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Parameter identification for portfolio optimization with a slow stochastic factor

  • Lei Hu ORCID logo and Dinghua Xu EMAIL logo
Published/Copyright: June 22, 2022
Journal of Inverse and Ill-posed Problems
From the journal Journal of Inverse and Ill-posed Problems Volume 30 Issue 6

Abstract

In this paper, we intend to identify two significant parameters – expected return and absolute risk aversion – in the Merton portfolio optimization problem under an exponential utility function where volatility is driven by a slow mean-reverting diffusion process. First, we find the approximate solution of the fully nonlinear Hamilton–Jacobi–Bellman equation for the Merton model by the stochastic asymptotic approximation method. Second, we estimate parameters – expected return and absolute risk aversion – through the approximate solution and prove the uniqueness and stability of the parameter identification problem. Finally, we provide an illustrative example to demonstrate the capacity and efficiency of our method.

Keywords: Portfolio optimization; slow stochastic factor; asymptotic approximation; parameter identification problem; inverse problems
MSC 2010: 35Q91; 35R30; 60J65; 65C05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871435

Award Identifier / Grant number: 11471287

Funding statement: This work was supported by the National Natural Science Foundation of China with grant no. 11871435 and 11471287.

A The accuracy of the approximation for 𝜑

The linear PDE (2.15) is rewritten as

(A.1) ( δ ⁢ M + δ ⁢ M 1 + L ) ⁢ φ = 0 , φ ⁢ ( T , z ) = 1 ,

where ℳ is defined in (2.5), and we define

M 1 = - ρ ⁢ g ⁢ λ ⁢ ∂ ∂ ⁡ z , L = ∂ ∂ ⁡ t - λ 2 2 ⁢ q ,

where λ ⁢ ( z ) := μ / σ ⁢ ( z ) . Similar to the method for the approximate solution of the value function in Section 2.2, we obtain the approximate solution of 𝜑, φ ≈ φ 0 ⁢ ( t , z ) + δ ⁢ φ 1 ⁢ ( t , z ) , where φ 0 and φ 1 are given by

φ 0 ⁢ ( t ) = e ⁢ x ⁢ p ⁢ ( - λ 2 2 ⁢ q ⁢ ( T - t ) ) , φ 1 ⁢ ( t , z ) = ρ 2 ⁢ q ⁢ ( T - t ) 2 ⁢ λ 2 ⁢ λ ′ ⁢ g ⁢ φ 0 .

With this choice of functions φ i , i = 0 , 1 , the following equations are satisfied:

(A.2) L ⁢ φ 0 = 0 ,
(A.3) L ⁢ φ 1 + M 1 ⁢ φ 0 = 0 .
Define the residual R = φ - ( φ 0 + δ ⁢ φ 1 ) . From equations (A.1), (A.2) and (A.3), one obtains

(A.4) ( δ ⁢ M + δ ⁢ M 1 + L ) ⁢ R = - δ ⁢ M ⁢ φ 0 - δ 3 2 ⁢ M 1 ⁢ φ 1 - δ 2 ⁢ M ⁢ φ 1 = δ ⁢ S δ ⁢ ( t , z ) ,

where the source term S δ ⁢ ( t , z ) can simply be computed using the equations for ℳ, M 1 , φ 0 and φ 1 . Using the terminal condition in (A.1) and the terminal values for φ i , i = 0 , 1 , one obtains that the residual function 𝑅 satisfies the terminal condition

(A.5) R ⁢ ( T , z ) = 0 .

Denoting by Z t δ the diffusion process with infinitesimal generator δ ⁢ M + δ ⁢ M 1 , the residual 𝑅, solution of the PDE problems (A.4) and (A.5), is given by the Feynman–Kac formula

R ⁢ ( t , z ) = δ ⁢ E ⁢ { ∫ t T e - 1 2 ⁢ q ⁢ ∫ t s λ 2 ⁢ ( Z u δ ) ⁢ d u ⁢ S δ ⁢ ( s , Z s δ ) ⁢ d s | Z t δ = z } .

Under our assumptions, one sees by direct computation that S δ is at most polynomially growing in 𝑧, and one obtains | R ⁢ ( t , z ) | ≤ δ ⁢ C , that is,

φ = φ 0 ⁢ ( t , z ) + δ ⁢ φ 1 ⁢ ( t , z ) + O ⁢ ( δ ) .

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Received: 2020-12-10
Revised: 2022-03-01
Accepted: 2022-04-10
Published Online: 2022-06-22
Published in Print: 2022-12-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Parameter identification for portfolio optimization with a slow stochastic factor
  3. Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model
  4. On the inverse gravimetry problem with minimal data
  5. On stable parameter estimation and short-term forecasting with quantified uncertainty with application to COVID-19 transmission
  6. Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems
  7. Convergence analysis of iteratively regularized Gauss–Newton method with frozen derivative in Banach spaces
  8. Identification of an unknown flexural rigidity of a cantilever Euler–Bernoulli beam from measured boundary deflection
  9. Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation
  10. Ill-posed problems and the conjugate gradient method: Optimal convergence rates in the presence of discretization and modelling errors
  11. Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs
https://doi.org/10.1515/jiip-2020-0156
Keywords for this article
Portfolio optimization; slow stochastic factor; asymptotic approximation; parameter identification problem; inverse problems

Articles in the same Issue

  1. Frontmatter
  2. Parameter identification for portfolio optimization with a slow stochastic factor
  3. Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model
  4. On the inverse gravimetry problem with minimal data
  5. On stable parameter estimation and short-term forecasting with quantified uncertainty with application to COVID-19 transmission
  6. Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems
  7. Convergence analysis of iteratively regularized Gauss–Newton method with frozen derivative in Banach spaces
  8. Identification of an unknown flexural rigidity of a cantilever Euler–Bernoulli beam from measured boundary deflection
  9. Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation
  10. Ill-posed problems and the conjugate gradient method: Optimal convergence rates in the presence of discretization and modelling errors
  11. Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs
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