Optimal Portfolio Choice in a Jump-Diffusion Model with Self-Exciting

We solve the optimal portfolio choice problem for an investor who can trade a risk-free asset and a risky asset. The investor faces both Brownian and jump risks and the jump is modeled by a Hawkes process so that occurrence of a jump in the risky asset price triggers more sequent jumps. We obtain the optimal portfolio by maximizing expectation of a constant relative risk aversion (CRRA) utility function of terminal wealth. The existence and uniqueness of a classical solution to the associated partial differential equation are proved, and the corresponding verification theorem is provided as well. Based on the theoretical results, we develop a numerical monotonic iteration algorithm and present an illustrative numerical example.


Introduction
Empirical studies suggest that asset price encounters jumps and its volatility is stochastic. Further studies show that jumps occur in clusters, that is, a sequence of jumps occur in short time following a (big) jump which occurs after a relatively long quiet period of time. The feature of clustered jumps can be caught by a type of stochastic process known as Hawkes process. In this paper, we model occurrence of jumps by a Hawkes process hence our model is an extension of well-known jump-diffusion models, e.g. [1]. Meanwhile, we assume that such Hawkes jumps may occur in asset price itself as well as in its volatility. As a result, our model merges with the vast literature of stochastic volatility. timal portfolio choice problem in Section 2. The Hawkes process is introduced in this section. In Section 3, we analyze the HJB equation. The existence and uniqueness of a classical solution to the equation are proved under some appropriate conditions. We also prove a verification theorem for the solution in Section 4.
Two illustrative examples and one numerical example are provided in Section 5. Conclusion and further discussion are in Section 6. An extension is supplied in Appendix.

Hawkes Process for Self-Exciting Jumps
where the intensity process t λ is given by the integrated form with the jump intensity described by (2.1) is called a Hawkes process [8]. It is known that the process is stationary if 1 β α < . A Hawkes process differs from a doubly stochastic Poisson process since its incre- is a local martingale. For more information and a formal definition of Hawkes process, we refer to [8]. Figure 1 illustrates a sample path of one self-exciting (Hawkes) process.
The Hawkes process has a feature of self-exciting which is ideal to model jumps in financial markets. As one jump occurs, the jump intensity is increased by the occurred jump through the mechanics of (2.1). Hence a sequent jump happens more likely in a unit time following. In other words, as a jump happens, it impacts on the jump intensity as well as on itself. As a result, one may see a sequence of jumps in a short frame of time after one (big) jump. Thus, jump is self-triggered through the channel described by (2.1) and jumps tend to be clustered. Of course, it may not be the only channel to generate clustered jumps, but the empirical studies of [9] show an evidence that this channel is convincing. Meanwhile, the mean-reversion property of (2.1) prevents the jump intensity from explosion given 0 β α ≤ < . Indeed, taking the expectation of (2.1) and us-

Asset Dynamics with Self-Exciting Jumps
We consider a market with a risk-free asset (bond) and a risky asset (stock

Optimal Portfolio Selection Problem
Now we turn to the Merton's problem: An investor invests in the risky asset and 0,T . In order to maximize the expected utility of the terminal wealth, the investor needs to find an optimal investment strategy.
Let t X be the wealth of portfolio at time t and t π be the proportion of wealth invested into the risky asset at time t. Then An investment strategy is an adapted stochastic process It is admissible if the associated wealth process is non-negative almost surely. The jump size t Y , in particular, is assumed to take a form of e 1 Gaussian random variable. This setting is popularly admitted in the literature of jump-diffusion model. See, for example, the seminal paper of [1]. As a result, an admissible strategy π shall satisfy the constraint 0 1 t π ≤ ≤ . Thus, shorting either stock or bond is not permitted. This constraint condition may be relaxed 1 according to distribution and support set of a specific jump size t Y . We denote all admissible strategies by  .
An optimal investment strategy is a strategy that maximizes the expected utility of the terminal wealth. That is, the objective of an investor is to find V and * π such that is the expectation conditioned on t X x = and t λ λ = . In this paper, we solve the problem for the CRRA utility We prove the existence and uniqueness of a classical solution to the associated HJB equation when . By a similar approach, our framework may be extended to the case of 0 p < regardless of an amount of efforts. The case of logarithm utility (corresponding to 0 p = ) has been studied in [2] while assuming constant volatility and risk premium. In Section 5, we will discuss an extension case of logarithm utility with stochastic volatility and stochastic risk premium, as an application of our general results.
The Hamilton-Jacobi-Bellman (HJB) equation associated with the above stochastic optimization problem can be derived as Here subscripts denote partial derivatives and Y is a random variable having the same distribution as t Y . In this paper, we shall make a full mathematical analysis of the HJB Equation There are several papers studying the case where there are jumps in volatilities, e.g. [5]. Different from only jumps in volatility, our model incorporates jumps in both of asset price and volatility, and in jump intensity as well. It is worth to mention that the HJB Equation (2.5) is more complicated than that in [5]. The framework of our model is the same as that in [2], except the setting of utility function. As well-known, CRRA utility functions generally involve much more difficult technical problems than the logarithm utility function.
Remark 2.1. The HJB Equation (2.5), together with the terminal condition (2.6), may have several solutions. From a view point of partial differential equation, it is necessary to prescribe an asymptotic behavior of the solution as λ → ∞ . Note that, as a suboptimal strategy, investing everything into the risk-free asset gives the lower bound rt prt Although it is very had to estimate an upper bound, we shall prove the existence of a unique solution of (2.5)-(2.6) under the following growth condition:

Scaling Invariance
Note that applying the same strategy for two initial portfolios with initial condition ( ) ( ) . Hence, optimal strategies do not depend on x, and we have the scaling  In fact, with integrability of the jump size Y the first order condition brings us The second term in the right hand side of the above formula is the hedging demand for the self-exciting jump risks.
In the rest of this section, we shall impose certain conditions on ( ) ( ) For the case of CRRA utility, [2] suggest to prove the existence of a solution by verifying a contracting mapping as [5]. Our attempts show that it is not a trivial task to do that, instead, we use a different analytic method to accomplish the mission.

