Cosmin Crucean: Perturbative QED on de Sitter Universe.SEENET-MTP
Lecture by dr Cosmin Crucean (Theoretical and Applied Physics, West University of Timisoara, Romania) on July 9, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
Parameter Estimation in Stochastic Differential Equations by Continuous Optim...SSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 8.
More info at http://summerschool.ssa.org.ua
Cosmin Crucean: Perturbative QED on de Sitter Universe.SEENET-MTP
Lecture by dr Cosmin Crucean (Theoretical and Applied Physics, West University of Timisoara, Romania) on July 9, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
Parameter Estimation in Stochastic Differential Equations by Continuous Optim...SSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 8.
More info at http://summerschool.ssa.org.ua
This presentation is the result of a one-week student group work during the Southerm-Summer School on Mathematical-Biology, held in São Paulo, BR, in January 2012, http://www.ictp-saifr.org/mathbio . As a follow-up of the subject pat of the group together with F. Lutscher (Univ. Ottawa) and R.M Coutinho (IFT-UNESP, Brazil) published a paper on ecological complexity on this subject, available at http://www.sciencedirect.com/science/article/pii/S1476945X12000773
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be an important problem. We develop an approach to the optimization of the discretization grids for finite-difference scheme. Using the suggested approach we are able to achieve the exponential convergence of the boundary Neumann- to-Dirichlet maps. It increases the convergence order without increasing the stencil size of the finite-difference scheme and preserves stability.
This presentation is the result of a one-week student group work during the Southerm-Summer School on Mathematical-Biology, held in São Paulo, BR, in January 2012, http://www.ictp-saifr.org/mathbio . As a follow-up of the subject pat of the group together with F. Lutscher (Univ. Ottawa) and R.M Coutinho (IFT-UNESP, Brazil) published a paper on ecological complexity on this subject, available at http://www.sciencedirect.com/science/article/pii/S1476945X12000773
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be an important problem. We develop an approach to the optimization of the discretization grids for finite-difference scheme. Using the suggested approach we are able to achieve the exponential convergence of the boundary Neumann- to-Dirichlet maps. It increases the convergence order without increasing the stencil size of the finite-difference scheme and preserves stability.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
1. 4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Prediction of Financial Processes
Parameter Estimation in
Stochastic Differential Equations
by Continuous Optimization
Gerhard-
Gerhard-Wilhelm Weber *
Vefa Gafarova, Nüket Erbil, Cem Ali Gökçen, Azer Kerimov
Institute of Applied Mathematics
Middle East Technical University, Ankara, Turkey
* Faculty of Economics, Management and Law, University of Siegen, Germany
Center for Research on Optimization and Control, University of Aveiro, Portugal
Pakize Taylan Dept. Mathematics, Dicle University, Diyarbakır, Turkey
2. Outline
• Stochastic Differential Equations
• Parameter Estimation
• Various Statistical Models
• C-MARS
• Accuracy vs. Stability
• Tikhonov Regularization
• Conic Quadratic Programming
• Nonlinear Regression
• Portfolio Optimization
• Outlook and Conclusion
4. Stochastic Differential Equations
dX t = a ( X t , t )dt + b( X t , t )dWt
drift and diffusion term
Wt N (0, t ) (t ∈ [0, T ])
Wiener process
5. Stochastic Differential Equations
dX t = a ( X t , t )dt + b( X t , t )dWt
drift and diffusion term
Ex.: price, wealth, interest rate, volatility
processes
Wt N (0, t ) (t ∈ [0, T ])
Wiener process
6. Regression
X = ( X1 , X 2 ,..., X m ) and output variable Y ;
T
Input vector
linear regression :
m
Y = E (Y X 1 ,..., X m ) + ε = β0 + ∑ X j β j + ε ,
j =1
β = ( β 0 , β1 ,..., β m ) which minimizes
T
2
( )
N
RSS ( β ) := ∑ yi − x β T
i
i =1
ˆ = ( X T X )−1 X T y ,
β
( )
−1
Cov( β) = X T X
ˆ σ2
7. Generalized Additive Models
( ) ( )
m
E Yi xi1 , xi 2 ,..., xi m = β0 + ∑ f j x i j
j =1
f j are estimated by a smoothing on a single coordinate.
