Chebyshev wavelet-based method for solving various stochastic optimal control problems and its application in finance

Document Type : Research Article

Authors

M. Yarahmadi

S. Yaghobipour

Department of Mathematics and Computer Sciences, Lorestan University, Lorestan, Iran.

10.22067/ijnao.2023.82445.1265

Abstract

In this paper, a computational method based on parameterizing state and control variables is presented for solving Stochastic Optimal Control (SOC) problems. By using Chebyshev wavelets with unknown coefficients, state and control variables are parameterized, and then a stochastic optimal control problem is converted to a stochastic optimization problem. The expected cost functional of the resulting stochastic optimization problem is approximated by sample average approximation thereby the problem can be solved by optimization methods more easily. For facilitating and guar-anteeing convergence of the presented method, a new theorem is proved. Finally, the proposed method is implemented based on a newly designed algorithm for solving one of the well-known problems in mathematical fi-nance, the Merton portfolio allocation problem in finite horizon. The simu-lation results illustrate the improvement of the constructed portfolio return.

Keywords

Stochastic optimal control

Chebyshev wavelets

Expansion

Optimal asset allocation

Subjects

• Optimal Control

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Introduction

The present problem is dealing with the nanofluid flow through the porous square cavity with temperature difference, which has a wide range of applications in recent years, such as geophysics, geothermal energy utilization, and many technologies. The bioconvection of the nanofluid containing gyrotactic microorganisms has a wide range of practical applications, such as chemical catalytic converters, buried electronic cables, pollutant dispersion in aquifers, food industrial forms, and so on. These types of many areas of applications are documented in these references 1024141533. The properties and utilization of the nanofluid were first introduced by Choi and Eastman 11 at ASME annual meeting. Many people have described the properties of nanofluids, such as 1332242327. In many electronic devices, like computers, boilers, converters, and so on, the angle of inclination to the surface affects the gravity force on the fluid, temperature gradient, and velocity of the fluid flow. In 34, the author expressed the free convection of the composite wall enclosure. Kuyper et al. 20 studied the effect of inclined angle on different flows in square cavity walls. Kuznetsov 21 explained the microscopic convection motion of the oxytactic microorganisms due to the temperature effect. Shermet and Pop 30 expressed that the result of the thermal movement of microorganisms in the nanofluid having gyrotactic microorganisms is a closed porous square cavity. Aziz, Khan, and Pop 5 presented the flow behavior of the nanofluid with gyrotactic microorganisms on a flat plate. At viscous dissipation, the behavior of oxytactic microorganisms in porous square cavities was discussed 23. Jamuna and Balla 17 discussed the behavior of the heat source and sink of the gyrotactic microorganisms in the square cavity. The activation energy effect on the gyrotactic microorganisms was discussed in 18. The influence of Soret and Dufour on free convection of the fluid flow in the inclined four-side closed walls was explained in 8. The MHD double-diffusion in the porous square enclosure with radiation and chemical reaction and the outcome inclination of the porous square cavity filled with gyrotactic microorganisms with the heat transformation was discussed in 67. Nanofluid movement in an inclined square cavity with gyrotactic microorganisms at MHD free convection was reported in 28. The effect of angle movement of the square adiabatic wall on mixed convection of the nanofluid was explored in 16. Aounallah et al. 4 explained the turbulent flow behavior of the nanofluid in an inclined square cavity on free convection, and Sheremet, Grosan, and Pop 28 investigated the free convective flow of nanofluid in inclined four-sided chamber with gyrotactic microorganisms. Tsai, Li, and Lin 31 discussed the inclination of the plate shield, and Aboueian-Jahromi, Hossein Nezhad, and Behzadmehr 1 studied the steady flow in inclined cylinders. Rajarathinam and Nithyadevi 26 examined the movement of Cu-water nanofluid in inclined cavity walls with pores. The thermosolutal Maragoni effects of the bioconvective fluid flow with gyrotactic microorganisms on inclined sheets were explained in 19. Recently, Varol, Oztop, and Koca 36 explained different fluids' laminar flow in the different inclined enclosures. Since, from the above literature survey, we note that many authors concentrated on the inclined angle of different geometries with convection of nanofluid with gyrotactic microorganisms. The novelty of this paper contains the square-shaped cavity enclosure with fluid containing nanoparticles and gyrotactic microorganisms. Galerkin's finite-element method is used to solve the nondimensional governing equations.