Basic Assumptions
First of all, we state some necessary assumptions as follow.
Recall that we assume ~e 1 To explain our idea in a clear manner, we assume that Intuitively, the assumption (3.8) claims that the excess return of the asset will be negative if the jump frequency is high enough. In the literature of jump-diffusion models, the excess return is usually assumed with a compensation of jump risk, see, e.g. [10]. Hence the assumption (3.8) is equivalent to say that ( ) µ λ may not be enough compensated when (negative) jumps occur at a high frequency. That does not sound unreasonable.
The assumption may be relaxed to We shall discuss this later in the Appendix.

An Integral Formulation
where, with A and B as in (3.4),  is a non-local operator defined by We shall use characteristic curves to convert (3.11) into an equivalent integral formulation. For this we introduce For notational simplicity, in the sequel, we write ( ) Let M be a positive constant to be determined. For fixed ( ) , D λ τ ∈ and along the characteristic curve Substituting the definition of  into the expression we obtain the integral formulation: for every ( ) , ,t λ τ Λ =Λ are as in (3.13).
In the sequel we shall choose an appropriate positive constant M and solve the above integral equation by a monotonic iteration technique.

Determination of the Constant M
First we consider the function Hence, we have the following: In addition, the following holds: Consequently, if (3.8) holds, then for λ * as in (3.10),

Monotonic Iteration
We now solve (3.14) by the following iteration: for each non-negative integer n and ( ) ,t Q λ ∈ . Then when ( )   We wish to prove a uniform convergence. For this, we introduce a norm ⋅ by   In addition, H is a solution of (3.14) and has the following properties:   This completes the proof.

Verification Theorem and Optimality of the Solution
That the optimal investment problem can be solved through the preceding results is based on a verification result which guarantees that the solution to the HJB equation is the value function corresponding to the optimal investment problem.

Proof of the Verification Theorem
Therefore, taking expectation of (4.7) we obtain This completes the verification theorem.
The above theorem provides a verification result for a general framework of stochastic volatility and double jump models.

Two Illustrative Cases and a Numerical Example
In the following, we present two illustrative cases. The first case of logarithm utility function is studied in [2], where µ and σ are constants. In the present paper, we extend the case to allow both of them stochastic. In the second case of

Logarithm Utility
When the utility function is logarithm, i.e. ( ) ln X λ λ = one finds that the optimal strategy does not depend on x.
Hence, setting Then we obtain a simpler first order condition for * π : Note that 0 π <  , so there exists a unique solution where L is the differential-difference operator associated with the self-exciting Hawkes process and F is a function associated with the asset dynamics defined by σ are constants, [2] study the optimal investment and consumption problem, and prove a verification result for the logarithm utility case.
In the present, we may allow both µ and σ be stochastic.
For this logarithm case, it is straightforward to show that if all jumps are negative, i.e. 0 Y < , then * 0 π λ ∂ ∂ < , hence we find the fact consistent with the phenomena known as flight-to-quality: as a market crash happens, all positions in risky assets are reduced.

The Case of Stochastic Volatility
We let This model is close to the one used in [10] to study effects of rare events, except that there is no diffusion term in the variance dynamics here. The dynamics of the variance becomes a type of OU process of subordinations if we extensively replace t N by a pure jump Lévy process with no drift and positive increments (subordination). Such a process is used to model volatility in [3], [4], or [5], where no simultaneous jump is assumed in their asset price.
The solution can be expressed as We consider the special case: which conflicts with the assumption (3.8). In this case, (5.4) becomes Note that given 0 0 µ > , we have

A Numerical Example
In our proof, the monotonic iteration (3.18) and (3.19) actually suggest an iteration algorithm to find the solution numerically. Figure 2 gives a numerical example. The upper edge of curves stands for the limit of iteration corresponding

Conclusions and Discussions
We study the optimal portfolio choice problem in a jump-diffusion model where the jump likelihood is increased by jump itself. We establish the existence and uniqueness of a classical solution to the corresponding HJB equation. A verification theorem which guarantees the optimality of the solution is proved. Our approach relies on a monotonic iteration procedure which naturally hints a numerical algorithm. We consider CRRA utilities and a stochastic investment opportunity set which may nest interesting models in the literature.
By similar steps, our work may be extended to the case of the risk aversion (1 p − ) greater than 1. It is also possible to extend our framework to a multi-dimension case like [2]. With multi-assets, jumps may be not only self-excited but also mutually excited. The latter feature may be suitable to study financial contagions (see, e.g. [9]). At last, as mentioned earlier, the assumption (3.8) can be relaxed to be