Standard convention : ( )
E f j ( xij ) = 0 .
• Backfitting algorithm (Gauss-Seidel)
ri j = yi − β 0 − ∑ f k ( xik ) ,
ˆ
k≠ j
it “cycles” and iterates.
8. Generalized Additive Models
• Given data ( yi , xi ) (i = 1,2,...,N ),
• penalized residual sum of squares
2
N m m b
PRSS (β 0 , f1 ,..., f m ) : = ∑ yi − β 0 − ∑ f j ( xij ) + ∑ µ j ∫
2
f j'' (t j ) dt j
i =1 j =1 j =1 a
µ j ≥ 0.
• New estimation methods for additive model with CQP :
9. Generalized Additive Models
min t
t , β0 , f
2
N m
subject to ∑
i=1
yi − β0 − ∑ f j ( xij ) ≤ t 2 , t ≥ 0,
j =1
2
∫ f j (t j ) dt j ≤ M j (j = 1, 2,..., m),
''
dj
splines: f j ( x) = ∑ θl j hl j ( x).
l =1
By discretizing, we get
min t
t , β0 , f
W ( β 0 , θ ) 2 ≤ t 2 , t ≥ 0,
2
subject to
2
V j ( β0 ,θ ) ≤ M j (j = 1,..., m).
2
10. Generalized Additive Models
min t
t , β0 , f
2
N m
subject to ∑
i=1
yi − β0 − ∑ f j ( xij ) ≤ t 2 , t ≥ 0,
j =1
2
∫ f j (t j ) dt j ≤ M j (j = 1, 2,..., m),
''
dj
splines: f j ( x) = ∑ θl j hl j ( x).
l =1
By discretizing, we get
min t
t , β0 , f
W ( β 0 , θ ) 2 ≤ t 2 , t ≥ 0,
2
subject to
2
V j ( β0 ,θ ) ≤ M j (j = 1,..., m).
2
11. Generalized Additive Models
min t
t , β0 , f
2
N m
subject to ∑
i=1
yi − β0 − ∑ f j ( xij ) ≤ t 2 , t ≥ 0,
j =1
2
∫ f j (t j ) dt j ≤ M j (j = 1, 2,..., m),
''
dj
splines: f j ( x) = ∑ θl j hl j ( x).
l =1
By discretizing, we get
min t
t , β0 , f
W ( β 0 , θ ) 2 ≤ t 2 , t ≥ 0,
2
subject to
2
V j ( β0 ,θ ) ≤ M j (j = 1,..., m).
2
13. MARS
y y
• • • • • •
• • • •
• • • • • • • •
• • • • • •
• • • • • •
• • •• • •
• • ••
c-(x,τ)=[−(x−τ)]+ c+(x,τ)=[+(x−τ)]+ c-(x,τ)=[−(x−τ)]+ c+(x,egressionx−τ)]+
rτ)=[+( w ith
τ x
τ x
14. C-MARS
N M max 2
∑( y − f (x ) ) + ∑ µ ∑ ∑
2
θ Drα, sψ m (t m ) d t m
∫
2
PRSS := i i m
2
m
i =1 m =1 α =1 r <s
α = (α1 ,α 2 ) r , s∈V ( m )
Tradeoff between both accuracy and complexity.
