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 ( Ω , F , ( F t ) t ≥ 0 , P ) , a counting process, { N t } t ≥ 0 , satisfies

P ( N t + Δ t − N t = 1 | F t ) = λ t Δ t + o ( Δ t ) ,

P ( N t + Δ t − N t > 1 | F t ) = o ( Δ t ) ,

where the intensity process λ t is given by the integrated form

λ t = e − α t λ 0 + ( 1 − e − α t ) λ ∞ + ∫ 0 t β e − α ( t − s ) d N s

or by the differentiation form

d λ t = α ( λ ∞ − λ t ) d t + β d N t ; (2.1)

here λ ∞ is the long-run average of the jump intensity corresponding to the jump; α > 0 is the decay rate driving the jump intensity back to the long-run average before a jump occurs; β ≥ 0 is a constant indicating non-negative impact of the jump occurrence on the jump intensity.

A jump process { N t } with the jump intensity described by (2.1) is called a Hawkes process [8] . It is known that the process is stationary if β α < 1 . A Hawkes process differs from a doubly stochastic Poisson process since its increments are not independent. { N t } is not Markovian but { ( N t , λ t ) } is. The compensated process N t − ∫ 0 t λ s d s 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 using E [ d N s ] = E [ λ s ] d s we obtain that

E [ λ t ] = λ 0 e ( β − α ) t + α λ ∞ α − β ( 1 − e ( β − α ) t ) → λ ∞ 1 − β α       as     t → ∞ .

We consider a market with a risk-free asset (bond) and a risky asset (stock). The price { B t } of the risk-free asset follows dynamics:

d B t = r B t d t ,

where r is the risk-free interest rate. The price { S t } of the risky asset follows a stochastic differential equation

d S t S t − = [ μ ( λ t ) + r ] d t + σ ( λ t ) d W t + Y t d N t (2.2)

where { N t } is a Hawkes process described in the preceding section, { W t } is a standard Brownian motion, and Y t > − 1 is a random jump size independent of the jump process and the Brownian motion. To be more precise, we define

∫ 0 t Y t d N t = ∑ 1 ≤ i ≤ N t y i

where { y i } i = 1 ∞ are i.i.d. random variables that are independent of either the Brownian motion or the jump process.

Note that the volatility σ and the risk premium μ are set to be functions of the jump intensity. Given appropriate conditions about the mappings (e.g. monotonic mappings), we may write λ = λ ( σ ) and μ = μ ( σ ) , so our model is compatible with many models studied in stochastic volatility literature.

Now we turn to the Merton’s problem: An investor invests in the risky asset and the risk-free asset in a time horizon [ 0, T ] . In order to maximize the expected utility of the terminal wealth, the investor needs to find an optimal investment strategy.

Let X t be the wealth of portfolio at time t and π t be the proportion of wealth invested into the risky asset at time t. Then

d X t X t − = π t d S t S t − + ( 1 − π t ) d B t B t = ( π t μ + r ) d t + π t σ d W t + π t Y t d N t . (2.3)

An investment strategy is an adapted stochastic process π = { π t } t ∈ [ 0 , T ] . It is admissible if the associated wealth process is non-negative almost surely. The jump size Y t , in particular, is assumed to take a form of e Z t − 1 , where Z t is a 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 ≤ π t ≤ 1 . 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 Y t . We denote all admissible strategies by A .

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

V ( x , λ , t ) = max π ∈ A E t x , λ [ U ( X T ) ] ,   π * = arg max π ∈ A E 0 x , λ [ U ( X T ) ] (2.4)

where E t x , λ is the expectation conditioned on X t = x and λ t = λ . In this paper, we solve the problem for the CRRA utility U ( x ) = x p / p . We prove the existence and uniqueness of a classical solution to the associated HJB equation when p ∈ ( 0,1 ) . By a similar approach, our framework may be extended to the case of p < 0 regardless of an amount of efforts. The case of logarithm utility (corresponding to p = 0 ) 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

α ( λ − λ ∞ ) V λ − V t + λ V = max π { ( r + π μ ) x V x + 1 2 π 2 σ 2 x 2 V x x + λ E [ V ( x ( 1 + π Y ) , λ + β , t ) ] } , (2.5)

V ( x , λ , T ) = U ( x ) . (2.6)

¹If a tight bound of Y is − a ≤ Y ≤ b , where a and b are positive constants, then { π t } is admissible if and only if − 1 / b ≤ π t ≤ 1 / a .

