This paper deals with Mckean-Vlasov backward stochastic differential equations with weak monotonicity coefficients.We first establish the existence and uniqueness of solutions to Mckean-Vlasov backward stochastic diff...This paper deals with Mckean-Vlasov backward stochastic differential equations with weak monotonicity coefficients.We first establish the existence and uniqueness of solutions to Mckean-Vlasov backward stochastic differential equations.Then we obtain a comparison theorem in one-dimensional situation.展开更多
In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error...In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.展开更多
This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–M...This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate.In particular,the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense,and this convergence is independent of the order in which the mass parameterμand the step size h tend to zero.展开更多
In this article,we study the meromorphic solutions of the following non-linear differential equation■where n and k are integers with n≥k≥3,P_(d)(z,f)is a differential polynomial in f of degree d≤n−1,p′js andα′j...In this article,we study the meromorphic solutions of the following non-linear differential equation■where n and k are integers with n≥k≥3,P_(d)(z,f)is a differential polynomial in f of degree d≤n−1,p′js andα′js are non-zero constants.We obtain the expressions of meromorphic solutions of the above equations under some restrictions onα′js.Some examples are given to illustrate the possibilities of our results.展开更多
In this paper,we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients f(z+1)-f(z-1)+a(z)S(f,z)=P(z,f(z))/Q(z,f(z)).We obtain ...In this paper,we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients f(z+1)-f(z-1)+a(z)S(f,z)=P(z,f(z))/Q(z,f(z)).We obtain the necessary conditions on the degree of R(z,f)for these delay differential equations and give a classification of the delay Schwarzian differential equations according to the multiplicities of the root of Q(z,f)on f.Finally,we provide some examples to illustrate that all cases occur.展开更多
In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both...In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.展开更多
This study investigates an option pricing method called g-pricing based on backward stochastic differential equations combined with deep learning.We adopted a datadriven approach to find a market-appropriate generator...This study investigates an option pricing method called g-pricing based on backward stochastic differential equations combined with deep learning.We adopted a datadriven approach to find a market-appropriate generator of the backward stochastic differential equation,which is achieved by leveraging the universal approximation capabilities of neural networks.Option pricing,which is the solution to the equation,is approximated using a recursive procedure.The empirical results for the S&P 500 index options show that the proposed deep learning g-pricing model has lower absolute errors than the classical Black–Scholes–Merton model for the same forward stochastic differential equations.The g-pricing mechanism has potential applications in option pricing.展开更多
In this paper,we study the existence of pseudo S-asymptotically ω-periodic mild solutions of abstract partial neutral differential equations in Banach spaces.By using the principle of Banach contractive mapping,the e...In this paper,we study the existence of pseudo S-asymptotically ω-periodic mild solutions of abstract partial neutral differential equations in Banach spaces.By using the principle of Banach contractive mapping,the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions of abstract partial neutral differential equations are obtained.To illustrate the ab-stract result,a concrete example is given.展开更多
Streptococcus suis(S.suis)is a major disease impacting pig farming globally.It can also be transferred to humans by eating raw pork.A comprehensive study was recently carried out to determine the indices throughmultip...Streptococcus suis(S.suis)is a major disease impacting pig farming globally.It can also be transferred to humans by eating raw pork.A comprehensive study was recently carried out to determine the indices throughmultiple geographic regions in China.Methods:The well-posed theorems were employed to conduct a thorough analysis of the model’s feasible features,including positivity,boundedness equilibria,reproduction number,and parameter sensitivity.Stochastic Euler,Runge Kutta,and EulerMaruyama are some of the numerical techniques used to replicate the behavior of the streptococcus suis infection in the pig population.However,the dynamic qualities of the suggested model cannot be restored using these techniques.Results:For the stochastic delay differential equations of the model,the non-standard finite difference approach in the sense of stochasticity is developed to avoid several problems such as negativity,unboundedness,inconsistency,and instability of the findings.Results from traditional stochastic methods either converge conditionally or diverge over time.The stochastic non-negative step size convergence nonstandard finite difference(NSFD)method unconditionally converges to the model’s true states.Conclusions:This study improves our understanding of the dynamics of streptococcus suis infection using versions of stochastic with delay approaches and opens up new avenues for the study of cognitive processes and neuronal analysis.Theplotted interaction behaviour and new solution comparison profiles.展开更多