{
V (m) := κ m | j = 1, 2,..., K m
j } ( )
Drα, sψ m (t m ) := ∂αψ m ∂α1 trm ∂α 2 tsm (t m )
t m := (tm1 , tm2 ,..., tm K )T
m
α = (α1 , α 2 )
α := α1 + α 2 , where α1 , α 2 ∈{0,1}
15. C-MARS
Tikhonov regularization:
2
PRSS = y −ψ (d ) θ + µ Lθ
2
2 2
Lθ 2
Conic quadratic programming:
y − ψ (d ) θ 2
min t,
t ,θ
subject to ψ (d ) θ − y 2 ≤ t ,
Lθ 2
≤ M
16. C-MARS
Tikhonov regularization:
2
PRSS = y −ψ (d ) θ + µ Lθ
2
2 2
Lθ 2
Conic quadratic programming:
y − ψ (d ) θ 2
min t,
t ,θ
subject to ψ (d ) θ − y 2 ≤ t ,
Lθ 2
≤ M
18. Stochastic Differential Equations Revisited
dX t = a ( X t , t )dt + b( X t , t )dWt
drift and diffusion term
Ex.: price, wealth, interest rate, volatility,
processes
Wt N (0, t ) (t ∈ [0, T ])
Wiener process
19. Stochastic Differential Equations
dX t = a ( X t , t )dt + b( X t , t )dWt
drift and diffusion term
Ex.: bioinformatics, biotechnology
(fermentation, population dynamics)
Universiti Teknologi Malaysia
Wt N (0, t ) (t ∈ [0, T ])
Wiener process
20. Stochastic Differential Equations Revisited
dX t = a ( X t , t )dt + b( X t , t )dWt
drift and diffusion term
Ex.: price, wealth, interest rate, volatility,
processes
Wt N (0, t ) (t ∈ [0, T ])
Wiener process
21. Stochastic Differential Equations
Milstein Scheme :
ˆ ˆ ˆ 1
2
ˆ (
X j +1 = X j + a ( X j , t j )(t j +1 − t j ) + b( X j , t j )(W j +1 − W j ) + (b′b)( X j , t j ) (W j +1 − W j ) 2 − (t j +1 − t j )
ˆ )
and, based on our finitely many data:
& ∆W j ( ∆W j ) 2
X j = a ( X j , t j ) + b( X j , t j ) + 1 2 (b ′b)( X j , t j ) − 1 .
hj hj
22. Stochastic Differential Equations
• step length h j = t j +1 − t j := ∆ t j
X j +1 − X j
, if j = 1, 2,..., N − 1
& hj
X j :=
X N − X N −1 , if j = N
hN
• Wt N (0, t ), ∆W j (independent), Var( ∆W j ) = ∆ t j
• ∆W j = Z j ∆ t j , Zj N (0,1)
( )
& Zj 1
X j = a ( X j , t j ) + b( X j , t j ) + (b′b)( X j , t j ) Z j2 − 1
hj 2
23. Stochastic Differential Equations
• More simple form:
X j = G j + H j c j + ( H j ′ H j )d j ,
&
where
G j := a( X j , t j ) , H j := b( X j , t j ),
c j := Z j hj , (
d j :=1 2 Z j2 − 1 . )
• Our problem:
∑( )
N 2
min X j − (G j + H j c j + ( H j′ H j )d j )
&
y 2
j =1
y is a vector which comprises a subset of all the parameters.
