Existence of solutions to initial value problem for a parabolic Monge–Ampère equation and application

A Transformation Method for Solving the Hamilton–jacobi–bellman Equation for a Constrained Dynamic Stochastic Optimal Allocation Problem

“…Macová andŠevčovič [30] or Ishimura andŠevčovič [20]). As a typical example leading to the stochastic dynamic optimization problem (1) in which the underlying stochastic process satisfies SDE (2) one can consider a problem of dynamic portfolio optimization in which the assets are labeled as i = 1, · · · , n, and associated with price processes {Y i t } t≥0 , each of them following a geometric Brownian [33,34], Browne [10], Bielecki and Pliska [7] or Songzhe [44]). The value of a portfolio with weightsθ =θ(y, t) is denoted by Yθ t .…”

Global existence of solutions to parabolic Monge–Ampère equations on Riemannian manifolds

“…The corresponding real equation was treated in several papers [3][4][5]. The existence and regularity of solutions to Monge-Ampère equations have been investigated by many mathematicians [7,2,8,13,10,15,9,17].…”

“…However, the case we investigate is significative to handle the problem with a general right side, as suggested in Remark 1.1. Our method is mainly based on references [7,2,8,13,10,15,9,17,11,6,14], that is, because Eq. (1.1) is a nonlinear parabolic equation, according to the standard theory [12], we can obtain that the solution exists for a short time.…”

Traveling wave solutions to the nonlinear evolution equation for the risk preference

“…Here we note that r is introduced with respect to the optimal value function. A similar transformation is considered in [12], where the transformation −V x /V xx is employed. Following [13], we make a change of variables x = e y (y = log x) and put r(y, τ ) = r(x, τ ); we infer that…”

“…We define f (r) := r 3 − (v + 1 − b)r 2 + Cr + a 2 , which is the factor of the denominator in (12). The condition (11) implies that f (r − ) = f (r + ) = 0.…”