Here subscripts denote partial derivatives and Y is a random variable having the same distribution as Y t . 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. [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

V ( x , λ , t ) ≥ U ( x e r t ) = U ( x ) e p r t . (2.7)

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 C > 0 .

1 C ≤ V ( x , λ , t ) U ( x ) ≤ C ,     ∀ λ ≥ 0 , t ∈ [ 0 , T ] . (2.8)

Note that applying the same strategy for two initial portfolios with initial condition ( X 0 , λ 0 ) = ( x , λ ) and ( X 0 , λ 0 ) = ( 1 , λ ) respectively, we find that the corresponding wealth of the two portfolios differ by a factor of x at any time t ∈ [ 0, T ] . Hence, optimal strategies do not depend on x, and we have the scaling invariance

V ( x , λ , t ) = U ( x ) H ( λ , T − t ) , (3.1)

where τ = T − t is the time to expiry and H ( λ , τ ) : = V ( 1 , x , T − τ ) / U ( 1 ) . Plugging (3.1) into (2.5) and using Remark 2.2 we obtain the equation for H:

H τ + α ( λ − λ ∞ ) H λ = max 0 ≤ π ≤ 1 { A ( λ , π ) H ( λ , τ ) + B ( λ , π ) H ( λ + β , τ ) } , (3.2)

H ( λ , 0 ) = 1 , (3.3)

where

A ( λ , π ) = p r + p μ ( λ ) π − p ( 1 − p ) σ 2 2 π 2 − λ ,     B ( λ , π ) = λ E [ ( 1 + π Y ) p ] . (3.4)

²We assume that the optimizer is achieved in the interior.

It is easy to check that both A and B are concave function of π . Hence, if H > 0 , then there exists a unique π * such that²

π * ( λ , τ ) = arg max 0 ≤ π ≤ 1 { A ( λ , π ) H ( λ , τ ) + B ( λ , π ) H ( λ + β , τ ) } . (3.5)

In fact, with integrability of the jump size Y the first order condition brings us that

π * = μ ( λ ) ( 1 − p ) σ ( λ ) 2 + λ ( 1 − p ) σ ( λ ) 2 E [ ( 1 + π * Y ) p − 1 Y ] H ( λ + β , τ ) H ( λ , τ ) . (3.6)

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 σ ( ⋅ ) , μ ( ⋅ ) 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, [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.

First of all, we state some necessary assumptions as follow.

p ∈ ( 0 , 1 ) ,   μ ( ⋅ ) , σ 2 ( ⋅ ) ∈ C 2 ( [ 0 , ∞ ) ) . (3.7)

Recall that we assume Y ~ Y t = e Z t − 1 . Thus Y > − 1 a.s., E [ Y 2 ] < ∞ .

To explain our idea in a clear manner, we assume that

lim sup λ → ∞ [ μ ( λ ) + λ E [ Y ] ] < 0. (3.8)

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

lim sup λ → ∞ μ ( λ ) + λ E [ Y ] σ ( λ ) < ∞ . (3.9)

We shall discuss this later in the Appendix.

Under (3.8) we define

λ * : = min { s ∈ [ λ ∞ , ∞ ) | μ ( λ ) + λ E [ Y ] ≤ 0 , ∀ λ ∈ [ s , ∞ ) } . (3.10)

Let τ = T − t . We study the problem

{ H τ + α ( λ − λ ∞ ) H λ = F [ H ( ⋅ , τ ) ] ( λ ) ∀   τ ∈ [ 0 , T ] , λ ≥ 0 , H ( λ , 0 ) = 1 ∀   λ ≥ 0 , (3.11)

where, with A and B as in (3.4), F is a non-local operator defined by

F [ u ] ( λ ) : = max 0 ≤ π ≤ 1 { A ( λ , π ) u ( λ ) + B ( λ , π ) u ( λ + β ) } .

Note that F satisfies, for any positive constant c and continuous functions u and v,

F [ c u ] = c F [ u ] ,     F [ u ] − F [ v ] ≤ F [ u − v ] . (3.12)

We shall use characteristic curves to convert (3.11) into an equivalent integral formulation. For this we introduce

D = [ 0 , ∞ ) × [ 0 , T ] ,     Λ ( λ , τ , t ) : = λ ∞ + ( λ − λ ∞ ) e α ( t − τ ) . (3.13)

For notational simplicity, in the sequel, we write Λ ( λ , τ , t ) simply as Λ .

