Abstract
We show how to solve Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance (Propositions 1 and 2), i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound Hawkes process (GCHP), and for a capital R(t) (risk process) of an insurance company with the amount of claims described by the risk model based on GCHP. The main approach in both cases is to use functional central limit theorem for the GCHP to approximate it with a diffusion process. Then we construct and solve Hamilton–Jacobi–Bellman (HJB) equation for the expected utility function. The novelty of the results consists of the new Hawkes-based models and in the new optimal investment results in finance and insurance for those models.
Highlights
Merton optimal investment and consumption stochastic problem is one of the most studied classical problem in finance (Merton 1969, 1971, 1990; Bjork 2009; Karatzas and Shreve 1998)
We will show how to solve the Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance, i.e., for a wealth portfolio X (t) consisting of a bond and a stock price described by general compound
2017b), and for a capital R(t) of an insurance company with the amount of claims described by risk model based on general compound Hawkes process (GCHP) (Swishchuk 2018; Swishchuk et al 2020)
Summary
Merton optimal investment and consumption stochastic problem is one of the most studied classical problem in finance (Merton 1969, 1971, 1990; Bjork 2009; Karatzas and Shreve 1998). We will show how to solve the Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance, i.e., for a wealth portfolio X (t) consisting of a bond and a stock price described by general compound. (2) Merton portfolio optimization problem in insurance aims to find an optimal investment for the capital R(t) of an insurance company at time t (R(t) is the risk model based on general compound Hawkes process (GCHP) (Swishchuk 2018; Swishchuk et al 2020), when an investor decides to invest some capital A(t) in risky assets (e.g., stocks) and the rest, (R(t) − A(t)) in risk-free assets (e.g., bonds or bank account).
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