Riccati Ordinary Differential Equation Research Articles

The Explicit Laplace Transform for the Wishart Process

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models

Over the past few years quadratic Backward Stochastic Differential Equations (BSDEs) have been a popular field of research. However there are only very few examples where explicit solutions for these equations are known. In this paper we consider a class of quadratic BSDEs involving affine processes and show that their solution can be reduced to solving a system of generalized Riccati ordinary differential equations. In other words we introduce a rich and flexible class of quadratic BSDEs which are analytically tractable, i.e. explicit up to the solution of an ODE. Our results also provide analytically tractable solutions to the problem of utility maximization and indifference pricing in multivariate affine stochastic volatility models. This generalizes univariate results of Kallsen and Muhle-Karbe (2010) and some results in the multivariate setting of Leippold and Trojani (2010) by establishing the full picture in the multivariate affine jump-diffusion setting. In particular we calculate the interesting quantity of the power utility indifference value of change of numeraire. Explicit examples in the Heston, Barndorff-Nielsen–Shephard and multivariate Heston setting are calculated.

Singular mean-field optimal control for forward-backward stochastic systems and applications to finance

In this paper, we study a class of singular stochastic optimal control problems for systems described by mean-field forward-backward stochastic differential equations, in which the coefficient depend not only on the state process but also its marginal law of the state process through its expected value. Moreover, the cost functional is also of mean-field type. The control variable has two components, the first being absolutely continuous and the second singular control. Necessary conditions for optimal control for this systems in the form of a Pontrygin maximum principle are established by means convex perturbation techniques for both continuous and singular parts. Our stochastic maximum principle differs from the classical one in the sense that here the adjoint equation has a mean-field type. The control domain is assumed to be convex. As an illustration of our results, we consider a mean-variance portfolio selection mixed with a recursive utility functional optimization problem involving singular control. The explicit expression of the optimal portfolio selection strategy is obtained in the state feedback form involving both state process and its marginal distribution, via the solutions of Riccati ordinary differential equations with time-inconsistent solution.

On mean-field stochastic maximum principle for near-optimal controls for Poisson jump diffusion with applications

In this paper, we study mean-field type stochastic control problems for systems described by mean-field stochastic differential equations with jump processes, in which the coefficients contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. We derive necessary as well as sufficient conditions of near-optimality for our model, using Ekeland’s variational principle, spike variation method and some estimates of the state and adjoint processes. Under certain concavity conditions with non-negative derivatives, we prove that the near-maximum condition on the Hamiltonian function in integral form is a sufficient condition for near-optimality. Our result differs from the classical one in the sense that here the adjoint equation has a mean-field type, while the second-order adjoint equation remains the same as in the classical case. As an application, our results are applied to a mean-variance portfolio selection where explicit expression of the near-optimal portfolio selection strategy is obtained in the state feedback form involving both state process and its marginal distribution, via the solutions of Riccati ordinary differential equations.

A study of the solution to the Riccati equation in term structure modelling

We study numerical (i.e. approximate) and analytical (i.e. exact) solutions to the system of Riccati ordinary differential equations (RODE) used in estimating the class of Gaussian affine term structure models (ATSM). We base our study on accuracy and convergence time, and find that usage of the approximate solution and exact solution to the RODE yield similar accuracy. Our results are consistent with the no-arbitrage condition specified in ATSM. We also show that exact solutions, when available, lead to faster convergence times than approximate solutions.

Optimal investing stopping in stochastic environment

PurposeThe purpose of this paper is to investigate how to determine optimal investing stopping time in a stochastic environment, such as with stochastic returns, stochastic interest rate and stochastic expected growth rate.Design/methodology/approachTransformation method was used for solving optimal stopping problem by providing a way to transform path‐dependent problem into a path‐independent one. Based on option pricing theory, optimal investing stopping time was thought of as an optimal executed timing problem of American‐style option.FindingsFirst, the authors transform a path‐dependent stop timing problem to a path‐independent one with transformation under very general conditions, to directly use the existing conclusion of optimal stopping time literature. Second, when dynamics of capital growth is homogeneous, the authors changed the two dimensional optimal stop timing problem into a single dimension problem based on the assumption of zero exercise costs. Third, the authors investigated the comparative dynamics about asset selling boundary on asset value, state variable and return predictability. With constant discount rate and growth rate, the optimal selling timing depends on the simple comparison between capital cost and growth rate.Originality/valueThe paper's contributions to analysis method may be as follows. The authors demonstrate how to transform a path‐dependent stopping problem into a path‐independent one under general conditions. The transform method in this article can be applied to other path‐dependent optimal stopping problems. In particular, a Riccati ordinary differential equation for the transformation is set up. In most examples commonly met in finance, the equation can be solved explicitly.

