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We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients. [ABSTRACT FROM AUTHOR]

This paper investigates an inverse problem of identifying the implied volatility in the financial products linked with gold price, which has important application infinancial derivatives pricing. Based on the optimal control framework, the existence and necessary condition of the minimizer for the control function are obtained. Since the optimal control problem is non-convex, one may not expect a unique solution in general. However, the local uniqueness of the solution is proved in this paper. In the end of the paper, an algorithm is proposed and some numerical experiments are given. Numerical results show that our algorithm is stable and effective. The results obtained in the paper are interesting and useful and can be extended to other inverse problems arising in financial market. [ABSTRACT FROM AUTHOR]

In this paper, we would like to give an indirect possible way to reconstruct the shape of rotating body by temperature measurements. The mathematical model is governed by a heat conduction equation in which the parameter to determine is denoted by an unknown coefficient of first-order derivative. Firstly, the uniqueness of the solution is proved by using the maximum principle. Then, on the basis of optimal control framework, the inverse problem is transformed into an optimization problem. To ensure the well-posedness of the considered problem, we prove the existence and derive the necessary condition of the minimizer. By carefully analysing the necessary condition, we obtain the local uniqueness and stability on the basis of some a-priori estimates of forward problem. Finally, some numerical implements are performed to show the validity of the proposed method. [ABSTRACT FROM AUTHOR]

This work studies a nonlinear inverse problem of reconstructing the diffusion coefficient in a parabolic‐elliptic system using the final measurement data, which has important application in a large field of applied science. Being different from other works, which are governed by single partial differential equations, the underlying mathematical model in this paper is a coupled parabolic‐elliptic system, which makes theoretical analysis rather difficult. On the basis of the optimal control framework, the identification problem is transformed into an optimization problem. Then the existence of the minimizer is proved, and the necessary condition that must be satisfied by the minimizer is also given. Since the optimal control problem is nonconvex, one may not expect a unique solution universally. However, the local uniqueness and stability of the minimizer are deduced in this paper. [ABSTRACT FROM AUTHOR]

This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well. [ABSTRACT FROM AUTHOR]

In this study, we investigate the well-posedness of the solution of an optimal control problem related to a nonlinear inverse coefficient problem. Problems of this type have important applications in several fields of applied science. Unlike other terminal control problems, the observation data are only given for a fixed direction rather than for the whole domain, which may make the conjugate theory for parabolic equations ineffective. Moreover, the coefficients in our model are singular, so we propose some weighted Sobolev spaces. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After deducing the necessary conditions that must be satisfied by the minimizer, we prove the uniqueness and stability of the minimizer. Following a minor modification of the cost functional and imposing some a priori regularity conditions on the forward operator, we obtain the convergence of the minimizer for the noisy input data considered in this study. The results obtained in this study are interesting and useful, and they can be extended to more general parabolic equations with singular coefficients. [ABSTRACT FROM AUTHOR]

The aim of this work is to reconstruct a spacewise dependent heat source in a degenerate heat equation using the final measurement temperature. Based on the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established, and the global uniqueness and stability of the minimizer are also obtained from the necessary condition. Finally, a numerical algorithm on the basis of the conjugate gradient method is designed to treat the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm designed in the paper is quite efficient and stable, and the unknown heat source is recovered very well. [ABSTRACT FROM AUTHOR]

This paper investigates the solution of a parameter identification problem associated with the two-dimensional heat equation with variable diffusion coefficient. The singularity of the diffusion coefficient results in a nonlinear inverse problem which makes theoretical analysis rather difficult. Using an optimal control method, we formulate the problem as a minimization problem and prove the existence and uniqueness of the solution in weighted Sobolev spaces. The necessary conditions for the existence of the minimizer are also given. The results can be extended to more general parabolic equations with singular coefficients. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the inverse problem of determining a pair in the reaction–diffusion equationwith initial and boundary conditionsfrom the final measurement data , which has important application in a large fields of applied science. Based on the optimal control framework, the existence, uniqueness and stability of the minimizer for the cost functional are established. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. The gradient iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples. [Copyright &y& Elsevier]

Abstract: The local well-posedness of the minimizer of an optimal control problem is studied in this paper. The optimization problem concerns an inverse problem of simultaneously reconstructing the initial temperature and heat radiative coefficient in a heat conduction equation. Being different from other ordinary optimization problems, the cost functional constructed in the paper is a binary functional which contains two independent variables and two independent regularization parameters. Particularly, since the status of the two unknown coefficients in the cost functional are different, the conjugate theory which is extensively used in single-parameter optimization problems cannot be applied for our problem. The necessary condition which must be satisfied by the minimizer is deduced. By assuming the terminal time T is relatively small, an estimate regarding the minimizer is obtained, from which the uniqueness and stability of the minimizer can be deduced immediately. [Copyright &y& Elsevier]

Abstract: This work studies an inverse problem of determining the first-order coefficient of degenerate parabolic equations using the measurement data specified at a fixed internal point. Being different from other ordinary parameter identification problems in parabolic equations, in our mathematical model there exists degeneracy on the lateral boundaries of the domain, which may cause the corresponding boundary conditions to go missing. By the contraction mapping principle, the uniqueness of the solution for the inverse problem is proved. A numerical algorithm on the basis of the predictor–corrector method is designed to obtain the numerical solution and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown function is recovered very well. The results obtained in the paper are interesting and useful, and can be extended to other more general inverse coefficient problems of degenerate PDEs. [Copyright &y& Elsevier]

