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.  . 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.
The contributions of the present paper are twofold. First, we solve the optimal investment problem and prove a verification result for a CRRA utility while  solves for a logarithm utility function. They do discuss the CRRA case but leave proofs to some references. However, our work shows that it is highly non-trivial to prove the existence and uniqueness of a classic solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is essential for solving the optimal portfolio choice problem and implementing numerical methods. Second, our model may incorporate stochastic volatility or stochastic risk premium. Thus it shall be powerful to explain more financial phenomena, e.g. flight-to-quality, under-diversification, and possibly disclose more economic insights.
Technically, in the strand of the relevant literature,  introduces an Ornstein-Uhlenbeck (OU) type process of subordinator to model volatility. After that, there are several works about optimal portfolio selection in a model with OU type processes of subordinators. For example,  solves the portfolio choice problem in a model where volatility is a linear combination of Ornstein-Uhlenbeck type processes of subordinators.  solves the optimal investment and consumption problem in a similar model where a state variable (economic factor) is an Ornstein-Uhlenbeck type process of subordinator. They deal with a more general model, compared to  . As a result, they arrive at a nonlinear partial integro-differential equation, instead of a linear one in the latter. In both of these two papers, there are no jump components in the dynamics of asset prices. In stark contrast to the literature we incorporate jumps in the asset price in the present paper and hence confront a new/different challenge (if not more difficult) when we solve the optimal portfolio choice problem.
In our general setting, the volatility is a function of the state variable (the jump intensity) following a Hawkes process. Hence the present paper may combine two strands of research: modeling volatility by an OU type of process of subordinator and modeling jumps by Hawkes processes together. There are several papers studying the case where there are jumps in volatilities, e.g.  , etc., while the present paper considers jumps in both of asset price and volatility, and in jump intensity as well. This leads to a more complicated HJB equation than in  . As a result, our model has the features of stochastic volatility and self-exciting, while the optimal investment problem is still solvable in sense of by an iteration method, given that the iteration is proved to converge correctly to the unique classical solution of the problem.
This paper is incremental to the few aforementioned papers and dedicated to solving several technical problems related to CRRA utility functions. Nevertheless, the literature on portfolio choice is vast and is growing quickly. We would like to refer to some of them, for example,  , which proposes a semivariance method for diversified portfolio selection;  , that discusses a portfolio adjusting problem. The both assume that security returns are subject to experts’ estimations.
The organization of the paper is as follows. We formulate our model and optimal 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.
2. Problem Formulation
2.1. Hawkes Process for Self-Exciting Jumps
A Hawkes process is a counting process with self-exciting feature. Roughly speaking, it is a compounded Poisson process with stochastic intensity.
Given a complete probability space , a counting process, , satisfies
where the intensity process is given by the integrated form
or by the differentiation form
here is the long-run average of the jump intensity corresponding to the jump; is the decay rate driving the jump intensity back to the long-run average before a jump occurs; is a constant indicating non-negative impact of the jump occurrence on the jump intensity.
A jump process with the jump intensity described by (2.1) is called a Hawkes process  . It is known that the process is stationary if . A Hawkes process differs from a doubly stochastic Poisson process since its increments are not independent. is not Markovian but is. The compensated process is a local martingale. For more information and a formal definition of Hawkes process, we refer to  . 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  show an evidence that this channel is convincing.
Figure 1. A sample path of Hawkes and stock price .
Meanwhile, the mean-reversion property of (2.1) prevents the jump intensity from explosion given . Indeed, taking the expectation of (2.1) and using we obtain that
2.2. Asset Dynamics with Self-Exciting Jumps
We consider a market with a risk-free asset (bond) and a risky asset (stock). The price of the risk-free asset follows dynamics:
where r is the risk-free interest rate. The price of the risky asset follows a stochastic differential equation
where is a Hawkes process described in the preceding section, is a standard Brownian motion, and is a random jump size independent of the jump process and the Brownian motion. To be more precise, we define
where are i.i.d. random variables that are independent of either the Brownian motion or the jump process.
