Merton Investment Problems in Finance and Insurance for the Hawkes-based Models

We show how to solve 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 Hawkes process (GCHP), and for a capital R(t) of an insurance company with the amount of claims described by the risk model based on GCHP. 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.


Introduction
Merton optimal investment and consumption stochastic problem is one of the most studied classical problem in finance ( [22,23,24,6,19]).In this paper, 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 Hawkes process (GCHP) ( [33,34,28]), and for a capital R(t) of an insurance company with the amount of claims described by risk model based on GCHP ( [30,35]).
Namely, we will show how to solve the following two portfolio investment problems: 1) Merton portfolio optimization problem in finance ( [22,23]) aims to find the optimal investment strategy for the investor with those two objects of investment, namely risk-less asset (e.g., a bond or bank account), paying a fixed rate of interest r, and a number of risky assets (e.g., stocks) whose price GCHP.In this way, in our case, we suppose that B t and S t follows the following dynamics, respectively: where G(t) := N (t)  i=1 a(X i ) is the GCHP, X i is a discrete-time Markov chain (MC) with finite or infinite states, r > 0 is the interest rate, and a(x) is a continuous and finite function on X.We note, that the justification of using HP in finance may be found in [7], and using GCHP that based on HP N(t) and a(X i ) may be found in [34,32].The model for a stock price S(t) that based on GCHP is a new and original in this paper.The investor starts with an initial amount of money, say X 0 = x, and wishes to decide how much money to invest in risky and risk-less assets to maximize the final wealth X t at the maturity T ; 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 actually the risk model based on general compound Hawkes process (GCHP) ( [30,35])), 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).We note, that the risk model R(t), based on GCHP, has the following representation: where R(t) = u is the initial capital, c > 0 is the premium rate, N(t) is the Hawkes process, X i is a discrete-time finite or infinite state Markov chain with state space X = {1, 2, 3, ..., N} or X = {1, 2, ..., N, ...}, respectively, and a(x) is a continuous and finite function on X.We note, that the justification for the Hawkes-based risk model in the form of the above equation may be found in [35].
The investor starts with an initial capital, say R(0) = u, and wishes to decide how much money to invest in risky and risk-less assets to maximize the capital R(t).
The Merton optimal investment and consumption stochastic problem in finance was first considered in the seminal papers of Merton, [22,23].General description of the problem and coverage of most today's problems and methods may be found in [9,10,6,24,19].
The first papers on stochastic optimal control in insurance appeared relatively recently, e.g., we would like to mention the papers written by Martin-Löf [21], Brockett and Xia [3], Browne [4], to name a few.Since then many papers and books were written on this topic including [1,18,26].Financial control methods applied in insurance, such as e.g. in the control and the management of the specific risk insurance companies, are described in [17].Risk theory for the compound Poisson process that is perturbed by diffusion was considered in [8].An analogue of the Cramer-Lundberg approximation in the optimal investment case was studied in [13].Asymptotic ruin probabilities and optimal investment were investigated in [11].Optimal risk distribution control model with application to insurance was studied in [36].Applications of stochastic processes in insurance and finance may be found in [25].Many mathematical methods and aspects in risk theory may be found in [5,12,14].
Hawkes process was first introduced in [15,16].Good introduction into Hawkes processes and their properties may be found in [20].GCHP and regime-switching GCHP were first introduced in [28] and studied in details using real data in [31,33,32,34].Risk model based on GCHP was first introduced in [27] and described in details in [30].Applications of the risk model based on GCHP to empirical data and optimal investment problem were considered in [35].
The paper is organized as follows.Section 2 is devoted to the definitions and properties of Hawkes process and general compound Hawkes processes, and LLN (Law of Large Numbers) and FCLT (Functional Central Limit Theorem) for them.Section 3 deals with Merton investment problem in finance for the stock price described by GCHP, and Section 4 deals with Merton investment problem in insurance for the risk model based on GCHP.Section 5 concludes and describes the future work.
2 General Compound Hawkes process

