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

Ishimura, Naoyuki; Ševčovič, Daniel

doi:10.1007/s13160-012-0087-8

Cited by 7 publications

(18 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for all x ∈ R, t ∈ [0, T ) subject to the terminal condition V(x, T ) := U(x) (see e.g. 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]).…”

Section: Problem Statementmentioning

confidence: 99%

“…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. Concerning numerical methods for solving the transformed quasi-linear parabolic PDE there are recent papers by Ishimura, Koleva and Vulkov [17,18,25,26] where they considered a simplified problem without inequality constraints on the optimal control function.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Kilianová¹,

Ševčovič²

2013

ANZIAM J.

View full text Add to dashboard Cite

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear HamiltonJacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical Hölder smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method.2000 Mathematics subject classification: Primary: 35K55, Secondary: 34E05 70H20 91B70 90C15 91B16.

show abstract

Section: Problem Statementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kilianová¹,

Ševčovič²

2013

ANZIAM J.

View full text Add to dashboard Cite

show abstract

“…the fixed policy iteration method for solving HJB equation (10) corresponds to the numerical solution of the transformed equation (12) by means of the semi-implicit scheme (30) in which α is approximated by its linearization at ϕ j−1 from the previous time step τ j−1 .…”

Section: 2mentioning

confidence: 99%

“…In contrast to the problem involving the terminal utility maximization only (cf. [1], [10], [12], [13]), the resulting transformed equation is a non-local quasi-linear parabolic equation containing non-local terms involving the intertemporal utility function. The non-local parabolic equation can be further transformed into a coupled system of two quasi-linear local parabolic equations.…”

Section: Introductionmentioning

confidence: 99%

Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton–Jacobi–Bellman equation

Kilianová

Ševčovič

2019

Japan J. Indust. Appl. Math.

Self Cite

View full text Add to dashboard Cite

In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) equation. We propose the so-called Riccati method for transformation of the fully nonlinear HJB equation into a quasi-linear parabolic equation with non-local terms involving the intertemporal utility function. As a numerical method we propose a semi-implicit scheme in time based on a finite volume approximation in the spatial variable. By analyzing an explicit traveling wave solution we show that the numerical method is of the second experimental order of convergence. As a practical application we compute optimal strategies for a portfolio investment problem motivated by market financial data of German DAX 30 Index and show the effect of considering intertemporal utility on optimal portfolio selection.AMS-MOS Classification: 35K55, 34E05, 70H20, 91B70, 90C15, 91B16

show abstract

“…In the context of solving the HJB equation, the Riccati transformation was proposed by Abe and Ishimura in [1] and later studied by Ishimura andŠevčovič [14], Xia [41], Macová andŠevčovič [24], Kilianová andŠevčovič [16], Kilianová and Trnovská [17]. The Riccati transformation of the value function V is defined as follows:…”

Section: The Riccati Transformation Of the Hjb Equation To A Quasi-limentioning

confidence: 99%

Expected utility maximization and conditional value-at-risk deviation-based Sharpe ratio in dynamic stochastic portfolio

Kilianová¹,

Ševčovič²

2018

Kybernetika

Self Cite

View full text Add to dashboard Cite

In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation (CV aRD) based Sharpe ratio for measuring risk-adjusted performance of a dynamic portfolio. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index and we evaluate and analyze the dependence of the CV aRD-based Sharpe ratio on the utility function and the associated risk aversion level.

show abstract

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

Cited by 7 publications

References 8 publications

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

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

Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton–Jacobi–Bellman equation

Expected utility maximization and conditional value-at-risk deviation-based Sharpe ratio in dynamic stochastic portfolio

Contact Info

Product

Resources

About