Large Time Behavior of Solutions to SemiLinear Equations with Quadratic Growth in the Gradient

Abstract

This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it. LARGE TIME BEHAVIOR OF SOLUTIONS TO SEMI-LINEAR EQUATIONS WITH QUADRATIC GROWTH IN THE GRADIENT SCOTT ROBERTSON AND HAO XINGAbstract. This paper studies the large time behavior of solutions to semi-linear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space. Two types of large time behavior are obtained: i) pointwise convergence of the solution and its gradient; ii) convergence of solutions to associated backward stochastic differential equations. When the state space is R d or the space of positive definite matrices, both types of convergence are obtained under growth conditions on coefficients. These large time convergence results have direct applications in risk sensitive control and long term portfolio choice problems.

Long-Term Optimal Investment in Matrix Valued Factor Models

“…First, Lemma A.1 below shows that the current assumptions (Assumptions 2.1, 2.3, 2.4, and 3.1) verify [48,. Therefore, [48, Propositions 2.5 and 2.7 and Theorem 3.9] can be used to study the long horizon problem.…”

“…Assumption 3.4 in [48]. Item (i) is clearly satisfied by the current specification of f and g. Item (ii) follows from Lemma A.3 below.…”

“…Using the framework developed in [48], we are able to consider general, nonaffine, matrix valued state variable models. Moreover, stochastic correlation between the state variable and risky assets can be treated, whereas a special (constant) correlation structure is needed to ensure the affine structure.…”

“…Moreover, stochastic correlation between the state variable and risky assets can be treated, whereas a special (constant) correlation structure is needed to ensure the affine structure. However, having only the analytic results in [48] is not enough. To link these analytic results to portfolio optimization problems, a verification result is essential.…”

“…To this end, our Proposition 3.6 extends [30, Theorem 3.1] to the standard class of nonnegative wealth processes. In fact, the framework introduced in [48] can also be used to study models with vector valued factor models; see [48, section 3.1]. However, we do not pursue this direction in the present article, and instead use matrix valued models to illustrate our methodology.…”