16 қатарлары түрінде жазып аламыз. Мұндағы,
Cөйлем 1. 10]. Келесі қасиеттер және ұғымдар семантикалық болып табылады
(1) тип;
(2) форкинг;
(3) -стабилділік;
(4) Ласкар рангі;
(5) Қатты тип;
(6) Морли тізбегі;
(7) Ортогональдылық, типтердің регулярлығы;
(8) I(,T)- спектр функциясы.
Осыдан Cөйлем 1- де көрсетілген барлық қасиеттер йонсондық теориялардың зерттеу аясында
семантикалық болады. Дәлелдеуі тривиалды және Сөйлем 1-де алынған негізгі түсініктердің йонсондық сәйкестіктерін қолдана отырып Сөйлем 1 – ден алынады.
Пайдаланылған әдебиеттер тізімі
1 Кейслер, Х. Дж. Основы теории модели / Кейслер Х. Дж. - Москва: Наука,1982. - 108 с.
2 Мустафин Т.Г., Нурмагамбетов Т.А. Введение в прикладную теорию моделей. – Караганда: Изд.
КарГУ, 1987. – 94 с.
3 Ешкеев А.Р., Касыметова М.Т. Йонсоновские теории и их классы моделей: монография. – Караганда: КарГУ, 2016. – 370 с.
4 Poizat B. A Course in Model Theory. Springer-Verlag New York, lnc. in 2000. – P. 445.
5 Биркгоф Г.,Барти Т. Современная прикладная алгебра. Издательство «Мир». Москва. 1976. – С.
400.
6 Ешкеев А.Р., Ульбрихт О.И. JSp-косемантичность и JSB свойство абелевых групп. Siberian Electronic Mathematical Reports, http://semr.math.nsc.ru, - Vol. 13(2016). - P.861-874.
7 Yeshkeyev A.R. On Jonsson stability and some of its generalizations// Journal of Mathematical Sciences, 2010. – Vol. 166. -– No 5. – P. 646-654.
8 Yeshkeyev A. R. The structure of lattices of positive existential formulae of (
- PJ)-theories //Scienceasia, Vol. 39, 2013. - P. 19-24 .9 Yeshkeyev A.R. The properties of central types with respect to enrichment by Jonsson set, 2017,.1 (85),36- 41.
10 Mustafin T.G. On similarities of complete theories / T.G.Mustafin // Logic Colloquium ’90. Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic. — Helsinki, 1990. — P. 259–265.
11 Yeshkeyev A.R. Jonsson’s Theories. Textbook, Karaganda State University, Karaganda, 2009.
12 Yeshkeyev A.R. The Properties of Positive Jonsson’s Theories and Their Models International Journal of Mathematics and Computation. – 2014. –Vol. 22.1. – P. 161-171.
ГРНТИ28.23.37 УДК 519.6
E.A. Kondakova1, O.I. Krivorotko2, S.I. Kabanikhin3, ZH.M. Bektemessov4
1,2,3
Novosibirsk state university; Institute of computational mathematics and mathematical geophysics of SB RAS, Novosibirsk, Russia,
4al-Farabi Kazakh national university, Almaty, Kazakhstan
THE OPTIMAL CONTROL METHOD FOR STOCHASTIC DIFFERENTIAL EQUATIONS IN FINANCE ECONOMY
Аңдатпа
Соңғы онжылдықта жаппай жаһандану нәтижесінде пайда болған экономикалық тәуекелдерді нақты бағалауға және инвестициялаудың оңтайлы стратегияларын іздестіруге деген қажеттілік пайда болды.
Осыған байланысты қазір стохастикалық дифференциалдық теңдеулер (СДУ) белсенді дамуда. Жұмыста Мертонның математикалық моделі қарастырылады (СДУ стандартты Винер процесімен). Басқару функциясын табудың кері міндеті зерттелуде. Ол үшін динамикалық бағдарламалау принципі және Гамильтон-Якоби-Беллман теңдеуінің шешімі қолданылады.
