scholarly journals Deep Learning for Constrained Utility Maximisation

2021 ◽  
Author(s):  
Ashley Davey ◽  
Harry Zheng
Keyword(s):  
Differential Equation ◽  
Deep Learning ◽  
Stochastic Control ◽  
Backward Stochastic Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Computational Cost ◽  
Utility Maximisation ◽  
Highly Nonlinear ◽  
Path Dependent ◽  
Hamilton Jacobi Bellman

AbstractThis paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms.

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

2013 ◽  
Vol 5 (04) ◽  
pp. 407-422 ◽  
Author(s):  
Matthew A. Beauregard ◽  
Qin Sheng
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Nonlinear Partial Differential Equation ◽  
Partial Differential ◽  
Spatial Adaptation ◽  
Monitor Function ◽  
Monitoring Strategies ◽  
Highly Nonlinear ◽  
Equidistribution Principle

AbstractFinite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-calledprimitiveregime, the rest belong to a later category of themodifiedtype, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation

2015 ◽  
Vol 5 (4) ◽  
pp. 387-404 ◽  
Author(s):  
Jie Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Convergence Analysis ◽  
Stochastic Control ◽  
Backward Stochastic Differential Equation ◽  
Order Convergence

AbstractConvergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

scholarly journals A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
I. Rusagara ◽  
C. Harley
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Transfer Coefficients ◽  
Partial Differential ◽  
One Dimensional ◽  
Heat Transfer Coefficients ◽  
Highly Nonlinear ◽  
Triangular Fin

The temperature profile for fins with temperature-dependent thermal conductivity and heat transfer coefficients will be considered. Assuming such forms for these coefficients leads to a highly nonlinear partial differential equation (PDE) which cannot easily be solved analytically. We establish a numerical balance rule which can assist in getting a well-balanced numerical scheme. When coupled with the zero-flux condition, this scheme can be used to solve this nonlinear partial differential equation (PDE) modelling the temperature distribution in a one-dimensional longitudinal triangular fin without requiring any additional assumptions or simplifications of the fin profile.

scholarly journals The Dynamic Programming Method of Stochastic Differential Game for Functional Forward-Backward Stochastic System

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Shaolin Ji ◽  
Chuanfeng Sun ◽  
Qingmeng Wei
Keyword(s):  
Dynamic Programming ◽  
Differential Game ◽  
Backward Stochastic Differential Equation ◽  
Transformation Method ◽  
Dynamic Programming Method ◽  
Stochastic Differential Game ◽  
Value Functions ◽  
Girsanov Transformation ◽  
Path Dependent ◽  
Hamilton Jacobi Bellman

This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differential equation (FBSDE). For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs). Applying the Girsanov transformation method introduced by Buckdahn and Li (2008), the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP), we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.

scholarly journals Application of perturbation theory to a P-wave eikonal equation in orthorhombic media

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. C309-C317 ◽  
Author(s):  
Alexey Stovas ◽  
Nabil Masmoudi ◽  
Tariq Alkhalifah
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Eikonal Equation ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equation ◽  
P Wave ◽  
Perturbation Approach ◽  
Highly Nonlinear ◽  
Order Polynomial ◽  
Zero Offset

The P-wave eikonal equation for orthorhombic (ORT) anisotropic media is a highly nonlinear partial differential equation requiring the solution of a sixth-order polynomial to obtain traveltimes, resulting in complex and time-consuming numerical solutions. To alleviate this complexity, we approximate the solution of this equation by applying a multiparametric perturbation approach. We also investigated the sensitivity of traveltime surfaces in ORT media with respect to three anelliptic parameters. As a result, a simple and accurate P-wave traveltime approximation valid for ORT media was derived. Two different possible anelliptic parameterizations were compared. One of the parameterizations includes anelliptic parameters defined at zero offset: [Formula: see text], [Formula: see text], and [Formula: see text]. Another parameterization includes anelliptic parameters defined for all symmetry planes: [Formula: see text], [Formula: see text], and [Formula: see text]. The azimuthal behavior of sensitivity coefficients with different parameterizations was used to analyze the crosstalk between anelliptic parameters.

A new modified polynomial-based optimal control design approach

2020 ◽  
pp. 095965182094689
Author(s):  
Morteza Nazari Monfared ◽  
Ahmad Fakharian ◽  
Mohammad Bagher Menhaj
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Immune Cell ◽  
Nonlinear Partial Differential Equation ◽  
Power Series Expansion ◽  
Feedback Controller ◽  
Main Concern ◽  
Partial Differential ◽  
State Dependent ◽  
Hamilton Jacobi Bellman

The main concern of this article is addressing a new modified approach to design a nonlinear optimal controller. The modification focuses on proposing a new approximate solution for the Hamilton–Jacobi–Bellman nonlinear partial differential equation. The introduced solution works based on the state-dependent power series expansion presentation of the involved functions in the Hamilton–Jacobi–Bellman partial differential equation. Applying this technique results in releasing a set of free state-dependent functions in the controller structure that can be adjusted to fulfill some special control missions in addition to the optimization objectives. They are formed based on the specific formulation of the candidate Lyapunov function. The proposed approach is exemplified for an intricate biological system, immunogenic tumor-immune cell interaction in the human body, to clarify the mechanism of designing the controller and adjusting the arrays of the free matrices. The closed-loop system by presented optimal state feedback controller meets the predefined optimization objectives without getting feedback from a hard-measurable state. It is achieved by adjusting the aforementioned released functions such that an optimal output feedback controller is obtained. To have some insights into the performance of the system and the effectiveness of the controller, the positiveness of the system’s states is proved and checked numerically by applying the differential transformation method to the system’s differential equations. Finally, to highlight the abilities of the proposed approach from different aspects, some simulations are carried out.

OPTIMAL STOCHASTIC CONTROL PROBLEM UNDER MODEL UNCERTAINTY WITH NONENTROPY PENALTY

2017 ◽  
Vol 20 (03) ◽  
pp. 1750015
Author(s):  
WAHID FAIDI ◽  
ANIS MATOUSSI ◽  
MOHAMED MNIF
Keyword(s):  
Differential Equation ◽  
Control Problem ◽  
Stochastic Control ◽  
Model Uncertainty ◽  
General Framework ◽  
Backward Stochastic Differential Equation ◽  
Terminal Condition ◽  
Penalty Term ◽  
Stochastic Control Problem ◽  
Continuous Filtration

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type [Formula: see text]-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty is studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.

Sign in / Sign up

Export Citation Format

Share Document