Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles

Impulsive and Stochastic Hybrid Systems in the Financial Sector

with Bubbles

Gerhard-Wilhelm Weber 1*, Büşra Temoçin 1, 2, Azer Kerimov 1, Diogo Pinheiro 3, Nuno Azevedo 3

1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 2 Department of Statistics, Ankara University, Ankara, Turkey 3 CEMAPRE, ISEG – Technical University of Lisbon, Lisbon, Portugal

* Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal Universiti Teknologi Malaysia, Skudai, Malaysia Advisor to EURO Conferences

6th International Summer School

National University of Technology of the Ukraine Kiev, Ukraine, August 8-20, 2011

http://www.ntu-kpi.kiev.ua/

http://www.iam.metu.edu.tr/EUROPT/index.html

• Stock Price Dynamics

• Lévy Processes

• Bubbles, Jumps and Insiders

• Stochastic Control of Stochastic Hybrid Systems I

• Merton’s Optimal Consumption – Investment Problem

• Hamilton-Jacobi-Bellman Equation

• Numerical Methods

• Stochastic Hybrid Systems II

• Conclusion

Outline


Stock Price Dynamics

d-fine, 2002

( , ) ( , ) t t t tdX a X t dt b X t dW



NIG Lévy Asset Price Model

Ö. Önalan, 2006

2 2

2 2 1

2 2

( ( ) )( ; , , , ) exp ( ) ,

( )NIG

K xf x x

x

where

22

1

0

( ) exp( ) .4 4

y yK y t t dt

t

Normal Inverse Gaussian Process (NIG) is the subclass of generalized

hyperbolic laws and has the following representation:



DAX Empirical and Simulation

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

Emp.

Sim.

NIG Lévy Asset Price Model

Ö. Önalan, 2006


Some basic Definition:

A process is said to be càdlàg (“continu à droite, limite à gauche”) or

RCLL (“right continuous with left limits”) is a function that is right-

continuous and has left limits in every point. Càdlàg functions are

important in the study of stochastic processes that admit (or even require)

jumps, unlike Brownian motion, which has continuous sample paths.

A process is adapted if and only if, for every realisation and every n,

this process is known at time n.

Lévy Processes


Definition: A cadlag adapted processes

defined on a probability space (Ώ, F, P) is said to be a Lévy processes, if it possesses the following properties:

( | 0) tL L t

Lévy Processes


Lévy Processes

(ί )

(ίί ) For is independent of , i.e., has independent increments.

(ίίί) For any is equal in distribution to (the distribution of does not depend on t ); has stationary increments.

(ίv) For every , , i.e., is stochastically continuous.

In the presence of ( ί ), ( ίί ), ( ίίί ), this is equivalent to the condition

0 0 1. P L

st LLts ,0sF

L

0 , t ss t L LstL

sst LL

L

, 0 0 s t and 0lim

st

tsLLP

L

0

lim 0.

tt

P L


There is strong interplay between Lévy processes and infinitely divisible distributions.

Proposition: If is a Lévy processes, then is infinitely divisible for each .

Proof : For any and any :

Together with the stationarity and independence of increments we conclude that the random variable is infinitely divisible.

L tL

0tI n

0t

2 ( 1)... . t t n t n t n t n t nL L L L L L

tL

Lévy Processes


Moreover, for all and all we define

hence, for rational t > 0: .

Ru 0t

( ) log ; ti u L

tΨ u E e

1( ) ( ) ttΨ u Ψ u

Lévy Processes


For every Lévy process, the following property holds:

1 ( ) ti u L tΨ uE e e

2

1exp 1 1 ( )2

,

iux

xu c

t ibu e iux v dxR

Lévy Processes

where is the characteristic exponent of . 1( ) ( ) u u XL 1


The triplet is called the Lévy or characteristic triplet and

is called the Lévy or characteristic exponent,

where : drift term,

: diffusion coefficient and

: Lévy measure.

vcb ,,

dxvxuiecu

buiu xxui 1

2

112

)(

Rb

Rc

v

Lévy Processes


The Lévy measure is a measure on which satisfies

.

0R

2 1 ( ) x v dxR

This means that a Lévy measure has no mass at the origin, but infinitely many jumps can occur around the origin.

The Lévy measure describes the expected number of jumps of a certain size in a time in interval length 1.

Lévy Processes


The sums of all jumps smaller than some does not converge. However, the sum of the jumps compensated by their mean does converge. This pecularity leads to the necessity of the compansator term .

If the Lévy measure is of the form , then f (x) is called the Lévy density.

In the same way as the instant volatility describes the local uncertainty of a diffusion, the Lévy density describes the local uncertainity of a pure jump process.

The Lévy-Khintchine Formula allows us to study the distributional properties of a Lévy process.