Abstract: The zero-coupon bond (ZCB) is a special kind of bond without coupons, which is purchased today at a certain price, while at maturity the bond is redeemed for a fixed price. In the ZCB world there is the important quantity which is often called the market price of risk and cannot be observed directly but has a major impact on the ZCB value. In this paper, the inverse problem of determining the market price of risk from the current market prices of ZCB is discussed. Being different from ordinary parameter identification problems in parabolic equations, the mathematical model in the paper belongs to the second order parabolic equations with non-negative characteristic form, i.e., there exists degeneracy on the lateral boundaries in the model. Based on the optimal control framework, the existence, uniqueness and stability of the minimizer for the cost functional are established. A necessary condition which is a coupled system of a parabolic equation and a parabolic variational inequality is deduced. The results obtained in the paper are interesting and useful, and may be applied to a variety of derivatives pricing problems. [Copyright &y& Elsevier]

Abstract: This paper studies an evolutional type inverse problem of identifying the radiative coefficient of heat conduction equation when the over-specified data is given. Problems of this type have important applications in several fields of applied science. Being different from other ordinary inverse coefficient problems, the unknown coefficient in this paper depends on both the space variable x and the time t. Based on the optimal control framework, the inverse problem is transformed into an optimization problem and a new cost functional is constructed in the paper. The existence, uniqueness and stability of the minimizer of the cost functional are proved, and the necessary conditions which must be satisfied by the minimizer are also given. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations. [Copyright &y& Elsevier]

Abstract: This paper deals with the determination of a pair in the nonlinear parabolic equation with initial and boundary conditions from the overspecified data . Based on the optimal control framework, the problem is transformed into a nonlinear optimization problem and the existence of the minimizer for the control functional is established. The necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. Since the optimal control problem is nonconvex, one may not expect a unique solution in general. However, the local uniqueness and stability of the solution are proved, which is also the main contribution of the paper. [Copyright &y& Elsevier]

Abstract: This work deals with an inverse problem of identifying the radiative coefficient of heat conduction equations from discrete measurement data. On the basis of the interpolation technique, we find a new way to reconstruct the unknown coefficient by using the optimization method. The results obtained in the work are interesting and useful and may be applied in a variety of fields. [Copyright &y& Elsevier]

Abstract: In the Black–Scholes world there is the important quantity of volatility which cannot be observed directly but has a major impact on the option value. In practice, traders usually work with what is known as implied volatility which is implied by option prices observed in the market. In this paper, we use an optimal control framework to discuss an inverse problem of determining the implied volatility when the average option premium, namely the average value of option premium corresponding with a fixed strike price and all possible maturities from the current time to a chosen future time, is known. The issue is converted into a terminal control problem by Green function method. The existence and uniqueness of the minimum of the control functional are addressed by the optimal control method, and the necessary condition which must be satisfied by the minimum is also given. The results obtained in the paper may be useful for those who engage in risk management or volatility trading. [Copyright &y& Elsevier]

An inverse source problem in degenerate parabolic equations is considered, where the diffusion coefficient are zero on the whole boundary. Based on the strong maximum principle for degenerate parabolic equations established in the paper, the uniqueness of the solution is proved. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies an inverse problem of identifying the coefficient of parabolic equation when the final observation is given, which has important application in a large fields of applied science. Based on the optimal control framework, the existence and necessary condition of the minimum for the control functional are established. Since the optimal control problem is nonconvex, one may not expect a unique solution. However, in this paper the solution is proved to be locally unique. After the necessary condition is transformed into an elliptic bilateral variational inequality, an algorithm and some numerical experiments are proposed in the paper. The numerical results show that the algorithm designed in this paper is stable and that the coefficient is recovered very well. [Copyright &y& Elsevier]

This paper investigates an inverse problem of simultaneously reconstructing the initial value and source coefficient in a degenerate parabolic equation. Problems of this type have important applications in several fields of applied science. Being different from other inverse coefficient problems in classical parabolic equations (non-degenerate), the principal coefficient in the mathematical model discussed in the paper may vanish on both extremities of the domain. On the basis of Carleman estimate, the uniqueness and conditional stability of solution for the original problem are established. Then an iteration algorithm of Landweber type is designed to obtain the numerical solution and some typical numerical experiments are also performed. Numerical results show that the proposed method is stable and the unknown coefficients are recovered quite well. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse problem of identifying the source coefficient in degenerate parabolic equations. On the basis of extremum principle, the uniqueness of the proposed problem is obtained. The difference scheme for the direct problem is got by using the finite volume method and the corresponding stability is also proved. An iteration scheme of the Landweber type is applied to handle the numerical solution. In the end, several numerical experiments are performed to test the validity of the scheme. The numerical results show that the method is stable and the unknown function is recovered very well when iterative parameter α and iterative times are matching appropriately. [ABSTRACT FROM AUTHOR]

Abstract: This work investigates the inverse problem of reconstructing a spacewise dependent heat source in a two-dimensional heat conduction equation using a final temperature measurement. Problems of this type have important applications in several fields of applied science. Under certain assumptions, this problem can be transformed into a one-dimensional problem where the heat source only depends on the variable r. However, being different from other one-dimensional inverse heat source problems, there exists singularity on the coefficient of our model, which may make the analysis more difficult, regardless of theoretical or numerical. The inverse problem is reduced to an operator equation of the first kind and the corresponding adjoint operator is deuced. For the two dimensional case, i.e., , theoretical analysis can be done by similar derivation. Based on the landweber regularization framework, an iterative algorithm is proposed to obtain the numerical solution. Some typical numerical examples are presented to show the validity of the inversion method. [Copyright &y& Elsevier]

Abstract: This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well. [Copyright &y& Elsevier]