Mathematical modeling

We consider the bioconvection flow in an inclined two-dimensional porous four-sided square cavity of dimension $L$ containing nanofluid with gyrotactic microorganisms. Let us assume that $ \delta $ is the inclination angle of the cavity wall with the horizontal surface. The vertical walls are maintained in various temperatures $T_C$ and $ T_H$, respectively ( $T_H$$>$$T_C$ ). The remaining walls were kept perfectly insulated. The direction of gravity force $g$ acts opposite to the vertical axis (Y-axis). The steady-state Darcy--Boussinesq approximation governing equations are $$ 1 \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, $$$$ \frac{\mu}{k}u=-\frac{\partial p}{\partial x}-[(\rho_p-\rho_f)(C-C_{\min})-(1-C_{\min})\rho_f\beta(T-T_C)+\gamma n \Delta\rho]g \sin \delta, $$$$ \frac{\mu}{k}v=-\frac{\partial p}{\partial y}-[(\rho_p-\rho_f)(C-C_{\min})-(1-C_{\min})\rho_f\beta(T-T_C)+\gamma n \Delta\rho]g \cos \delta, $$$$ \begin{split} u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha_m(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2})+\tau D_B(\frac{\partial C}{\partial x}\frac{\partial T}{\partial x}+\frac{\partial C}{\partial y}\frac{\partial T}{\partial y})\\ +\frac{\tau D_T}{T_C}[(\frac{\partial T}{\partial x})^2+(\frac{\partial T}{\partial y})^2], \end{split} $$$$ u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_m(\frac{\partial^2 C}{\partial x^2}+\frac{\partial^2 C}{\partial y^2})+\frac{D_T}{T_c}(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}), $$$$ \frac{\partial}{\partial x}(un+\tilde u n-D_n\frac{\partial n}{\partial x})+\frac{\partial}{\partial y}(vn+\tilde v n-D_n\frac{\partial n}{\partial y})=0. $$ Here $\gamma$ is the mean volume for microorganisms, $\Delta\rho=\rho_{cell}-\rho_f$ is the density difference of cell, $T_C$ is the cold wall temperature, $T_H$ is the hot wall temperature, $\alpha_m$ is porous medium thermal diffusivity, $C$ is the concentration of nanoparticles, $C_0$ is nanoparticles average density, $ C_{\min}$ is minimum concentration of oxygen essential for microorganisms, $C_p$ is specific heat at constant pressure, $D_n$ is microorganisms diffusion coefficient, $ D_B$ is the Brownian diffusion constant, $D_T$ is the thermophoretic diffusion coefficient, $n$ is motile density number of microorganisms, $g$ is the gravity force, chemotaxis constant is $b$, and the maximum speed of cell swims is $w_C$. The average swimming velocities of microorganisms are $\widetilde{u}$ and $\widetilde{v}$ given as $$ 2 \widetilde{u}=\frac{bw_C}{\Delta C}\frac{\partial C}{\partial x}, \hspace{4mm} \widetilde{v}=\frac{bw_C}{\Delta C}\frac{\partial C}{\partial y}. $$ Consider dimensional stream function $\psi$. Then $u$ and $v$ in $x$ and $y$ directions are considered as $u=\frac{\partial\psi}{\partial y}$ and $v=\frac{\partial\psi}{\partial x}$ by introducing the boundedless variables $X=\frac{x}{H}$, $Y=\frac{y}{H}$, $\Psi=\frac{\psi}{\alpha_m}$,$\theta=\frac{T-T_C}{T_H-T_C}$, $\phi=\frac{C-C_{\min}}{\Delta