Hawkes Process
Definition (One-dimensional Hawkes Process) ( [15,16]).The onedimensional Hawkes process is a point process N(t) which is characterized by its intensity λ(t) with respect to its natural filtration: where λ > 0, and the response function or self-exciting function µ(t) is a positive function and satisfies +∞ 0 µ(s)ds := μ < 1.If (t 1 , t 2 , ..., t k ) denotes the observed sequence of past arrival times of the point process up to time t, the Hawkes conditional intensity is The Hawkes process is a self-exciting simple point process first introduced by A. Hawkes in 1971 ([15, 16]).The future evolution of a self-exciting point process is influenced by the timing of past events.
The process is non-Markovian except for some very special cases (e.g., exponential self-exiting function µ(t)).Thus, the Hawkes process depends on the entire past history and has a long memory.
The Hawkes process has wide applications in neuroscience, seismology, genome analysis, finance, insurance, and many other fields.
The constant λ is called the background intensity and the function µ(t) is sometimes also called the excitation function.
We suppose that µ(t) = 0 to avoid the trivial case, which is, a homogeneous Poisson process.Thus, the Hawkes process is a non-Markovian extension of the Poisson process.
The interpretation of the above equation for λ(t) is that the events occur according to an intensity with a background intensity λ which increases by µ(0) at each new event then decays back to the background intensity value according to the function µ(t).
Choosing µ(0) > 0 leads to a jolt in the intensity at each new event, and this feature is often called a self-exciting feature, in other words, because an arrival causes the conditional intensity function λ(t) in ( 1)-( 2) to increase then the process is said to be self-exciting.
The following LLN and CLT for HP may be found in [2].The convergences are considered in weak sense for the Skorokhod topology.
Remarks 1 and 2 above give the ideas about the averaged and diffusion approximated HP on a large time interval.
This general model is rich enough to: • incorporate non-exponential distribution of inter-arrival times of orders in HFT or claims in insurance (hidden in N(t)) • incorporate the dependence of orders or claims (via MC X i ) • incorporate clustering of of orders in HFT or claims (properties of N(t)) • incorporate order or claim price changes different from one single number (in a(X i )).
This model is also very general to include: -in finance: • compound Poisson process: k=1 X k , where N(t) is a Poisson process and a(X k ) = X k are i.i.d.r.v.
• compound Hawkes process ( [31]): k=1 X k , where N(t) is a Hawkes process and a(X k ) = X k are i.i.d.r.v.
• compound Markov renewal process: k=1 a(X k ), where N(t) is a renewal process and X k is a Markov chain; -in insurance: • classical Cramer-Lundberg model: a(X i ) = X i , X i are i.i.d.r.v., and µ(t) = 0 (then N(t) is a poisson process); • Sparre-Andersen model: a(X i ) = X i , X i are i.i.d.r.v., µ(t) = 0, and N(t) is a renewal process; • Markov-modulated model: , where X(t) is a MC; we call this model regime-switching risk model based on GCHP ( [33,28]).
Remark 3. The formulas for a * and σ look much simpler in the case of two-state Markov chain X i = {−δ, +δ} : (p, p ′ ) are transition probabilities of Markov chain X k , and From CLT for HP, sec.3.1, and from Theorem 1 above follows the following FCLT for GCHP (pure jump diffusion limit).