24. Stochastic Differential Equations
g
2 2 dp
G j = a( X j , t j ) = α 0 + ∑ f p (U j , p ) = α 0 + ∑∑ α lp B p (U j , p )
l
p =1 p =1 l =1
2 2 d rh
H j c j = b( X j , t j )c j = β 0 + ∑ g r (U j ,r ) = β 0 + ∑∑ β rm Crm (U j ,r )
r =1 r =1 m =1
2 2 d sf
Fj d j = b′b( X j , t j )d j = ϕ0 + ∑ hs (U j , s ) = ϕ0 + ∑∑ ϕ sn Dsn (U j , s )
s =1 s =1 n =1
where
U j = (U j ,1 , U j ,2 ) := ( X j , t j ) ;
• k th order base spline Bη ,k : a polynomial of degree k − 1, with knots, say x η ,
1, xη ≤ x < xη +1
Bη ,1 ( x) =
0, otherwise
x − xη xη + k − x
Bη ,k ( x) = Bη ,k −1 ( x) + Bη +1,k −1 ( x)
xη + k −1 − xη xη + k − xη +1
25. Stochastic Differential Equations
• penalized sum of squares PRRS
∑{ Xj ( j j j) }
N 2
& − G + H c + F d 2 + λ f ′′(U ) 2 dU
PRSS (θ , f , g , h) : =
j =1
j j ∑ p∫ p p pp =1
2 2
+ ∑ µr ∫ [ gr (U r )] dU r +∑ϕs ∫ [ hs′′(U s )] dU s
′′
2 2
r =1 s =1
bκ
• λ p , µ r , ϕ s ≥ 0 (smoothing parameters), ∫ = ∫ (κ = p, r , s )
aκ
• large values of λ p , µ r , ϕ s yield smoother curves,
smaller ones allow more fluctuation
∑{ X j − ( G j + H j c j + Fj d j ) }
N 2
& =
j =1
2
N
& 2 dp
h
2 dr
g
2 ds
f
∑ X j − α 0 + ∑∑ α p Bp (U j , p ) + β0 + ∑∑1 βr Cr (U j ,r ) + ϕ0 + ∑∑ ϕs Ds (U j ,s )
j =1
l l m m n n
p =1 l =1 r =1 m = s =1 n =1
26. Stochastic Differential Equations
θ = (α , β , ϕ ) ( ) ( )
T T g T
, α = α0 ,α ,α α p = α , α ,..., α ( p = 1, 2),
T T T T T 1 2 dp
1 2 , p p p
β = ( β0 , β , β ) ( )
T
T T
T
1 2 , β r = β , β ,..., β
1
r
2
r
d rh
r (r = 1, 2),
(
ϕ = (ϕ0 , ϕ1T , ϕ 2 ) , ϕ s = ϕ s , ϕ s2 ,..., ϕ sd )
T T
( s = 1, 2).
f
T 1 s
∑{ } ( )
N T
• Then, &
X j − Ajθ
2
& − Aθ 2 .
= X A = A1T , A2 ,..., AN
T T
( )
j =1 2 T
& & & &
X = X 1 , X 2 ,..., X N
• Furthermore,
b 2 N −1 2
∫ f p′′ (U p ) dU p ≅
a
∑ f p′′ (U jp ) (U j +1, p − U jp )
j =1
2
dp l l
g
N −1
= ∑ ∑ α p B p′′ (U jP )u j .
j =1 l =1
27. Appendix Stochastic Differential Equations
b 2 N −1 2
∫ f p′′ (U p ) dU p ≅ ∑ B j ′′u jα p = AP α p
2
p B
( p = 1, 2)
a
j =1
2
( )
T
Ap := B1p′′T u1 , B2p′′T u2 ,..., BN −1′′T u N −1
B p
u j := U j +1, p − U j , p ( j = 1, 2,..., N − 1).
b N −1 2
[ gr′′(U r )] dU r ≅ ∑ C rj ′′v j β r = ArC β r
2
∫ (r = 1, 2)
2
a j =1
2
( )
T
ArC := C1r′′T v1 , C2 ′′T v2 ,..., CN −1′′T vN −1
r r
v j := U j +1,r − U j ,r ( j = 1, 2,..., N − 1).
b 2 N −1 2
h ′′ (U ) dU ≅ D s′′ w ϕ = A Dϕ
∫ s s s ∑ j j s
2
s s ( s = 1, 2)
2
a j =1
( )
T
A := D ′′ w1 , D2 ′′T w2 ,..., DN −1′′T wN −1
s
D s
1
s T s
w j := U j +1, s − U j , s ( j = 1, 2,..., N − 1).