C}$, and $N=\frac{n}{n_0}$, where $n_0$ is the microorganism averaged density. Substituting above unbounded variables into equation 12$$ 3 \begin{split} \frac{\partial^2\Psi}{\partial X^2}+\frac{\partial^2\Psi}{\partial Y^2}=RaNr(\frac{\partial \phi}{\partial X}\cos \delta -\frac{\partial \phi}{\partial Y}\sin \delta)-Ra(\frac{\partial \theta}{\partial X}\cos \delta-\frac{\partial \theta}{\partial Y}\sin \delta)\\ + RaRb(\frac{\partial N}{\partial X}\cos \delta -\frac{\partial N}{\partial Y}\sin \delta), \end{split} $$$$ \begin{split} (\frac{\partial \Psi}{\partial Y}\frac{\partial \theta}{\partial X}-\frac{\partial \Psi}{\partial X}\frac{\partial \theta}{\partial Y})=(\frac{\partial ^2 \theta}{\partial X}+\frac{\partial ^2 \theta}{\partial Y})+Nb(\frac{\partial \phi}{\partial X}\frac{\partial \theta}{\partial X}+\frac{\partial \phi}{\partial Y}\frac{\partial \theta}{\partial Y})\\+Nt[(\frac{\partial \theta}{\partial X})^2+(\frac{\partial \theta}{\partial Y})^2], \end{split} $$$$ Le(\frac{\partial\Psi}{\partial Y}\frac{\partial \phi}{\partial X}-\frac{\partial\Psi}{\partial X}\frac{\partial \phi}{\partial Y})=\frac{\partial ^2 \phi}{\partial X^2}+\frac{\partial ^2 \phi}{\partial Y^2}+\frac{Nt}{Nb}(\frac{\partial ^2 \theta}{\partial X^2}+ \frac{\partial ^2 \theta}{\partial Y^2}), $$$$ 4 \frac{\partial \Psi}{\partial X}\frac{\partial N}{\partial Y}-\frac{\partial \Psi}{\partial Y}\frac{\partial N}{\partial X}+\frac{PrPe}{Sc}(\frac{\partial ^2 \phi}{\partial X^2}+\frac{\partial ^2 \phi}{\partial Y^2})=\frac{Pr}{Sc}(\frac{\partial^2 N}{\partial X^2}+\frac{\partial^2 N}{\partial Y^2}), $$ where $Ra=\frac{gK \beta (1-C_0) \Delta T L}{v \alpha_m}$, $Rb=\frac{\gamma \Delta \rho n_0}{\rho_f \beta (1-C_0) \Delta T}$, $Le=\frac{\alpha_m}{D_B}$, $Pe=\frac{bw_C}{D_n}$, $Nb=\frac{\tau D_B\Delta C}{\alpha_m}$, $Nt=\frac{\tau D_T(T_H-T_C)}{\alpha_mT_C},$$\Pr=\frac{\mu_f}{\rho_f\alpha_f}$, $Sc=\frac{\mu_f}{\rho_fD_n}$, and $Nr=\frac{(\rho_p-\rho_f)C_0}{\rho_f\beta (1-C_0)\Delta T}$. The dimensionless form of conditions at boundary is expressed in Figure 5 We have $\Psi=0$ for all sides, $\phi=1$, $\theta=1$, $N=1$ at $X=0$, $\phi=1$, $\theta=0$, $N=1$ at $X=1$, $\phi=1$, $\frac{\partial \theta}{\partial Y}=0$, $Pe.N\frac{\partial \phi}{\partial Y}=\frac{\partial N}{\partial Y}$ at $Y=0$, and \begin{equation*} \frac{\partial \phi}{\partial Y}=\frac{\partial \theta}{\partial Y}=\frac{\partial N}{\partial Y}=0\qquad at \ Y=1. \end{equation*} Physical geometry and coordinate system. 5 Local solid Nusselt number, Sherwood number of nano particles and Sherwood microorganism are defined as $Nu_Y=-(\frac{\partial \theta}{\partial X})_{X=0,1}$, $Sh_Y=-(\frac{\partial \phi}{\partial X})_{X=0,1}$, and $Nn_Y=-(\frac{\partial N}{\partial X})_{X=0,1}$. The average quantities of Nusselt number, nanoparticle Sherwood number, and microorganism Sherwood number is defined as $Nu_{avg}=$$\int_{0}^{1}Nu_Y \, dY,$$ $$Sh_{avg}=$$\int_{0}^{1}Sh_Y \, dY,$$ $$Nn_{avg}=$$\int_{0}^{1}Nn_Y \, dY.$$ $

