2,537 research outputs found
Turnpike Property and Convergence Rate for an Investment Model with General Utility Functions
In this paper we aim to address two questions faced by a long-term investor
with a power-type utility at high levels of wealth: one is whether the turnpike
property still holds for a general utility that is not necessarily
differentiable or strictly concave, the other is whether the error and the
convergence rate of the turnpike property can be estimated. We give positive
answers to both questions. To achieve these results, we first show that there
is a classical solution to the HJB equation and give a representation of the
solution in terms of the dual function of the solution to the dual HJB
equation. We demonstrate the usefulness of that representation with some
nontrivial examples that would be difficult to solve with the trial and error
method. We then combine the dual method and the partial differential equation
method to give a direct proof to the turnpike property and to estimate the
error and the convergence rate of the optimal policy when the utility function
is continuously differentiable and strictly concave. We finally relax the
conditions of the utility function and provide some sufficient conditions that
guarantee the turnpike property and the convergence rate in terms of both
primal and dual utility functions.Comment: 29 page
Constrained Quadratic Risk Minimization via Forward and Backward Stochastic Differential Equations
In this paper we study a continuous-time stochastic linear quadratic control
problem arising from mathematical finance. We model the asset dynamics with
random market coefficients and portfolio strategies with convex constraints.
Following the convex duality approach, we show that the necessary and
sufficient optimality conditions for both the primal and dual problems can be
written in terms of processes satisfying a system of FBSDEs together with other
conditions. We characterise explicitly the optimal wealth and portfolio
processes as functions of adjoint processes from the dual FBSDEs in a dynamic
fashion and vice versa. We apply the results to solve quadratic risk
minimization problems with cone-constraints and derive the explicit
representations of solutions to the extended stochastic Riccati equations for
such problems.Comment: 22 page
Basket Options Valuation for a Local Volatility Jump-Diffusion Model with the Asymptotic Expansion Method
In this paper we discuss the basket options valuation for a jump-diffusion
model. The underlying asset prices follow some correlated local volatility
diffusion processes with systematic jumps. We derive a forward partial integral
differential equation (PIDE) for general stochastic processes and use the
asymptotic expansion method to approximate the conditional expectation of the
stochastic variance associated with the basket value process. The numerical
tests show that the suggested method is fast and accurate in comparison with
the Monte Carlo and other methods in most cases.Comment: 16 pages, 4 table
Intensity Process for a Pure Jump L\'evy Structural Model with Incomplete Information
In this paper we discuss a credit risk model with a pure jump L\'evy process
for the asset value and an unobservable random barrier. The default time is the
first time when the asset value falls below the barrier. Using the
indistinguishability of the intensity process and the likelihood process, we
prove the existence of the intensity process of the default time and find its
explicit representation in terms of the distance between the asset value and
its running minimal value. We apply the result to find the instantaneous credit
spread process and illustrate it with a numerical example.Comment: 15 pages, 2 figure
Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems
In this paper we prove that there exists a smooth classical solution to the
HJB equation for a large class of constrained problems with utility functions
that are not necessarily differentiable or strictly concave. The value function
is smooth if admissible controls satisfy an integrability condition or if it is
continuous on the closure of its domain. The key idea is to work on the dual
control problem and the dual HJB equation. We construct a smooth, strictly
convex solution to the dual HJB equation and show that its conjugate function
is a smooth, strictly concave solution to the primal HJB equation satisfying
the terminal and boundary conditions.Comment: 18 page
- …