Түйін сөздер: стохастикалық дифференциалдық теңдеулер, оңтайлы басқару, Мертон міндеті, экономика, динамикалық бағдарламалау, Гамильтон-Якоби-Беллман теңдеуі.
Е.А. Кондакова1, О.И. Криворотко2, С.И. Кабанихин3, Ж.М. Bektemessov4
1,2,3 Новосибирский государственный университет;
Институт вычислительной математики и математической геофизики,
4Казахский национальный университет им. al-Farabi, Алматы, Казахстан МЕТОД ОПТИМАЛЬНОГО КОНТРОЛЯ ДЛЯ СТОХАСТИЧЕСКОГО ДИФФЕРЕНЦИАЛЬНОГОУРАВНЕНИЯ В ФИНАНСОВОЙ ЭКОНОМИКЕ
Аннотация
В последние десятилетия в результате повсеместной глобализации появилась потребность в точной оценке возникающих экономических рисков и поиске оптимальных стратегий инвестирования. В связи с этим сейчас активно развиваются стохастические дифференциальные уравнения (СДУ). В работе рассматривается математическая модель Мертона (СДУ со стандартным винеровским процессом).
Исследуется обратная задача нахождения функции управления. Для этого применяется принцип динамического программирования и решение уравнения Гамильтона-Якоби-Беллмана.
Ключевые слова: стохастические дифференциальные уравнения, оптимальное управление, задача Мертона, экономика, динамическое программирование, уравнение Гамильтона-Якоби-Беллмана.
Е.А. Кондакова1, О.И. Криворотко2, С.И. Кабанихин3, Ж.М. Бектемесов4
1,2,3 Новосібір мемлекеттік университеті;
Есептеу математикасы және математикалық геофизика институты,
4Қазақ ұлттық университеті. әл-Фараби, Алматы, Қазақстан
СТОХАСТИКАЛЫҚ ДИФФЕРЕНЦИАЛДЫ ОҢТАЙЛЫ БАСҚАРУ ӘДІСІ ҚАРЖЫ ЭКОНОМИКАДАҒЫ ТАЛАПТАР
Annotation
In recent decades, as a result of widespread globalization, there is a need for an accurate assessment of emerging economic risks and finding optimal investment strategies. In connection with this, stochastic differential equations (SDE) are now actively developing. In the paper, the mathematical model of Merton (SDE with a standard Wiener process) is considered. The inverse problem of finding the control function is investigated. For this, the principle of dynamic programming and the solution of the Hamilton-Jacobi-Bellman equation are applied.
Keywords: stochastic differential equations, optimal control, Merton problem, economy, dynamic programming, Hamilton-Jacobi-Bellman equation.
Mathematical models in the financial economy are divided into deterministic (systems of nonlinear ordinary differential equations (ODE) [1] and parabolic equations[2]), and also stochastic ones, which are described by systems of stochastic differential equations (SDE) [3]. The problems of controlling stochastic dynamical systems are widely encountered in practice and are the subject of deep mathematical research [4].