Numerical method

To find the numerical solution to 349. In this method, a two-dimensional field is divided into small triangular parts, in which each part is named an element. Over each element, assume a piecewise trial function. Let $\Psi$, $\theta$, $\phi$, and $N$ be approximated by $\Psi=\sum_{i=1}^{3} {\Psi_{i} \xi _{i}} $, $\theta=\sum_{i=1}^{3} {\theta_{i} \xi _{i}} $, $\phi=\sum_{i=1}^{3} {\phi_{i} \xi _{i}} $, and $N=\sum_{i=1}^{3} {N_{i} \xi _{i}} $, where $\xi_i$ is the linear interpolating functions over each triangular element. The FEM model matrix is as follows: $\begin{matrix} [L^{11}]&[L^{12}]&[L^{13}]&[L^{14}]\\ [L^{21}]&[L^{22}]&[L^{23}]&[L^{24}]\\ [L^{31}]&[L^{32}]&[L^{33}]&[L^{34}]\\ [L^{41}]&[L^{42}]&[L^{43}]&[L^{44}]\\ \end{matrix} % \begin{array} \{\Psi\}\\ \{T\}\\ \{C\}\\ \{N\}\\ \end{array} = \begin{array} \{M^1\}\\ \{M^2\}\\ \{M^3\}\\ \{M^4\}\\ \end{array}$ , \\ where \\ $L^{11}=\iint_{\Omega_e} [\frac{\partial \xi_j}{\partial X}\frac{\partial \xi_i}{\partial X}+\frac{\partial \xi_j}{\partial Y}\frac{\partial \xi_i}{\partial Y}] \,dx\,dy$,\\ $L^{12}=-Ra\iint_{\Omega_e}(\xi_j\frac{\partial\xi_i}{\partial X}\cos\delta-\xi_j\frac{\partial\xi_i}{\partial Y}\sin\delta)\,dX\,dY$,\\ $L^{13}=RaNr\iint_{\Omega_e}(\xi_j\frac{\partial\xi_i}{\partial X}\cos\delta-\xi_j\frac{\partial\xi_i}{\partial Y}\sin\delta)\,dX\,dY$,\\ $L^{14}=RaRb\iint_{\Omega_e}(\xi_j\frac{\partial\xi_i}{\partial X}\cos\delta-\xi_j\frac{\partial\xi_i}{\partial Y}\sin\delta)\,dX\,dY$,\\ $M^1=0$,\\ $L^{21}=0$,\\ {\small$L^{22}=\iint_{\Omega_e}[\bar{\frac{\partial\Psi}{\partial Y}}\xi_j\frac{\partial\xi_i}{\partial X}-\bar{\frac{\partial\Psi}{\partial X}}\xi_j\frac{\partial\xi_i}{\partial Y}+\frac{\partial\xi_i}{\partial X}\frac{\partial\xi_j}{\partial X}+\frac{\partial\xi_i}{\partial Y}\frac{\partial\xi_i}{\partial Y}-Nt(\bar{\frac{\partial\theta}{\partial X}}\xi_j\frac{\partial\xi_i}{\partial X}+\bar{\frac{\partial\theta}{\partial Y}}\xi_j\frac{\partial\xi_i}{\partial Y})]\,dX\,dY $},\\ $L^{23}=-Nb\ \iint_{\Omega_e}[\bar{\frac{\partial\theta}{\partial X}}\xi_j\frac{\partial\xi_i}{\partial X}+\bar{\frac{\partial\theta}{\partial Y}}\xi_j\frac{\partial\xi_i}{\partial Y}]\,dX\,dY $, \\ $L^{24}=0 $, $ M^2=0$,\\ $L^{31}=0 $, \\ $L^{32}=-\frac{Nt}{Nb}\iint_{\Omega_e}[\frac{\partial\xi_i}{\partial X}\frac{\partial\xi_j}{\partial X}+\frac{\partial\xi_i}{\partial Y}\frac{\partial\xi_j}{\partial Y}]\,dX\,dY$,\\ $L^{33}=\ \iint_{\Omega_e}[Le\left(\bar{\frac{\partial\Psi}{\partial Y}}\frac{\partial\phi}{\partial X}-\bar{\frac{\partial\Psi}{\partial X}}\frac{\partial\phi}{\partial Y}\right)+\frac{\partial\xi_i}{\partial X}\frac{\partial\xi_j}{\partial X}+\frac{\partial\xi_i}{\partial Y}\frac{\partial\xi_j}{\partial Y}]\,dX\,dY $ , \\ $L^{34}=0$, $ M^3=0$,\\ $L^{41}=0$, $ L^{42}=0$, \\ $L^{43}=\iint_{\Omega_e}\frac{PePr}{Sc}(\frac{\partial\xi_i}{\partial X}\frac{\partial\xi_j}{\partial X}+\frac{\partial\xi_i}{\partial Y}\frac{\partial\xi_j}{\partial Y})\,dX\,dY$, \\ $L^{44}=\iint_{\Omega_e}[\bar{\frac{\partial\Psi}{\partial Y}}\xi_j\frac{\partial\xi_i}{\partial X}-\bar{\frac{\partial\Psi}{\partial X}}\xi_i\frac{\partial\xi_j}{\partial Y}+\frac{Pr}{Sc}(\frac{\partial\xi_i}{\partial X}\frac{\partial\xi_j}{\partial X}+\frac{\partial\xi_i}{\partial Y}\frac{\partial\xi_j}{\partial Y})]\,dX\,dY$, \\ $M^4=0$. \\ To linearize the system of equations, the functions are incorporated, which are assumed to be known. After applying the boundary conditions, a matrix of system of linear equations is formed, which is solved by using the Gauss--Seidel iteration method. The convergence of the solution is assumed when the relative error for each variable between two consecutive iterations is observed below the convergence criteria such that $|\psi ^{n+1} -\psi ^n| \leq 10^{-5}$, where $n$ is the number of iterations and $\psi$ stands for $\Psi,\theta,C$. To choose the grid size, the grid independence test is performed for $21\times 21, 41 \times 41, 61 \times 61,71 \times 71, 81 \times 81, 91 \times 91$ grid sizes. The grid independence test reveals that the grid size $81 \times 81 $ is sufficient to study in the the bioconvection phenomena.