Merton Investment Problem in Finance for the Hawkes-based Model
Let us consider Merton portfolio optimization problem.We suppose that B t and S t follows the following dynamics, respectively: where G(t) := N (t) i=1 a(X i ) is the GCHP, r > 0 is the interest rate.We note, that the justification of using HP in finance may be found in [7], and using GCHP that based on HP N(t) and a(X i ) may be found in [34,32].The model for a stock price S(t) that based on GCHP is a new and original in this paper.
The investor starts with an initial amount of money, say X 0 = x, and wishes to decide how much money to invest in risky and risk-less assets to maximize the expected utility of the terminal wealth X t at the maturity T, i.e., X T .
We denote by n(t) := (n B (t), n S (t)) an investor portfolio, where n B (t) and n S (t) are the amounts in cash invested in the bonds and the risky assets, respectively.The value X(t) at time t of such portfolio is We suppose that our portfolio is admissible, i.e., X(t) ≥ 0, a.s., 0 ≤ t ≤ T, and self-financing, i.e., Suppose that G(t) := N (t) i=1 a(X i ) follows FCLT when t → +∞, ( [30,35]), thus G(t) can be approximated as (see sec.3.3) where μ := +∞ 0 µ(s)ds < 1, a * is a average of a(x) over stationary distribution of MC X i , λ is a background intensity, σ > 0 is defined in sec.3.3.For exponential decaying intensity μ = α/β.
Thus, S(t) in ( 1) can be presented in the following way: Using Itô formula we can get: Then the change of the wealth process X t can be rewritten in the following way, taking into account ( 1)-( 4): ( Let π(t) := n S (t)/X(t) be the portion of wealth invested in the assets/stocks at time t.
Then, from ( 1)-( 5), we have the following expression for dX(t) : Finally, after replacing X(t) with X π (t), to stress the dependence of X(t) on π t , from (6) we have the following equation for dX π (t) : Our main goal is to solve the following optimization problem: meaning to maximize the wealth/value function or performance criterion where U(x) is a utility function.To find optimal π, we follow the standard procedure in this case (see [6,19]).For the utility function we take the logarithmic one, U(x) = log(x).Therefore, we have to maximize max π E[log(X π T )|X 0 = x], .Solving the equation (7) and maximizing non-martingale term in the exponent for the solution, we can find the optimal investment solution π * (t) : and σ * and a * are defined in sec.3.3.
Remark 6.As we can see from the expression for π * (t), the optimal investment solution depends on all parameters of the Hawkes-based model, namely, Hawkes process's parameters λ and µ(t), Markov chain X i and function a(x) through a * .for the Hawkes-based Risk Model Let us consider R(t) as the risk model based on GCHP, namely, Here, X i (claim sizes) is a finite state Markov chain, a(x) is a continuous and bounded function on X = {1, 2, ..., N}-space state for X i , and N(t) is a Hawkes process with intensity λ(t) > 0, independent of X i , and satisfying: Here, µ(t) is self-exiting function.
We note, that the justification for the Hawkes=based risk model in the form of the above equation may be found in [35].
Since we will consider optimization for a first insurer, thus we will concentrate on problems with infinite planning horizon.
Let A(t) be an amount invested in a risky asset, and suppose that the price S(t) of the risky asset follows GBM, i.e., dS(t) = S(t)(adt + bdW (t)), where a is a real constant, b > 0.
Further, the leftover, R(t) − A(t) > 0, is invested in a bank account (or bonds) with interest rate r > 0, thus

Merton Investment Problem in Insurance
Let β(t) := A(t)/S(t) be the number of assets held at time t.Then the position of the insurer has the following dynamics: Therefore, the dynamics for R(t) is (taking into account all above equations for S(t), d(R(t) A (t)) and dR(t)): Here, G(t) = N (t) i=1 a(X i ).Let π(t) := A(t)/R(t) be the fraction of the total wealth R(t) invested in the risky assets.
Then, we can rewrite the equation for dR(t) in the following way (we use notation R π (t) to stress dependence of R(t) from π): where W (t) is a standard Brownian motion.As for the control at time t we will take the function π(t), i.e., the fraction of the total wealth R π (t) which should be invested in risky assets.
We will show how to find the optimal strategy π(t), which maximizes our expected utility function, E[U(R π (t))|R π (0) = r], where U(r) is a special utility function.
Here, W 1 (t) is a Wiener process independent of W (t) (the case for correlated W 1 (t) with W (t) i.e., such that [W (t), W 1 (t)] = ρdt, can be considered as well with some modifications).
We suppose that c > a * λ 1−μ (safety loading condition).After substituting (2) into (1) for dR(t) we get: where W 2 (t) is a standard Wiener process independent of W (t) and W 1 (t).Generator for R π (t) is (here, R π (0) = x) Thus, we have to maximize E r [U(R π (t))], where U(r) is a utility function.
The HJB equation has the following form: We take the exponential utility function U(r) = −e −pr , p > 0.
As we can see, the optimal control π(t) = π does not depend on t, thus is a constant, and contains all initial parameters of the risk model based on GCHP.
Remark 7. As we can see from the expression for π(t), the optimal control depends not only from interest rate r, but also from all parameters of the Hawkes-based model, namely, Hawkes process's parameters λ and µ(t), Markov chain X i and function a(x) through a * , and the asset's parameters a and b.
Merton Investment Problem for Poisson-based Risk Model in Insurance.
Corollary: The optimal control π P (t) for Poisson Risk Model is: .

Discussion
The future work will be devoted to numerical example based on real data, simulations and considering the case without diffusion approximation for R(t), by creating HJB equation for initial risk model R(t) = R(0) + ct − i=1 a(X i ) and by solving this HJB equation for initial risk model base on GCHP.Probably we cannot avoid here numerical methods, because the HJB equation in this case cannot be solved exactly with a close form solution.
We described in this paper how to solve 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 Hawkes process (GCHP), and for a capital R(t) of an insurance company with the amount of claims described by the risk model based on GCHP.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.