In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where ...In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.展开更多
To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem...To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Two oscillatory criteria of solutions for systems of parabolic differential equations with deviating arguments are obtained.展开更多
The nonlinear mixed-effects model with stochastic differential equations （SDEs） is used to model the population pharmacokinetic （PPK） data that are extended from ordinary differential equations （ODEs） by adding ...The nonlinear mixed-effects model with stochastic differential equations （SDEs） is used to model the population pharmacokinetic （PPK） data that are extended from ordinary differential equations （ODEs） by adding a stochastic term to the state equation. Compared with the ODEs, the SDEs can model correlated residuals which are ubiquitous in actual pharmacokinetic problems. The Bayesian estimation is provided for nonlinear mixed-effects models based on stochastic differential equations. Combining the Gibbs and the Metropolis-Hastings algorithms, the population and individual parameter values are given through the parameter posterior predictive distributions. The analysis and simulation results show that the performance of the Bayesian estimation for mixed-effects SDEs model and analysis of population pharmacokinetic data is reliable. The results suggest that the proposed method is feasible for population pharmacokinetic data.展开更多
The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are ...The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are established for the existence of periodic solutions and some previous results are extended.展开更多
J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).
Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain...Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain some results.展开更多
The oscillatory behavior of neutral differential equation with positive and negative coefficients is investigated by mathematics analysis technique and the fixed point principle. Some sufficient conditions for oscilla...The oscillatory behavior of neutral differential equation with positive and negative coefficients is investigated by mathematics analysis technique and the fixed point principle. Some sufficient conditions for oscillation of neutral differential equation with positive and negative coefficients are obtained.展开更多
In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa, G. Gundersen and J.K. Langley, Li C...In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa, G. Gundersen and J.K. Langley, Li Chun-hong.展开更多
In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions ...In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.展开更多
基金Supported by the National Natural Science Foundation of China(12001074)the Research Innovation Program of Graduate Students in Hunan Province(CX20220258)+1 种基金the Research Innovation Program of Graduate Students of Central South University(1053320214147)the Key Scientific Research Project of Higher Education Institutions in Henan Province(25B110025)。
文摘This paper deals with Mckean-Vlasov backward stochastic differential equations with weak monotonicity coefficients.We first establish the existence and uniqueness of solutions to Mckean-Vlasov backward stochastic differential equations.Then we obtain a comparison theorem in one-dimensional situation.
文摘In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.
基金supported by the PhD Research Startup Foundation of Hubei University of Economics(Grand No.XJ23BS42).
文摘This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate.In particular,the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense,and this convergence is independent of the order in which the mass parameterμand the step size h tend to zero.
基金supported by the National Natural Science Foundation of China(No.12001117)the Guangdong Basic and Applied Basic Research Foundation(No.2021A1515110654).
文摘In this article,we study the meromorphic solutions of the following non-linear differential equation■where n and k are integers with n≥k≥3,P_(d)(z,f)is a differential polynomial in f of degree d≤n−1,p′js andα′js are non-zero constants.We obtain the expressions of meromorphic solutions of the above equations under some restrictions onα′js.Some examples are given to illustrate the possibilities of our results.
基金supported by the National Natural Science Foundation of China(12201420,12231013,11701382 and 11971288)the Guangdong Basic and Applied Basic Research Foundation,China(2021A1515010054)and NCAMS.
文摘In this paper,we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients f(z+1)-f(z-1)+a(z)S(f,z)=P(z,f(z))/Q(z,f(z)).We obtain the necessary conditions on the degree of R(z,f)for these delay differential equations and give a classification of the delay Schwarzian differential equations according to the multiplicities of the root of Q(z,f)on f.Finally,we provide some examples to illustrate that all cases occur.
基金Supported by the National Natural Science Foundation of China (Grant No. 12301521)the Natural Science Foundation of Shanxi Province (Grant No. 20210302124081)。
文摘In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.
基金supported by Taishan Scholar Project of Shandong Province of China(Grant tstp20240803)the National Key R&D Program of China(Grant No.2023YFA1008903)the Major Fundamental Research Project of Shandong Province of China(Grant No.ZR2023ZD33).