0

11 x

iux

v dx f x dx

Lévy Processes


Another key concept, the Lévy-Ito Decomposition Theorem, allows one to describe the structure of a Lévy process sample path.Lévy-Ito Decomposition Theorem:Let be a Lévy process, the distribution of parametrized by

Then decomposes as , where is a Brownian motion, and is an independent Poisson point process with intensity measure , and is a martingale with jumps

Lévy Processes

X 1X 2( , , ).

X t t t tX t W J M tW( 0) t t t tX X X

11

st s Xs t

J X tM

1 .1 tt t XM X


Stochastic Control of Hybrid Systems

, 0,1,..., , where dX t t NR

[0, ):,

tX t t

• is a pure jump process taking values in ,

• is a switching jump-diffusion between jumps.

t

,X t t

M.K. Ghosh and A. Bagchi, 2004

Stochastic Hybrid Model I



0 0

for

( ) ( ( ), ( )) ( ( ), ( )) ( ) ( ( ), ( ), ) ( , ),

( ) ( ( ), ( ), ) ( , ),(1)

0,

(0) , (0) .

dX t X t t dt X t t dW t g X t t u N dt du

d t h X t t u N dt d

X

u

t

X

R

R

The asset price dynamics is given by the following system:

The price process switches from one jump-diffusion path to another as the discrete component moves from one step to another.

X t

t




is defined as : d Nh R R R

, if ,, ,

0, otherwise,

ijj i u xh x i u

: Poisson random measure,

: Poisson process on corresponding

to the given Poisson random measure,

: domain of

( )

( ) ( , , )

( ),

| ( ) ,

{ ( ), ( , ) | , ([0, ]), ( )}

N

N

N

N P

D N

D t D N t U

W s N A B s t A T B R

F

F B B




Under boundedness, measurability and Lipschitz conditions a pathwise unique solution of (1) exists.

Considering the SDE

0 0 0

0 0

( ) ( ( ), ) ( ( ) ( ),), t t

Y t X Y s ds Y s dW s

the unique solution in time interval is found as

1

1

1 1 0 1 1

0 1

1

0 1 0 1 1

( ), if 0 ,( )

( ) ( ( ), , ( )), if ,

, if 0 ,( )

( ( ), , ( )), if ,

Y t tX t

Y g Y N t

tt

h Y N t

11 1 0,

,t

X t t

10,



Merton’s Consumption Investment Problem

An investor with a finite lifetime must determine the amounts that will be consumed and the fraction of wealth that will be invested in a stock portfolio,

so as to maximize expected lifetime utility.

Assuming a relative consumption rate

the wealth process evolves with the following SDE:

tct

0

( , ) ( , )( ( ), ( )) ( ( ), ( )) ( ) ( ( ), ( )) ( ( ( ), ( )) ( )) ( ( ), ( ))

( ( ), ( )) ( ( ), ( )) ( ) ( ) ( ( ), ( ), ) ( , .)

c cd X X t t X X t t r t X t t X t t r t X t t dt

X t t X t t d B t t g X t t u N dt duR

( , )

( )( ) ,

( )c

c tt

X t



The objective is

Assuming a logarithmic utility function, which is iso-elastic, the optimization problem becomes

D. David and Y. Yolcu Okur, 2009

,

0( , )max ( , ) ( ( ), )

subject to ( ) 0, ( ) > 0 ( [0, ]), (0) =

T

csc

E U c s ds g X T T

c t X t t T X .

( ) ( ) ( , )

0( , )max ln ( ) ln ( ) .

Tt T c

cE e c t dt Ke X T



We assume a constant relative risk aversion (CRRA) utility function

where

Hence, the objective takes the following form

1

1, ,

x

U x t

,

,0

( [0, ])

max , ( ( ), ) (2)

subject to ( ) 0, ( ) > 0 , (0) =

T

s T cs

c

t T

E e U c s ds e U X T T

c t X t X .

0, \ 1 .


Hamilton-Jacobi-Bellman Equation

In applying the dynamic programming approach, we solve the Hamilton-Jacobi-Bellman (HJB) equation associated with the utility maximization problem (2).

From (W. Fleming and R. Rishel, 1975) we have that the corresponding HJB equation is given by

subject to the terminal conditionwhere the value function is given by

0,

2

, max , ,

1,

23

nt t

Tt t t t

c

T Tt t

J t U c J t J t r r c

J t

R

, ,J T u J

,

,

, sup , ( ( ), ) .

T

s T cs

c t

J t E e U c s ds e U X T T



In solving the HJB equation (3), the static optimization problem

can be handled separately to reduce the HJB equation (3) to a nonlinear partial differential equation of only.

Introducing the Lagrange function as

where is the Lagrange multiplier.