Result and discussion

The present equations 34 \caption{Streamlines for the inclination angle $\delta=0^{\circ}-{180}^{\circ}$} 7 Streamlines are presented in Figure 7 Isotherms are demonstrated in Figure 8 Isotherms for the inclination angle $\delta=0^{\circ}-{180}^{\circ}$ 8 Isoconcentration of nanoparticle volume fraction for the inclination angle $\delta=0^{\circ}-{180}^{\circ}$ 9 Microorganism isoconcentrations for the inclination angle $\delta=0^{\circ}-{180}^{\circ}$ 10 Representation of (a) Average Nusselt number (b) Average nanoparticle Sherwood number (c) Average Microorganism Sherwood number for angle versus $Rb$ and angle versus $Nt$ 11 Nanoparticle isoconcentrations of the fluid for various angles are expressed in Figure 9 Figure 10 In Figure 11

Conclusion

The effect of porous square cavity inclination with the horizontal surface with nanofluid and gyrotactic microorganisms was analyzed with the streamlines, isotherms, nanoparticle volume fraction, and microorganism isoconcentration from $0^{\circ}$ to$ {180}^{\circ}$. 1. The velocity of the nanofluid flow is high at the angle $ {30}^{\circ}$, ${120}^{\circ}$ and in the remaining angles, the flow intensity is low. 2. The temperature distribution of the nanofluid is affected by the square cavity inclination. 3. Nanoparticle isoconcentration and microorganism isoconcentrations are high at the ${30}^{\circ}$ and $ {120}^{\circ}$, and at the remaining angles, the value is low. 4. Thermophoresis parameter increases $ {Nu}_{avg}$, $ {Sh}_{avg}$, and ${Nn}_{avg}$ from $0^{\circ}\leq\delta\leq{180}^{\circ}$. Also, $Nt$ increases ${Nu}_{avg}$ from $0^{\circ}\leq\delta\leq{180}^{\circ}$, but at $0^{\circ}$ and ${\ 180}^{\circ}$, the value is low. 5. Bioconvection Rayleigh number increases ${Nu}_{avg}$, $ {Sh}_{avg}$, and ${Nn}_{avg}$. \section*{Acknowledgements} Authors are grateful to there anonymous referees and editor for their constructive comments. $$ E = mc^2 $$ (1.1) (1) $$ E = mc^2 $$ (2) $$ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 $$ (3) $$ \begin{aligned} a &= b + c \\ d &= e + f \end{aligned} $$ As shown in (1), ...

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