Discretization
Difference Equations
Time Series Models
Diffusion Process
Main References:
T Ozaki
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 Introduction T The Markov Process: Based on
o
Van Kapen up to the FokkesPlack Equation
The components of Fn may be complex or:
depends on the imaginary
phase of the component part of Fn , Fn .
%
c
Fn = jFn je i ,
j j
Fn = jFn je i
j jd
z
cfw_
z
cf
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 An Introduction T Markov Chains
o
We are dealing with a discrete state space and time is
discrete called a discrete parameter process. The
process is Markovian, so that:
prob (Xn+1 = k  X0 = h,. Xn = j )
= prob (Xn+1 = k  Xn = j )
is the transition p
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 Estimation of Spectra
There are two parts to the discussion
A) Basic (asymptotic) results
B) Practical considerations for reducing bias and
variance of estimates.
The estimates of spectra as you might guess de
pend on whether one is estimating discret
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 Some Supplemental Notes On The Markov And
Diffusion Process: Feb/25/95
These notes are based on Stochastic Models in Biology by N.S. Goel and N. RichterDyn. They will supplement the class notes that you already have. They
cover essentially the same gro
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 An Introduction T Differential Equations Leading
o
T The Kdv Equations
o
Begin with ordinary differential equations, end with p.d.e.s.
We summarize the linear ordinary differential equation topics you have already had.
a) u au = 0
(1st Order Equation)
u
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
Abstract
1 Cross Spectral Analysis
References.
Brockwell & Davis
Bloomfield
Granger + Hatanaka
Recall the spectral/A: : relationship for a single
C F:
r:
v:
1
1X
h
e i ! h )
(
f (! ) =
2
h=
1
!
is the auo c
()
t orrelaton fu ton h ) =
i
nc
i
(
f (is t
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 Some Notes On Continuity And Differentiability
1.1 Lebesgue Decomposition Theorem
Any probability distribution function, F (x), may be written in the form:
F (x) = 1F1 (x) + 2 F2 (x) + 3F3 (x)
f 1, 2 , 3g ! 0 i 1, 1 + 2 + 3 = 1
F1 (x) is absolutely cont
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 Complex Demodulation
We consider the following nonstationary, but deterministic problem.
X t= R tc os(t+ t
)
R tis a slowly changing
amplitude (relative to the
1
period )
t is a slowly changing phase, relative to the pe1
riod .
The idea of complex dem
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 A Detailed Algebraic Development Of The
Chapman Kolmogorov Equation And Its Ancillary
Equations
Reference: Gardiner: Handbook of Stochastic methods, SpringerVerlay.
Notation will be modified slightly in interests of conformity to previous results.
We
Nonlinear Dynamics and Stochastic Differential Equations
ECON GA 3105

Spring 2010
1 The Center Manifold Theorem, Dimension Of
Limit Sets, And Relaxation Oscillations
The definition of manifolds is important, since most dynamical systems live on manifolds, not in Euclidean
n space.
By combining the ideas of local linearty with global
no