2

0,

1max , ,

2

nt t

T T Tt t t t t

cU c J t r r c J t

R

J

22

, , , , , ,

1, , , (4)

2

Tt t

Tt t t

L t c t t J t r r c

J t u c t c



The first-order necessary conditions with respect to and , respectively, of the static optimization problem with Lagrangean (4) are given by

, c

21, ... .

2 TtJ r J t

D. Akume and G.-W. Weber, 2010



Simultaneous resolution of these first-order conditions yields the optimal solutions and . Substituting these into (3) gives the PDE

with terminal condition

which can then be solved for the optimal value function

, opt optc opt

1

2

,, , , , ,

1

1, , , 0, (5)

2

optTopt opt

t

Topt T opt

c tJ t J t J t t r r c t

J t t t

1

1, ,

J T

, .optJ T



Because of the nonlinearity in and , the first-order conditions together with the HJB equation are a nonlinear system.

So the PDE equation (5) has no analytic solution and numerical methods such as Newton‘s method or Sequential Quadratic Programming (SQP) are required to solve for , , and iteratively.

opt optc

,opt t ,optc t ,opt t ,optJ t

D. Akume and G.-W. Weber, 2010


Bubbles, Jumps and Insiders


Notation:

0

0

( , , ), ( , ) ( , ),

( ) = ( , ) 0, ( , ) ( )( , ) ( , ) ( ) ,

:= ( ) | , [0, ] , := ( ) | ( (0, )), [0, ] ,

= ,

[0, ],

B B

t

B Nt t

B Nt t t

t t

P P P P

t z N dt dz N dt dz N v dt dz N dt dz v dz dt

B s s t t T N t t T

t T

R

R

F F

F

F N F B N

F F F

F FG

v v



0

0

( , , ), ( , ) ( , ),

( ) = ( , ) 0, ( , ) ( )( , ) ( , ) ( ) ,

:= ( ) | , [0, ] , := ( ) | ( (0, )), [0, ] ,

= ,

[0, ],

B B

t

B Nt t

B Nt t t

t t

P P P P

t z N dt dz N dt dz N v dt dz N dt dz v dz dt

B s s t t T N t t T

t T

R

R

F F

F

F N F B N

F F F

F FG

v v

Bubble!



0

00 0

0 0

0 1 01

0 0

( ) ( )( , ) ( ) lim ( , )

( , ) ( , ) lim ( , ) ( , )

( compact, ( IN),

( ) (0) ( )

, )

( ) ( ) ( , ) ( , ) [0, ]

n

nU

n n n F n nn

t t

B t B tt d B t

X t X s ds s d B s s z N

t dt

t N d t dz t z N dt dz

U U U n U U

d s dz t T

R

R R

0

0

0

( ) ( ) ( ) ( ) ( , ) ( , )

d X t t dt t d B t t z N d t dz

R

R

Forward Integrals

Forward Process




0

0

21

2( ) ( ( )) ( ) ( ( )) ( ) ( ) ( ( ) ( , ))

( ( )) ( ( )) ( , ) (

I

) ( ( )) ( ) ( )

( ( ) ( , )) (

f ( ( )

( )) ( ,

( ) :

.)

)

d Y t f ' X t t f '' X t t t f X t t z

f X t f ' X t t z dz dt f ' X t t d B t

f X t t z f ' X t N d t dz

Y t f X t

R

R

F

It Formula for Forward Integrals

o




0

0 0

0

1 1

1

( ) ( ) ( ) ,

(0) 1,

( ) ( ) ( ) ( ) ( ) ( , ) ( , ) ,

(0) 0,

dS t r t S t dt

S

dS t S t t dt t dB t t z N dt dz

S

R

0

( , ) ( )

( , )

,( ) ( ) ( ) ( ( ) ( )) ( ) ( ) ( ) ( )

( ) ( , ) ( ,

(0) .

, ) ( )

c c

c

d X t X t r t t r t t dt t t d B t

t t z N d t dz

X

c t dtR




0

( , ) (

( , )

, )( ) ( ) ( ) ( ) ( ( ) ( )) ( )

( ) ( ) ( ) (

( )If ( ) , then

( )

,) ( , ) ( , )

c

c cd X t X t r t t t r t t dt

t t d B t t t z N d t dz

c tt

X t

R

( ) ( , ) ( ) ( , )

( , ) 0

( *, *) : sup IE ln ( ) ( ) ln ( ) .

T

t c T cJ e t X t dt Ke X TA




( )

( ) ( )

*( ) is an optimal solution :

( *, ) ( , ) ( , ) .