28. Stochastic Differential Equations
2 2 2
& − Aθ 2 + λ A Bα 2 + µ AC β
∑ p p p ∑ r r r + ∑ ϕ s AsDϕ s
2 2
PRSS (θ , f , g , h) = X
2 2 2 2
p =1 r =1 s =1
Let us assume that λ p = µr = ϕ s =: µ = δ :
2
•
2
&
PRSS (θ , f , g , h) ≈ X − Aθ + δ 2 Lθ 2 ,
2
2
where L is a 6( N − 1) × m matrix:
0 A1B 0 0 0 0 0 0 0
0 0 A2B 0 0 0 0 0 0
0
θ = (α T , β T , ϕ T ) .
T
0 0 0 A1C 0 0 0 0
L := ,
0 0 0 0 0 A2C 0 0 0
0 0 0 0 0 0 0 A1D 0
0 0 0 0 0 0 0 0 A2D
29. Stochastic Differential Equations
2
min &
X − Aθ + µ Lθ
2
θ 2
2
Tikhonov regularization
min t,
t ,θ
subject to &
Aθ − X ≤ t,
2
Lθ 2
≤ M
Conic quadratic programming
30. Stochastic Differential Equations
min t
t ,θ
0N A t −X
&
subject to χ := T
+ ,
0m
θ 0
1 primal problem
06( N −1) L t 06( N −1)
η := + ,
0 0T θ M
m
χ ∈ LN +1 , η ∈ L6( N −1)+1
{
LN + 1 := x = ( x1 , x2 ,..., xN )T ∈ R N +1 | xN+1 ≥ x12 + x2 + ... + xN
2 2
}
&
(
max ( X T , 0) κ 1 + 0T N −1) , − M κ 2
6( )
0T 1 0T N −1) 0 1
κ1 + T κ2 = ,
N 6(
subject to T dual problem
A 0m L 0m 0m
κ 1 ∈ LN +1 , κ 2 ∈ L6 ( N −1)+1
31. Stochastic Differential Equations
(t , θ , χ ,η , κ1 , κ 2 ) is a primal dual optimal solution if and only if
0N A t −X &
χ := T
+ ,
0m θ 0
1
06( N −1) L t 06( N −1)
η := +
0 0T θ M
m
0T 1 0T N −1) 0 1
κ1 + T κ2 =
N 6(
T
A 0m L 0m 0m
κ 1T χ = 0, κ 2 η = 0
T
κ 1 ∈ LN +1 , κ 2 ∈ L6( N −1)+1
χ ∈ LN +1 , η ∈ L6( N −1)+1.
32. Stochastic Differential Equations
Ex.:
dVt = (θtT ( µ − rt ) + rt )Vt dt − ct dt + θtT σVt dWt ,
drt = α ⋅ ( R − rt ) dt + σ t ⋅ rt τ ⋅ dWt ,
dX t = µ ( t , X t , Zt ) dt + σ ( t , X t , Zt ) dWt .
nonlinear regression
33. Nonlinear Regression
2
∑ j ( j )
N
min f ( β ) = d − g x ,β
j =1
N
=: ∑ f j2 ( β )
j =1
F ( β ) := ( f1 ( β ),..., f N ( β ) )
T
min f ( β ) = F T ( β ) F ( β )
34. Nonlinear Regression
β k +1 := β k + qk
• Gauss-Newton method :
∇F ( β )∇T F ( β )q = −∇F ( β ) F ( β )
• Levenberg-Marquardt method :
λ ≥0
( )
∇F ( β )∇T F (β ) + λ I p q = −∇F ( β ) F ( β )
35. Nonlinear Regression
alternative solution
min t,
t,q
subject to ( ∇F (β )∇ T
)
F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) )
2
≤ t , t ≥ 0,
|| Lq || 2 ≤ M
conic quadratic programming
36. Nonlinear Regression
alternative solution
min t,
t,q
subject to ( ∇F (β )∇ T
)
F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) )
2
≤ t , t ≥ 0,
|| Lq || 2 ≤ M
conic quadratic programming
interior point methods
37. Nonlinear Regression
alternative solution
min t,
t,q
subject to ( ∇F (β )∇ T
)
F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) )
2
≤ t , t ≥ 0,
|| Lq || 2 ≤ M
1
min Q(q) := f ( β ) + qT ∇F ( β ) F ( β ) + qT ∇F ( β )∇T F ( β )q
q 2
subject to q 2 ≤∆
trust region
38. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
or stochastic control
39. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Parameter Estimation
Optimization Problem
Representation Problem
or stochastic control
40. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
Parameter Estimation
or stochastic control
41. Portfolio Optimization
max utility ! or
min costs !
