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Licensed Unlicensed Requires Authentication Published online by De Gruyter June 22, 2022

Parameter identification for portfolio optimization with a slow stochastic factor

Lei Hu ORCID logo and Dinghua Xu

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.

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

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