Let : (( *, ).)

t

Ts T

t

et

e ds

J

Ke

J J

J

A




*

*

* *2 2

* *

( , ) 0, ( ) 0 for almost all ( , )

: optimal portfolios of the insider (generator of a ),

: optimal portfolios of the uninformed (honest) agent

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( )

bu

(

b

)

)

e

( (

bl

i

h

i h

i h

t z t t z

t r t t t r tt t

t t t

t

* *

( ) 2 ( )

( )

( ) ( )

* * 2

0 0 0

1 1IE ( ) I

) ,

( , ) ( , ) E ( )2 2

i h

t

Ts T

t

i

T TT

h h

t

it

et

e ds K

e s dsdt K ec d

e

s sJ J c

Brownian Motion Case



0 0 0

1 1

Assumptions :

( , ) , ( ) 0 for almost all ( , )

( ( ), ( )),

Uninformed agent has access to filtration ( [0, ]),

Lévy measure is given by ( ,

(1)

(2) ) = ( ) ( (

t t

t t t

t z z t t z

' B T T T T

' t T

ds dz dz ds dF F

G F

F G G

V

( )* * * * *

2 2( ) ( )

) : unit point mass at 1),

( ) is given by ( ) = ( ) ( : Poisson process of intensity ).

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) (

(3

)

1( ) ( ) ( )

2 ( )

)

t

i h h i h Ts T

t

z

t t Q t t Q

t t r t et t t t t

t t te ds Ke

t t r tt

2

2 2

0 0

0 0

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) 4 ( ) ( ),

IE ( ) ( ) | IE ( ) ( ) |( ) ( )

t t

t t t

t r t t t t t t

B T B t Q T Q tt t

T t T t

G G

Mixed Case



Mixed Case

* *

( ) 2

0 0

( )2

0 0

( )

*

2

0

* 1IE ( ) ( ) ( )

2

( ) ( )IE ln 1 ( , )

( ) ( ) ( )

1IE (

( ,

) ( )

) ( , )

( )2

IE

i h

T tt

T tt

i i h h

TT

J c J c e s s s dsdt

z s se ds dz dt

s s r s z

K e s s s ds

K e

0R

l lG

( )

20

( ) ( )ln 1 ( , )

( ) ( ) ( )

T

T z s sds dz

s s r s z0R

l lG



Stochastic Hybrid Model II

0 0

,

( ( ) | ( )

(

, ( ), ( ), ) ( ( )) O

) ( ( ), ( )) ( ( ), ( )) (

( ) ,

(0) , 0

)

( )

n n n

nijP t t j t i X

dX t b X t t dt X t t dW t

s s s t X t t t i j

X X




1

IN

0 1 2 ( )

{ ( ), ( , ) | , ([0, ]), ( )},

inf{ 0 | ( ) },

,

0 ... ..., ,

( ) , if ( ( ), ( ))

n

n

n

m m

n n

nt

nt t

m

W s p A B s t A t B

t X t A

t n X t t S

RF B B

F FV




10

20

( )

( ) ( ( ), ( ), ( )) ( ( ), ( ), ( )) ( ) ( )

( ( ), ( ), ( )) ( ),

( ) ( ( ), ( ), ( ) ) ( , )

( ( ), ( ), ( )) ( ) ( ) ,

( ) ( (

m m m m mm

m m m m mm

t

dX t b X t t t g X X t dt

X t t t dW t

d t h X t t t u p dt du

g X t dt

d t h X

R

0

), ( ), ( )) ( )

mm m m m

mtI


Conclusion

i. The emerging financial sector needs more and more advanced methods from mathematics and OR.

ii. NIG-Levy process mean a better approximation for the stock price process.

iii. Impulsive and hybrid behaviour is turning out to become very import in modelling and decision making in finance, but also in ecomomics, natural and social sciences and engineering.


References

1. J. Nocedal, and S. Wright, Martingale methods in financial modeling, Springer Verlag, Heidelberg, 1999.

2. W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, Berlin, 1975.

3. J. Amendinger, P. Imkeller and M. Schweizer, Additional logarithmic utility of an insider, Stochastic Processes and Their Applications, 75 (1998) 263–286.

4. D. Akume and G.-W. Weber, Risk-constrained dynamic portfolio Management, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 17 (2010) 113-129.

5. D. David and Y. Okur, Optimal consumption and portfolio for an Insider in a market with jumps, Communications on Stochastic Analysis Vol. 3, No. 1 (2009) 101-117.

6. Ö. Önalan, Martingale Measures for NIG Lévy Processes with Applications to Mathematical Finance, International Research Journal of Finance and EconomicsISSN 1450-2887 Issue 34 (2009).


References

7. D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004.

8. J. Bertoin, Lévy Processes, Cambridge University Press, 1996.

9. O.E. Barndorff-Nielsen, Normal Inverse Gaussian Processes and the Modelling of Stock Returns, Research Report ,Department Theoretical Statistics, Aarhus University. 1995.


Impulsive and Stochastic Hybrid Systems in the Financial Sector with Bubbles

Education

ankara university

university of aveiro

university of siegen

stochastic hybrid systems

malaysia advisor

ukraine kiev

nuno azevedo3

azer kerimov1