Predictive mathematical model of time saved on descents in road cycling achieved through reduction in aerodynamic drag area

This paper presents empirically-based simulation results of descent time to completion in road cycling. A mathematical model is formulated to predict time saved on road cycling descents where a cyclist’s position is static through manipulation of aerodynamic drag area, system parameters and initial conditions. Road cyclists often adopt drastic static riding positions on long descents in order to minimize aerodynamic drag and optimize performance, measured as race time to completion. In those riding positions bicycle control is compromised and the risk of fall and injury increases. The aims of this study were to empirically determine whether there is a difference in aerodynamic drag area associated with the ‘Top-Tube’ descending riding position compared to ‘Normal’ descending riding position, and to investigate the effect of the difference on time to completion of road descents. Two elite male Australian time-trial cyclists were tested in an open jet wind tunnel. Drag force was measured using a force platform and a custom air bearing drag measurement system at 50 Hz at wind velocity of 15.6 m/s. Athletes were tested in their ‘Normal’ descending position and in the ‘Top-Tube’ position. Based on Newtonian–Lagrangian equations of motion of the cyclist–bicycle system, an analogue mathematical model as a non-linear Riccati ordinary differential equation was developed to enable prediction of velocity and time to completion of a road cycling course descent of known length and gradient as measures of performance for an athlete of known mass and empirically determined drag area in a descending position. Previously, proposed models of cycling performance have been based on physiological, anthropometric, and mechanical power output. No general closed-form time-to-completion mathematical model for cycling was found in the literature. Analytical solutions allow for a concise investigation of a dynamical system model behaviour that is not as readily available with a numerical solution. Wind tunnel testing showed up to 25% reduction in drag area for changes to cyclists riding position from their ‘Normal’ to the ‘Top-Tube’ dropped descending position. The analytical solution to the nonlinear Riccati differential equation showed that large time savings as a result of reduction of drag area can be made on road cycling race descents. In the example scenario simulated here, 29.2 s may be saved on a 5 km descent of 10% gradient with 25% reduction in aerodynamic drag area (CdA). The ‘Top-Tube’ riding position is associated with a large reduction in aerodynamic drag area in road descents compared to conventional descending riding position. Our model enables the prediction of time to completion on descents. This may assist cyclists to assess the trade-off between undertaking increased risk associated with drastic rider descending position and the potential for improved performance in the context of race tactics and strategy.

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Borner-Durr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.

Planar nonautonomous polynomial equations: The Riccati equation

We give a few sufficient conditions for the existence of two periodic solutions of the Riccati ordinary differential equation in the plane. We give also examples of the equation without periodic solutions.

SOLVABLE AFFINE TERM STRUCTURE MODELS

AnAffine Term Structure Model(ATSM) is said to be solvable if the pricing problem has an explicit solution, i.e., the corresponding Riccati ordinary differential equations have a regular globally integrable flow. We identify the parametric restrictions which are necessary and sufficient for an ATSM with continuous paths, to be solvable in a state space, where, the domain of positive factors, has the geometry of a symmetric cone. This class of state spaces includes as special cases those introduced byDuffie and Kan (1996), and Wishart term structure processes discussed byGourieroux and Sufana (2003). For all solvable models we provide the procedure to find the explicit solution of the Riccati ODE.

The Riccati Equation in Mathematical Finance

This paper uses ideas from symbolic computation to classify solutions to an important class of problems in mathematical finance and thus provides a linkage between these two fields. We show that Kovacic’s concept of closed-form solutions to the Riccati ordinary differential equation can be used to provide a precise mathematical definition that is useful in certain financial models. We extend this definition to a broader class of problems and discuss how these ideas can be usefully applied to practical problems in the finance area. We provide a specific application by developing a new implementation of the Cox–Ingersoll–Ross interest-rate model that may be of practical interest.

NEW SUFFICIENT CONDITIONS FOR THE APPROXIMATION OF DISTINCT SOLUTIONS OF THE QUADRATIC EQUATION IN BANACH SPACES


 
 
 Using the "theory of majorants" we provide new sufficient conditions for the approximation of distinct solutions of the quadratic equation in Banach spaces. Our results are applied to a Riccati ordinary differential equation.