Note that the volatility and the risk premium are set to be functions of the jump intensity. Given appropriate conditions about the mappings (e.g. monotonic mappings), we may write and , so our model is compatible with many models studied in stochastic volatility literature.
2.3. Optimal Portfolio Selection Problem
Now we turn to the Merton’s problem: An investor invests in the risky asset and the risk-free asset in a time horizon . In order to maximize the expected utility of the terminal wealth, the investor needs to find an optimal investment strategy.
Let be the wealth of portfolio at time t and 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 , in particular, is assumed to take a form of , where is a Gaussian random variable. This setting is popularly admitted in the literature of jump-diffusion model. See, for example, the seminal paper of  . As a result, an admissible strategy shall satisfy the constraint . Thus, shorting either stock or bond is not permitted. This constraint condition may be relaxed1 according to distribution and support set of a specific jump size . 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 such that
where is the expectation conditioned on and . 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 regardless of an amount of efforts. The case of logarithm utility (corresponding to ) has been studied in  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
1If a tight bound of Y is , where a and b are positive constants, then is admissible if and only if .
Here subscripts denote partial derivatives and Y is a random variable having the same distribution as . In this paper, we shall make a full mathematical analysis of the HJB Equation (2.5) subject to the terminal condition (2.6), and a certain growth condition such as (2.8) below.
There are several papers studying the case where there are jumps in volatilities, e.g.  . 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  . The framework of our model is the same as that in  , 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 . Note that, as a suboptimal strategy, investing everything into the risk-free asset gives the lower bound
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: For some constant .
3. Mathematical Analysis of the HJB Equation
3.1. Scaling Invariance
Note that applying the same strategy for two initial portfolios with initial condition and respectively, we find that the corresponding wealth of the two portfolios differ by a factor of x at any time . Hence, optimal strategies do not depend on x, and we have the scaling invariance
where is the time to expiry and . Plugging (3.1) into (2.5) and using Remark 2.2 we obtain the equation for H:
2We assume that the optimizer is achieved in the interior.
It is easy to check that both A and B are concave function of . Hence, if , then there exists a unique such that2
In fact, with integrability of the jump size Y the first order condition brings us that
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 and Y and show that the HJB Equation (3.2) subject to the initial condition (3.3) and a certain growth condition admits a unique solution. In the next section we prove a verification result showing that the solution obtained solves the optimal investment problem.
For the case of CRRA utility,  suggest to prove the existence of a solution by verifying a contracting mapping as  . Our attempts show that it is not a trivial task to do that, instead, we use a different analytic method to accomplish the mission.
3.2. Basic Assumptions
First of all, we state some necessary assumptions as follow.
Recall that we assume . Thus a.s., .
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.  . Hence the assumption (3.8) is equivalent to say that 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.
Under (3.8) we define
3.3. An Integral Formulation
Let . We study the problem
where, with A and B as in (3.4), is a non-local operator defined by
Note that satisfies, for any positive constant c and continuous functions u and v,
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 simply as .
Let M be a positive constant to be determined. For fixed and along the characteristic curve , we obtain from (3.11) that
Integration over gives
Substituting the definition of into the expression we obtain the integral formulation:
for every where D and 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.
3.4. Determination of the Constant M
First we consider the function
Direct calculation yields
Hence, we have the following:
Lemma 1. For each , there exists a unique such that
In addition, the following holds:
1) If , then and ;
2) If , then ;
3) If , then . In addition, is smooth on I.
Consequently, if (3.8) holds, then for as in (3.10),
We now define
This implies that
3.5. Monotonic Iteration
We now solve (3.14) by the following iteration: for each non-negative integer n and , we define iteratively by
Since for every and, we see that is a well-defined family of continuous functions on D. We introduce
Lemma 2. For each integer and every,.
Proof. We use a mathematical induction. Assume that for all. Then when and, we have
Thus, and for all. Consequently,. Also, by (3.16),. It then follows from (3.19) that
Hence, by mathematical induction, for every and every non-negative integer n. This completes the proof.
In the sequel, we focus on the case. Note that
Lemma 3. For each integer, on D.