文摘This study investigates an option pricing method called g-pricing based on backward stochastic differential equations combined with deep learning.We adopted a datadriven approach to find a market-appropriate generator of the backward stochastic differential equation,which is achieved by leveraging the universal approximation capabilities of neural networks.Option pricing,which is the solution to the equation,is approximated using a recursive procedure.The empirical results for the S&P 500 index options show that the proposed deep learning g-pricing model has lower absolute errors than the classical Black–Scholes–Merton model for the same forward stochastic differential equations.The g-pricing mechanism has potential applications in option pricing.
基金Supported by the National Natural Science Foundation of China(11226337)the Science and Technology Research Projects of Henan Education Committee(22B110017).
文摘In this paper,we study the existence of pseudo S-asymptotically ω-periodic mild solutions of abstract partial neutral differential equations in Banach spaces.By using the principle of Banach contractive mapping,the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions of abstract partial neutral differential equations are obtained.To illustrate the ab-stract result,a concrete example is given.
基金supported by the Deanship of Scientific Research,Vice Presidency for Graduate Studies and Scientific Research,King Faisal University,Saudi Arabia[KFU250259].
文摘Streptococcus suis(S.suis)is a major disease impacting pig farming globally.It can also be transferred to humans by eating raw pork.A comprehensive study was recently carried out to determine the indices throughmultiple geographic regions in China.Methods:The well-posed theorems were employed to conduct a thorough analysis of the model’s feasible features,including positivity,boundedness equilibria,reproduction number,and parameter sensitivity.Stochastic Euler,Runge Kutta,and EulerMaruyama are some of the numerical techniques used to replicate the behavior of the streptococcus suis infection in the pig population.However,the dynamic qualities of the suggested model cannot be restored using these techniques.Results:For the stochastic delay differential equations of the model,the non-standard finite difference approach in the sense of stochasticity is developed to avoid several problems such as negativity,unboundedness,inconsistency,and instability of the findings.Results from traditional stochastic methods either converge conditionally or diverge over time.The stochastic non-negative step size convergence nonstandard finite difference(NSFD)method unconditionally converges to the model’s true states.Conclusions:This study improves our understanding of the dynamics of streptococcus suis infection using versions of stochastic with delay approaches and opens up new avenues for the study of cognitive processes and neuronal analysis.Theplotted interaction behaviour and new solution comparison profiles.
文摘In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.
文摘To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Two oscillatory criteria of solutions for systems of parabolic differential equations with deviating arguments are obtained.
基金The National Natural Science Foundation of China(No.11171065,81130068)the Natural Science Foundation of Jiangsu Province(No.BK2011058)the Fundamental Research Funds for the Central Universities(No.JKPZ2013015)
文摘The nonlinear mixed-effects model with stochastic differential equations （SDEs） is used to model the population pharmacokinetic （PPK） data that are extended from ordinary differential equations （ODEs） by adding a stochastic term to the state equation. Compared with the ODEs, the SDEs can model correlated residuals which are ubiquitous in actual pharmacokinetic problems. The Bayesian estimation is provided for nonlinear mixed-effects models based on stochastic differential equations. Combining the Gibbs and the Metropolis-Hastings algorithms, the population and individual parameter values are given through the parameter posterior predictive distributions. The analysis and simulation results show that the performance of the Bayesian estimation for mixed-effects SDEs model and analysis of population pharmacokinetic data is reliable. The results suggest that the proposed method is feasible for population pharmacokinetic data.
文摘The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are established for the existence of periodic solutions and some previous results are extended.
文摘J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).
文摘Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain some results.
文摘The oscillatory behavior of neutral differential equation with positive and negative coefficients is investigated by mathematics analysis technique and the fixed point principle. Some sufficient conditions for oscillation of neutral differential equation with positive and negative coefficients are obtained.
基金This work is supported by the National Natural Science Foundation of China(10161006)the Natural Science Foundation of Jiangxi Prov(001109)Korea Research Foundation Grant(KRF-2001-015-DP0015)
文摘In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa, G. Gundersen and J.K. Langley, Li Chun-hong.
文摘In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.
