Search

Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified. [ABSTRACT FROM AUTHOR]

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified.

The price V of a contingent claim in finance, insurance and economics is defined as an expectation of a stochastic expression. If the underlying uncertainty is modeled as a strong Markov process X, the Feynman–Kac theorem suggests that V is the unique solution of a boundary problem for a parabolic equation. In the case of PDO with constant symbols, simple probabilistic tools explained in this paper can be used to explicitly calculate expectations under very weak conditions on the process and study the regularity of the solution. Assuming that the Feynman–Kac theorem holds, and a more general boundary problem can be localized, the local results can be used to study the existence and regularity of solutions, and derive efficient numerical methods. In the paper, difficulties for the realization of this program are analyzed, several outstanding problems are listed, and several closely efficient methods are outlined. [ABSTRACT FROM AUTHOR]

The price V of a contingent claim in finance, insurance and economics is defined as an expectation of a stochastic expression. If the underlying uncertainty is modeled as a strong Markov process X, the Feynman–Kac theorem suggests that V is the unique solution of a boundary problem for a parabolic equation. In the case of PDO with constant symbols, simple probabilistic tools explained in this paper can be used to explicitly calculate expectations under very weak conditions on the process and study the regularity of the solution. Assuming that the Feynman–Kac theorem holds, and a more general boundary problem can be localized, the local results can be used to study the existence and regularity of solutions, and derive efficient numerical methods. In the paper, difficulties for the realization of this program are analyzed, several outstanding problems are listed, and several closely efficient methods are outlined.

Random walk path methods including walk on spheres and walk on rectangles have been used to solve elliptic and parabolic partial differential equations (PDEs). These methods are able to provide not only the pointwise solutions to the linear PDEs but also contributions of boundaries and all source/sink terms as an analytical solution does. However, due to difficulty in dealing with heterogeneity, these methods cannot be applied to groundwater flow problems in highly heterogeneous aquifers. A novel method called walk on grid (WOG) is proposed based on lattice random walk to overcome the difficulty. WOG algorithm is verified in a 1D homogeneous transient problem, a 1D heterogeneous steady-state problem, a 2D heterogeneous transient problem and a 3D optimization problem. It is demonstrated that WOG is effective in solving groundwater flow problems in highly heterogeneous confined aquifers. Probabilities of walkers arriving at prescribed boundaries (terminal weights) and source counts may be useful for characterization of medium heterogeneity. WOG method sheds a new light on solving the PDEs of complicated groundwater problems in a changing environment and on analyzing medium heterogeneity. [ABSTRACT FROM AUTHOR]

In this article, we account for the liquidity risk in the underlying assets when pricing European exchange options, which has not been considered in the literature. An Ornstein–Uhlenbeck process with the mean-reversion property is selected to model the market liquidity risk, whose impacts on the underlying assets are assumed to be imposed with a liquidity discount factor. Under this framework, we develop a simplified approach to obtain the pricing formula, a distinguishing feature of which is that it does not require any numeraire change. We first transform the exchange option's pricing PDE into a pricing PDE of the European vanilla option, which is then solved in closed-form using the characteristic function approach. Finally, we present accuracy tests and sensitivity analysis to demonstrate the correctness of the formula and the influence of introducing the liquidity risk on exchange options, respectively. • We account for the liquidity risk for pricing European exchange options. • We develop a novel approach when deriving the closed-form solution. • We carry out sensitivity analysis to study the effect of liquidity risk. [ABSTRACT FROM AUTHOR]

We prove a Feynman–Kac-type theorem for jump-diffusion models in random environments. We consider the Cauchy and Dirichlet problems. Our results enable us to calculate some conditional expectations using related partial integro-differential equations (PIDEs) and vice versa to solve some PIDEs by stochastic methods. So, the results may have many applications. We illustrate the use of our results on an example of a generalized exponential Lévy model with regime-switching. [ABSTRACT FROM AUTHOR]

We present the idea of intertwining of two diffusions by Feynman–Kac operators. We present implications of the method and give its applications. The examples give new results on stochastic processes including a generalized squared Bessel processes. We present a version of the method and its applications to PDE of the second order. A new dependence between diffusions and solutions of hyperbolic PDE is presented. The version of Feynman–Kac representation for hyperbolic PDE is given. It is presented the simple form of Laplace transform of wave equation with axial symmetry. [ABSTRACT FROM AUTHOR]

The Feynman–Kac theorem and Bogolyubov inequality are applied to obtain a lower bound and an upper bound to the free energy of the s–d Hamiltonian with locally smeared interactions between electrons and impurities. The two bounds, which express in terms of the free energy of impurities in a mean field and electrons in a field of barriers and wells localized at the impurity sites, are almost equal if the impurity concentration is sufficiently small or s–d coupling sufficiently weak. [Copyright &y& Elsevier]