Let M be a positive constant to be determined. For fixed ( λ , τ ) ∈ D and along the characteristic curve { ( Λ ( λ , τ , t ) , t ) } 0 ≤ t ≤ τ , we obtain from (3.11) that

d d t [ e M t H ( Λ ( λ , τ , t ) , t ) ] = e M t [ M H ( Λ , t ) + H τ ( Λ , t ) + α ( Λ − λ ∞ ) H λ ( Λ , t ) ] = e M t { M H ( Λ , t ) + F [ H ( ⋅ , t ) ] ( Λ ) } ,   ∀ t ∈ [ 0 , τ ] .

Integration over t ∈ [ 0, τ ] gives

H ( λ , τ ) = e − M τ + ∫ 0 τ { M H ( Λ ( λ , τ , t ) , t ) + F [ H ( ⋅ , t ) ] ( Λ ( λ , τ , t ) ) } e M ( t − τ ) d t .

Substituting the definition of F into the expression we obtain the integral formulation:

H = e − M τ + ∫ 0 τ max 0 ≤ π ≤ 1 { [ M + A ( Λ , π ) ] H ( Λ , t ) + B ( Λ , π ) H ( Λ + β , t ) } e M ( t − τ ) d t , (3.14)

for every ( λ , τ ) ∈ D where D and Λ : = Λ ( λ , τ , t ) 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.

First we consider the function

f ( λ , π ) : = A ( λ , π ) + B ( λ , π ) ,     ∀   λ ≥ 0 , π ∈ [ 0 , 1 ] .

Direct calculation yields

f ( λ , 0 ) = p r ,

f π ( λ , π ) = p [ μ ( λ ) − ( 1 − p ) π σ 2 ( λ ) + λ E [ y ( 1 + π y ) p − 1 ] ] ,

f π π ( λ , π ) = − p ( 1 − p ) { σ 2 ( λ ) + λ E [ y 2 ( 1 + π y ) p − 2 ] } < 0.

Hence, we have the following:

Lemma 1. For each λ ≥ 0 , there exists a unique π * ( λ ) ∈ [ 0,1 ] such that

ζ ( λ ) : = max 0 ≤ π ≤ 1 f ( λ , π ) = f ( λ , π * ( λ ) ) ≥ f ( λ , 0 ) = p r . (3.15)

In addition, the following holds:

1) If λ ∈ I 0 : = { λ ≥ 0 | μ ( λ ) + λ E [ y ] ≤ 0 } , then π * ( λ ) = 0 and ζ ( λ ) = p r ;

2) If λ ∈ I 1 : = { λ ≥ 0 | μ ( λ ) − ( 1 − p ) σ 2 ( λ ) + λ E [ y ( 1 + y ) p − 1 ] ≥ 0 } , then π * ( λ ) = 1 ;

3) If λ ∈ I : = [ 0 , ∞ ) \ ( I 1 ∪ I 2 ) , then π * ( λ ) ∈ ( 0,1 ) . In addition, π * ( ⋅ ) is smooth on I.

Consequently, if (3.8) holds, then for λ ∗ as in (3.10),

π * ( λ ) = 0 ,     ζ ( λ ) = p r ,     ∀ λ ≥ λ ∗ . (3.16)

We now define

M = max 0 ≤ π ≤ 1,0 ≤ λ ≤ λ ∞ + ( λ ∗ − λ ∞ ) e α T { | A ( λ , π ) | } .

This implies that

M + A ( λ , π ) ≥ 0 ,     ∀ π ∈ [ 0,1 ] , λ ∈ [ 0, λ ∞ + ( λ ∗ − λ ∞ ) e α T ] . (3.17)

We now solve (3.14) by the following iteration: for each non-negative integer n and ( λ , τ ) ∈ D , we define iteratively H n by

H 0 ( λ , τ ) : = e p r τ , (3.18)

H n + 1 ( λ , τ ) : = e − M τ + ∫ 0 τ max 0 ≤ π ≤ 1 { [ M + A ( Λ , π ) ] H n ( Λ , t )             + B ( Λ , π ) H n ( Λ + β , t ) } e M ( t − τ ) d t . (3.19)

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

Hence,

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.

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:

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,

where

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

Thus,

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

, a.e.

Hence, using estimate (3.22) for, we find that

We can calculate, when and,

Hence,

where

Next, we calculate for and,

Note that, for,

Hence,

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.

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. [9] ):

where satisfy the Riccati equation:

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

Thus,

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.

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, [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 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.

We let and. Substituting the conditions into the model and setting, we arrive at

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 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.

Given the objective problem and the utility function, we let and derive the HJB equation:

The solution can be expressed as

and solve

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.

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.