In the paper, inverse problems for the stochastic differential equation of the Merton problem with the Wiener process [5] are numerically investigated.Inverse problems consist in determination of the function on the right-hand side (control function) [6]. Merton's task is to model a financial market with two assets: risky asset-shares and risk- free asset-bonds:
. ) ( )
( )
( ) (
, ) ( ) ( )
( ) ( ) (
dt t m dt t l dW dt
t Y t dY
dt t c dt t m dt t l dt t rX t dX
Here, X(t)is the money invested in bonds, Y(t)is the investments at the stock, ris the interest rate for the bond price,l(t) is the rate of transfer from the bond holdings to the stock, m(t)is the rate of transfer from the stock holdings to the bond, c(t)is the rate of consumption, is the expected rate of return, 𝜎 is the rate of return variation, Wis the standard Wiener process, which is a mathematical model for the one-dimensional Brownian motion. We
change variables
), (
) ) ( ( ), ( ) ( )
( Z t
t t Y t Y t X t
Z
then the problem can be written in the form:
( ( )( )) ( )
( ) . )( )
(t Z t r t r dt t dW c t dt
dZ
HereZ(t)is the wealth of the investor at timet, (t)is the fraction of total wealth held in stock. It is believed that the funds between stocks and bonds are transferred instantly and without losses. The restriction Z(t)0is necessary to exclude the situation when the investor goes bankrupt and his debt is refinanced through further borrowing.The investor invests the fraction of (t)from total wealth held in stock, and the remaining part of
) (
1 t in the bonds.Our goal will be to find the optimal investment strategy
( ) , ( )
, ( ) , ( ) 0)
(t * t * c t * * t c* t
(control function), which will maximize the functional:
), 1 0
( , ) 1 (
0
E
pc t dt pJ
e
t pwhere is the rate for discounting.
For the study, the dynamic programming principle (method 1) and the Hamilton-Jacobi-Bellman equation (method 2)are used. We obtain an analytical form of optimal control formulas using the principle of dynamic programming:
), 1 ) (
( * * 2
p t r
. ) ) (
1 ( 2
) ( 1
) 1
( 2 *
2
* Z t
p r rp p
t p
c
The Hamilton-Jacobi-Bellman equation for the Merton problem takes the form:
0.2 1
sup 1 2
2 2
2
Z z c z
r z r
Z pc t
t p
e
Solving it, we find formulas for optimal control. The results obtained by methods (1)and (2) are consistent. A numerical algorithm was developed based on dynamic programming to solve the optimal control problem for the stochastic differential equations and to find the optimal investment strategy.
To solve a direct problem, a set of parameters [7]and the initial datapresented in the table were taken:
) 0 (
Z
r
p100 0.07 0.12 0.4 0.1 0.5
We used the Euler-Bernstein difference scheme (the scheme has a first degree of approximation) for the solution of the direct problem:
, )
,
~( ) ,
( 1
1
k k k k k k
k Z a t Z t Z t
Z
where
k1 is a sequence of independent identically distributed random variables with zero mean and unit variance.The stability of the direct problem for the Merton model is analyzed. For this, a mathematicalexpectation is found for the variations of each model parameter.In numerical analysis, we have obtained that the graph of the parameter , unlike the others, has a uneven character (see Fig. 1). This means that for small variations of the parameter , significant changes are made to the solution of the direct problem. That is, the parameter is sensitive to small variations in the data. Itfollowsthatthe problem is also unstable.
a) b)
c) d)
e)
Figure 1.The mathematical expectation for the parameters:
a)p, b) , c)
r
, d)
, e) .The paper was supported by the project "Numerical methods of identifiability of inverse and ill-posed problems of natural science" (No. AP05134121) and by the President Fellowship of Russian Federation (MK-1214.2017.1).
References:
1 Shi J. Application of Alternative ODE in Finance and Economics Research //Business School - New Brunswick and Newark Rutgers University, NJ, 2010.
2 Solow R.M. A Contribution to the Theory of Economic Growth // The Quarterly Journal of Economics, Vol.70, No.1, 1956, P. 65-94.
3 Soner H.M. Stochastic Optimal Control in Finance // 2004, Oxford.
4 Fleming D., Rishel V. Optimal control of deterministic and stochastic systems// M., Mir,1978, P. 318.
5 Merton R.C. et al. Theory of rational option pricing // World Scientific, 1971.
6 Kabanikhin S.I. Definitions and examples of inverse and ill-posed problem // Journal of Inverse and Ill-Posed Problems, Vol. 16, No. 4, 2008, P. 317-357.
7 M.H.A. Davis, A.R. Norman, Portfolio selection with transaction costs // Mathematics of operations research, Vol. 15, No. 4, 1990.
МРНТИ 27.29.19