Topic
Riccati equation
About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.
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TL;DR: This paper derives closed-form expressions for the optimal portfolios and efficient frontier in terms of the solution of the so-calledstochastic Riccati equation (SRE) associated with the quadratic hedging and mean-variance problems.
Abstract: This paper concerns the problems ofquadratic hedging andpricing, andmean-variance portfolio selection in anincomplete market setting with continuous trading, multiple assets, and Brownian information. In particular, we assume throughout that the parameters describing the market model may be random processes. We approach these problems from the perspective oflinear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs); that is, we focus on the so-calledstochastic Riccati equation (SRE) associated with the problem. Excepting certain special cases, solvability of the SRE remains an open question. Our primary theoretical contribution is a proof of existence and uniqueness of solutions of the SRE associated with the quadratic hedging and mean-variance problems. In addition, we derive closed-form expressions for the optimal portfolios and efficient frontier in terms of the solution of the SRE. A generalization of theMutual Fund Theorem is also obtained.
183 citations
TL;DR: In this article, the authors considered the problem of finding a positive definite solution of the matrix equation X + A ∗ X -1 A = Q in a special case of the discrete-time Riccati equation.
182 citations
TL;DR: In this article, the authors established the existence of solutions of the initial value problem under the most general conditions on u(0), i.e., u(t) need only be such that R to the minus (2 divided by m-1 +N) sum (determinant x or = R) to the power of u(x)) dx is bounded independently of R or = 1.
Abstract: : This work establishes existence of solutions of the initial-value problem u(t) = delta (determinant u to m-1 power), u(x,0), = u(0)(x), where m 1, under the most general conditions on u(0). Namely, u(0) need only be such that R to the minus (2 divided by m-1 +N) sum (determinant x or = R) to the power of u(0)(x)) dx is bounded independently of R or = 1. Aronson and Caffarelli have shown this requirement to be necessary. Many auxiliary results are given in the form of estimates on solutions, uniqueness and continuous dependence theorems, etc. While the results may be viewed as 'technical' in that the main points consist of estimates of various sorts, the equation treated is of broad practical interest and the estimates reflect basic properties of the equation. The results obtained are the only ones known to the authors wherein the solvability of a realistic nonlinear initial value problem for a partial differential equation is now understood as completely as in the case of the heat equation.
182 citations
TL;DR: In this paper, three four-step methods for the numerical solution of the radial Schrodinger equation are compared theoretically and experimentally, and the results show that all of them are equivalent.
181 citations
TL;DR: In this article, the convergence and properties of the Riccati difference equation are studied for systems which are not necessarily stabilizable (in the filtering sense), particularly those having uncontrollable roots on the unit circle.
Abstract: This paper studies the convergence and properties of the solutions of the Riccati difference equation Special emphasis is given to systems which are not necessarily stabilizable (in the filtering sense), particularly those having uncontrollable roots on the unit circle Besides generalizing and unifying previous work, the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components
180 citations