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

Abstract

A singular nonlinear partial differential equation (PDE) is introduced, which can be interpreted as the evolution of the risk preference in the optimal investment problem under the random risk process. The unknown quantity is related to the Arrow-Pratt coefficient of relative risk aversion with respect to the optimal value function. We show the existence of monotone traveling wave solutions and the nonexistence of non-monotone such solutions, which are suitable from the standpoint of financial economics.

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

“…Following the methodology of the Riccati transformation first proposed by Abe and Ishimura in [1] and later studied by Ishimura et al [19,21], Xia [47], or Macová and Ševčovič [30] for problems without inequality constraints, and further analyzed by Ishimura and Ševčovič [20], we introduce the following transformation:…”

“…for any ϕ ∈ int(I M ) where a M , b M ∈ R n and a M > 0, b M ≥ 0 and c M ∈ R are constants calculated using the same formulas as in (19) and (20), where data (columns and rows) from Σ and µ corresponding to the active indices in the particular set M are removed.…”

“…In fact, the Riccati transformation is the logarithmic derivative of the derivative of the value function. Here we apply the Riccati transformation proposed and analyzed in a series of papers by Ishimura et al [1,19,21]. In the context of a class of HJB equations with range constraints, such a transformation has been analyzed recently by Ishimura and Ševčovič in [20] where also a traveling wave solution to the HJB equation has been constructed.…”

On traveling wave solutions to Hamilton-Jacobi-Bellman equation with inequality constraints

“…Proof We follow the ideas from [5]. If we assume to the contrary z ′ (ξ 0 ) = 0 then, as a consequence of the uniqueness of solutions to ODEs (37) and ( 14), we obtain z(ξ) ≡ const for all ξ ∈ R. Consequently, the profile v is constant, a contradiction.…”

“…x V < 0 and ∂ x V > 0. Proposition 3 Suppose that the terminal condition V (x, T ) ≡ u(x) is a smooth function and there exist constants λ ± > 0 such that λ − < −u ′′ (x)/u ′ (x) < λ + for all x ∈ R. Then λ − /ω < ϕ(x, t) < λ + /ω for all x ∈ R and t ∈ [0, T ] where ϕ is a solution to equation (5) or ( 18) and satisfying the terminal condition ϕ(x, T ) = −(1/ω)u ′′ (x)/u ′ (x), x ∈ R.…”