Proof. We need only consider the case. When, we obtain from (3.18), (3.19), and the definition of in (3.15) that
Thus,. Next using (3.20) we can show by a mathematical induction that on. This completes the proof.
Now we establish an upper bound. We define
Lemma 4. For each integer and,.
Proof. We use an induction argument. Assume that the assertion holds for some. Then for, by (3.19)-(3.20) we obtain
Thus, the assertion of the Lemma holds for. This completes the proof.
3.6. Solution of the Integral Equation
The family is a bounded monotonic family, so exists. We wish to prove a uniform convergence. For this, we introduce a norm by
Note that when,
Hence, using we obtain, when and,
By a mathematical induction, one can derive that
Hence, we have the following:
Lemma 5. There exists a function such that
In addition, H is a solution of (3.14) and has the following properties:
3.7. Lipschitz Continuity
For each, we define
For, we consider the function
Lemma 6. Let and. Then for each there exist a unique such that
In addition, is Lipschitz continuous on, , and
Proof. When, we have so
It remains to consider the case. Note that for,
Thus, there exists a unique such that on attains its maximum at. Note that
Here we define if; in this case we have for any. Hence, as, we see that is continuous and that I is an open set. When, by the implicitly function theorem,
Since for almost every, by the continuity of, we hence know that and. This completes the proof.
We now calculate the norm of. Notice that
When, we have so. When and,
Hence, we have
Lemma 7. Under the condition of Lemma 6, we have
Now, applying this estimate for (3.19) and using and we obtain
Lemma 8. For every non-negative integer n, and for each,
Proof. Clearly, the assertion (3.25) holds when. Assume that (3.25) holds for some. Then we have
Thus, by mathematical induction, the assertion of the Lemma holds.
Similarly, using (3.24), we can obtain an estimate for. We omit the details.
Now assume that
Differentiating with respect to we obtain
Hence, using estimate (3.22) for, we find that
We can calculate, when and,
Next, we calculate for and,
Note that, for,
It then follows that
Thus, we obtain
Hence, we have the following lemma.
Lemma 9. There exists a constant such that for each non-negative integer n,
Proof. The estimate for follows from a mathematical induction. The estimate for and follows by differentiating (3.24).
Now sending, obtain the following:
Theorem 1. Assume (3.7) and (3.8). Then the problem (3.11) admits a classical solution H that has the following properties:
1) For each and,;
2) For each,
Theorem 2. There exists a unique solution of (3.11) in the following class
Proof. Let H and be two solutions of in (3.11) in. Suppose is not true. Then there exists such that. Set
Then. Now for each we define
Then. Since, we see that. Denote by the characteristic curve passing through. Then
Hence, denoting we obtain
This implies that
Now define. Then we derive that
This implies that, for,
which contradicts the boundedness of. Thus we must have. Similarly, we can show that, so. This completes the proof.
4. 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.
Theorem 3. Suppose. Assume that solves (2)--(3). Set and set by (5). Then, the value function defined in (2.4); also the optimal investment strategy is given by.
4.1. Preliminary Results
Lemma 10. Let be positive constants and be stochastic process satisfying
Then: 1) With, for each constant and,
where is the solutions of the o.d.e.
2) Assume that. Denote by the unique positive root of. Then where a) if, then for every, b) if, then at time
Proof. The assertion of the Lemma follows from a general result for the joint characteristic function of and (see e.g.  ):
where satisfy the Riccati equation:
4.2. Proof of the Verification Theorem
Let be an admissible strategy and be the resulting wealth of the portfolio. Let be the solution of HJB equation obtained in the previous section. Subject to, by Itô Lemma, we have
Under the condition, since and H is bounded, we can verify that
Therefore, taking expectation of (4.7) we obtain
After taking the supreme over all admissible strategies, we then obtain
Finally, one can check that if we take, the above gives us an identity. Thus,
This completes the verification theorem.
The above theorem provides a verification result for a general framework of stochastic volatility and double jump models.
