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We propose to combine the Carleman estimate and the Newton method to solve an inverse problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse problem for determination of initial condition is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed. [ABSTRACT FROM AUTHOR]

We propose an efficient valuation approach for guaranteed minimum benefits embedded in variable annuity (VA) contracts, where the price of underlying asset is modeled by a mean reversion process with jumps. We consider the case in which early surrender is permitted, and an intensity-based framework is used to capture surrender behavior and other asset price volatilities. To analyze the contract within this complex stochastic setting, we rely on the Fourier cosine series expansion method to approximate the expectation of the value function, which serves as an estimation of the value of VA benefits. We conduct numerical examples to demonstrate the efficiency and accuracy of the proposed method. Sensitivity analysis of involved parameters is given. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of Hadamard Finite Part (HFP) integrals H σ (t ; u) = ⨎ 0 T | sin π (x − t) T | σ u (x) d x , 0 < t < T , σ < − 1 , σ ⁄ ∈ Z , u (x) being sufficiently differentiable and T -periodic on R. Thus σ = − (m + δ) , m ∈ { 1 , 2 , 3 , ... } , 0 < δ < 1. For each such σ , we regularize H σ (t ; u) , and show that H σ (t ; u) = H σ + 2 r (t ; U σ) , r = ⌊ (m + 1) / 2 ⌋ , where U σ (x) = ∑ k = 0 r a k u (2 k) (x) for some constants a k , H σ + 2 r (t ; U σ) being a regular integral. We then propose to approximate H σ (t ; u) by the quadrature formula Q σ , n (t ; u) ≡ H σ (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q σ , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q σ , n (t ; u) as n → ∞ and prove that it enjoys spectral convergence when u ∈ C ∞ (R). We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals H σ (t ; u) with σ < − 1 and σ ⁄ ∈ Z apply, with no changes, to the regular singular integrals H σ (t ; u) with σ > − 1 and σ ⁄ ∈ Z. We illustrate the effectiveness of Q σ , n (t ; u) with numerical examples both for σ < − 1 and σ > − 1. Finally, we show that the HFP or regular integral ⨎ 0 T f (x) d x of any T -periodic integrand f (x) that has algebraic singularities of the form | x − t + k T | σ , 0 < t < T , k = 0 , ± 1 , ± 2 , ... , with σ ⁄ ∈ Z , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed as H σ (t ; u) , where u (x) is a T -periodic and sufficiently differentiable function of x on R that can be computed from f (x). Therefore, ⨎ 0 T f (x) d x can be computed efficiently using our new numerical quadrature formulas Q σ , n (t ; u) on the individual H σ (t ; u). Again, only 2 n function evaluations, namely, u (x n , k) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]

In this paper, we consider the pricing of the ratchet equity-indexed annuity with the incorporation of mortality risk by an efficient and accurate method under the general time-changed Lévy processes. The pricing of the contract is a two-dimensional problem: we apply the complex Fourier series method to the log return dimension of the asset and high-order quadrature approximation to the other dimensions. Depending on the subordinating dynamic of the volatility, we use the time-changed Lévy process to model stochastic volatility via the random time change in the Lévy process, which compensates for limitations in the common Lévy process. By combining the recursive algorithm and complex Fourier series method, we obtain the approximate price of the contract. Error analysis proves that the total error can achieve an exponential convergence rate, and numerical examples demonstrate the accuracy and efficiency of the complex Fourier series expansion method. [ABSTRACT FROM AUTHOR]

For any circular analytic signal f + (e i t) ∈ H 2 (T) , if the sequence { a k } k = 0 ∞ in the unit disc D satisfies the hyperbolic non-separability condition ∑ k = 0 ∞ (1 − | a k |) = ∞ , it is known that the first N + 1 term approximation function based on the Takenaka–Malmquist system (S N + 1 f +) (e i t) = ∑ k = 0 N 〈 f + (e i t) , B k (e i t) 〉 B k (e i t) , can approximate the analytic signal f + (e i t) in the L 2 -norm sense, where B k (e i t) = 1 − | a k | 2 1 − a k ¯ e i t ∏ l = 0 k − 1 e i t − a l 1 − a l ¯ e i t , k ∈ N is the Takenaka–Malmquist system which can be seen as a generalization of Fourier series. We find that (S N + 1 f +) (e i t) can also be represented by the standard basis , Lagrange basis and weighted SVD basis. Different bases have different merits. The standard basis is simple. The Lagrange basis has a neat and compact form, which is very convenient for theoretical analysis. With the Takenaka–Malmquist system, the first N + 1 term approximation function (S N + 1 f +) (e i t) is able to write in an iterative form. The Takenaka–Malmquist system has the advantage of inheritance, and the point a k ∈ D can be easily selected step by step through the greedy principle. In this paper, we focus our attention on the weighted SVD basis. We give some orthogonal properties of the weighted SVD basis. It demonstrates that any real signal f (e i t) ∈ L 2 (T) can be precisely reconstructed within the given error range by the weighted SVD basis decomposition. The experimental results reveal that the algorithm based on weighted SVD basis is more stable and accurate. Furthermore, compared with classical Fourier decomposition and adaptive Fourier decomposition algorithms, the experimental results show that the weighted SVD basis decomposition is more insensitive to noise. This will provide a new method for signal denoising. [ABSTRACT FROM AUTHOR]

This paper studies the valuation problem of the defined benefit (DB) underpin guarantee. We consider that the salary process follows a geometric Brownian motion, and the stochastic price index process of the funds in the defined contribution (DC) account is modeled by a regime-switching Lévy process. Under this framework, the explicit valuation formula of the DB underpin option is derived by the Fourier cosine series expansion (COS) method, and the corresponding error analysis is provided. Numerous simulation experiments are performed to illustrate the accuracy and efficiency of the proposed method. In addition, the convergence of this method and its sensitivity with respect to various model paraments are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we study the valuation problem of life-contingent lookback options embedded in variable annuity with guaranteed minimum death benefit (GMDB). Specifically, the underlying asset price process is assumed to be an exponential regime-switching Lévy process, which is observed periodically. The Fourier cosine series expansion method is applied to compute exponential moments of the discretely monitored maximum and minimum of the regime-switching Lévy process. Furthermore, some explicit pricing formulas for the life-contingent lookback options embedded in GMDB products are derived. Finally, numerical experiments confirm the accuracy and efficiency of our method. [ABSTRACT FROM AUTHOR]