martingale method:
Optimization Problem
Representation Problem
Parameter Estimation
or stochastic control
42. References
Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004.
Boyd, S., and Vandenberghe, L., Convex Optimization, Cambridge University Press, 2004.
Buja, A., Hastie, T., and Tibshirani, R., Linear smoothers and additive models, The Ann. Stat. 17, 2 (1989)
453-510.
Fox, J., Nonparametric regression, Appendix to an R and S-Plus Companion to Applied Regression,
Sage Publications, 2002.
Friedman, J.H., Multivariate adaptive regression splines, Annals of Statistics 19, 1 (1991) 1-141.
Friedman, J.H., and Stuetzle, W., Projection pursuit regression, J. Amer. Statist Assoc. 76 (1981) 817-823.
Hastie, T., and Tibshirani, R., Generalized additive models, Statist. Science 1, 3 (1986) 297-310.
Hastie, T., and Tibshirani, R., Generalized additive models: some applications, J. Amer. Statist. Assoc.
82, 398 (1987) 371-386.
Hastie, T., Tibshirani, R., and Friedman, J.H., The Element of Statistical Learning, Springer, 2001.
Hastie, T.J., and Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990.
Kloeden, P.E, Platen, E., and Schurz, H., Numerical Solution of SDE Through Computer Experiments,
Springer Verlag, New York, 1994.
Korn, R., and Korn, E., Options Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics,
Oxford University Press, 2001.
Nash, G., and Sofer, A., Linear and Nonlinear Programming, McGraw-Hill, New York, 1996.
Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology (2002).
43. References
Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005).
Nesterov, Y.E , and Nemirovskii, A.S., Interior Point Methods in Convex Programming, SIAM, 1993.
Önalan, Ö., Martingale measures for NIG Lévy processes with applications to mathematical finance,
presentation in: Advanced Mathematical Methods for Finance, Side, Antalya, Turkey, April 26-29, 2006.
Taylan, P., Weber G.-W., and Kropat, E., Approximation of stochastic differential equations by additive
models using splines and conic programming, International Journal of Computing Anticipatory Systems 21
(2008) 341-352.
Taylan, P., Weber, G.-W., and A. Beck, New approaches to regression by generalized additive models
and continuous optimization for modern applications in finance, science and techology, in the special issue
in honour of Prof. Dr. Alexander Rubinov, of Optimization 56, 5-6 (2007) 1-24.
Taylan, P., Weber, G.-W., and Yerlikaya, F., A new approach to multivariate adaptive regression spline
by using Tikhonov regularization and continuous optimization, to appear in TOP, Selected Papers at the
Occasion of 20th EURO Mini Conference (Neringa, Lithuania, May 20-23, 2008) 317- 322.
Seydel, R., Tools for Computational Finance, Springer, Universitext, 2004.
Stone, C.J., Additive regression and other nonparametric models, Annals of Statistics 13, 2 (1985) 689-705.
Weber, G.-W., Taylan, P., Akteke-Öztürk, B., and Uğur, Ö., Mathematical and data mining contributions
dynamics and optimization of gene-environment networks, in the special issue Organization in Matter
from Quarks to Proteins of Electronic Journal of Theoretical Physics.
Weber, G.-W., Taylan, P., Yıldırak, K., and Görgülü, Z.K., Financial regression and organization, to appear
in the Special Issue on Optimization in Finance, of DCDIS-B (Dynamics of Continuous, Discrete and
Impulsive Systems (Series B)).