5. 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  , where and are constants. In the present paper, we extend the case to allow both of them stochastic. In the second case of, the Equation (3.2) has a solution in form of. This type of model is used to study effects of rare events in  . They do not show the verification of this solution though.
5.1. Logarithm Utility
When the utility function is logarithm, i.e., the solution is more explicit. Applying the same strategy for initial condition and one finds that the optimal strategy does not depend on x. Hence, setting we have. Then we obtain a simpler first order condition for:
Note that, so there exists a unique solution such that. The PDE for becomes
where is the differential-difference operator associated with the self-exciting Hawkes process and F is a function associated with the asset dynamics defined by
By Feynman-Kac formula, can be written as an expectation of a function of the process:
When and are constants,  study the optimal investment and consumption problem, and prove a verification result for the logarithm utility case. In the present, we may allow both and be stochastic.
For this logarithm case, it is straightforward to show that if all jumps are negative, i.e., then, 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.
5.2. The Case of Stochastic Volatility
We let and. Substituting the conditions into the model and setting, we arrive at
This model is close to the one used in  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 by a pure jump Lévy process with no drift and positive increments (subordination). Such a process is used to model volatility in  ,  , or  , where no simultaneous jump is assumed in their asset price.
Given the objective problem and the utility function, we let and derive the HJB equation:
The solution can be expressed as
with. The optimal strategy is given by
When the assumption (3.8) or holds, we know a classical solution to the HJB equation exists by the results in the preceding sections. In such a case, classical solutions to the ordinary differential equations (5.4) exist. We show below that could be explosive in a finite time horizon if the assumption is not satisfied.
We consider the special case:. Assume which conflicts with the assumption (3.8). In this case, (5.4) becomes
where. Note that given, we have
We suppose and choose such that
then we have
Hence blows up during where is given by
As a conclusion, the optimal portfolio choice problem in the stochastic volatility model may meet the issue of exploding in a finite time horizon. This paper provides a sufficient condition under which a classical solution exists and exploding does not happen. Our results shall be useful in studies of a stochastic volatility model or alike.
5.3. 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 to and, respectively. The figure indicates that the iteration converges monotonically. The following parameters are assumed for this example.
where in makes the total excess return goes to negative as jump occurs at a high frequency. A constant, e.g. satisfying the condition (3.8) as well results in a figure with the same pattern. The time horizon T is 0.2, and the iteration procedure is stopped as soon as
where ‘s are points of a partition of.
Figure 2. This figure shows the numerical iteration algorithm generates sequences converging monotonically to the limits that correspond to and, respectively.
6. 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 () greater than 1. It is also possible to extend our framework to a multi-dimension case like  . 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.  ). At last, as mentioned earlier, the assumption (3.8) can be relaxed to be
We provide a proof of existence of a continuous solution under this relaxed condition in the appendix. However, we need certain structural conditions on derivatives of and, in order to follow a similar line of proofs presented in Section 3 and show the regularity and uniqueness of the solution. We will investigate these conditions in the future.
The authors acknowledge support from NSFC (No. 11371280 and No. 71771142) and NSF (DMS-1516344).
Appendix. Existence of a Continuous Solution under a Relaxed Condition
In this appendix, we establish the existence of a solution H of (3.11) under the condition (3.9) instead of the condition (3.8). For each, we define
We write the partial differential equation in (3.11) as
We define a family iteratively as follows:. If is defined, we define as the unique solution of the linear problem
In term of characteristic curves, the linear problem for is well-posed.
1) Note that
It then follows by a comparison principle for the equation for that.
2) Next, assume that is an integer and. Then
Here we use the fact that and. Thus, by comparison principle, we have. Consequently, by mathematical induction, is an increasing family.
3) Note that, since is a concave function,
Note that if we have. Thus, under the assumption
Now assume that. Then we have
It then follows by comparison principle that.
Thus we have, for each integer,
Then is a bounded monotonic sequence of continuous functions. It follows from the Dini theorem that the sequence converges uniformly on compact subsets of D; so is continuous. Under certain structural conditions on derivatives of and we may follow a similar line of proofs presented in Section 3 to show the regularity and uniqueness of the solution.