Risk-Sensitive Asset Management under a Wishart Autoregressive Factor Model ()
1. Introduction
1.1. Risk-Sensitive Asset Management
Consider a continuous-time financial market that consists of one riskless asset and
risky assets. The price process of the riskless asset
and that of the
risky assets
,
, where
denotes the transpose of a vector or matrix, are semimartingales defined on a filtered probability space
. Define the wealth process
of a self-financing investor governed by the following stochastic differential equation (SDE):
(1.1)
where
is the initial wealth of the investor, and
,
is the dynamic investment strategy of the investor. Let
(1.2)
be the growth rate of the wealth at time
. For given constants
and
, define the risksensitized expected value of
by
![](https://www.scirp.org/html/8-1490151\96415b42-a3c0-4ef7-9f31-b86ff08a487b.jpg)
which is rewritten as
![](https://www.scirp.org/html/8-1490151\b0c09a88-767f-4189-9e93-719d13dc33e4.jpg)
and interpreted as the certainty equivalent value of
with respect to the exponential criterion function
. We are interested in maximizing
, that is,
(1.3)
which we call the risk-sensitive asset management problem. Here,
is a space of admissible investment strategies and is a subset of
, the totality of
-dimensional
-progressively measurable processes
on the time interval
such that
almost surely.
Remark 1.1 The risk-sensitive asset management problem (1.3) has been well-studied under a linear-Gaussian market model, for example, by [1-7]. In those works, the price processes
are given by the solutions to the following system of SDEs:
(1.4)
on a filtered probability space
endowed with the
-dimensional
-Brownian motion
,
. Here,
denotes the diagonal matrix whose
th element is equal to the
th element
of
,
,
, and
(1.5)
with
,
,
,
,
,
, and
. We reformulate (1.3) with (1.1), (1.2), (1.4), and (1.5) as a linear exponential quadratic Gaussian stochastic control problem, and the optimal investment strategy (portfolio)
for (1.3) is represented explicitly:
![](https://www.scirp.org/html/8-1490151\b2358c2a-f17f-4667-aeff-11c76bbcf406.jpg)
Here,
is the solution to a matrix differential Riccati equation, and
is the solution to a linear differential equation, including
.
Remark 1.2 Intuitively, recalling the cumulant expansion,
![](https://www.scirp.org/html/8-1490151\72750dfa-a858-41c8-a9a5-3c6afdef5a87.jpg)
where
denotes variance, we interpret (1.3) as a risk-sensitized optimization of the expected growth rate maximization,
![](https://www.scirp.org/html/8-1490151\48beff45-d99d-473a-992b-e079acdd670f.jpg)
1.2. Wishart Factor Model
The main aim of the present paper is to introduce a simple and tractable market model that satisfies the following requirements:
• The model describes the stochasticity of the covariance structure of
, interest rates, and mean-return rates of
.
• The model admits an explicit representation of the optimal investment strategy for (1.3).
For the purpose, we employ a Wishart autoregressive process as a stochastic factor, which is positive-definite symmetric matrix-valued. Such matrix-valued processes have been introduced and studied by [8], and recently, generalizations have been intensively studied, for example, see [9,10], and the references therein. Moreover, these processes are now extensively utilized for financial modeling. We can refer to the examples given below.
• Modeling of multivariate stochastic volatility (covariance) under the risk-neutral probability: see [11-16].
• Modeling of multivariate asset price process under physical probability with stochastic covariance and mean-return rates: see [14,17,18].
• Modeling of (term structure of) interest rates and stochastic intensity for credit risk: see [14,17,19,20].
Our market model defined by (2.1)-(2.4) in Section 2 is an extension of the model employed by [18], (see Example 2.1), who studied the expected CRRA-utility ma ximization of terminal wealth, which is essentially equivalent to (1.3). A main contribution of the present paper is a rigorous mathematical analysis of portfolio optimization problem (1.3) under a flexible Wishart autoregressive stochastic factor model: We strengthen the mathematical results in [18] by formulating an appropriate space of admissible trading strategies (see (3.5)) and showing a verification theorem for the candidate of the optimal strategy (see Theorem 3.1), both of which are omitted in [18].
In the next section, we introduce our market model with a Wishart autoregressive factor and present preliminary calculations of the associated Hamilton-JacobiBellman (HJB) equation for solving risk-sensitive asset management problem (1.3). In Section 3, we introduce our main results. In Section 4, we show the proof of the main theorem after preparing lemmas.
2. Setup
2.1. Market Model with Wishart Autoregressive Factor
Let
be a filtered probability space endowed with the
-dimensional
-Brownian motion
,
, and the
-dimensional
-Brownian motion
,
, which is independent of
. Using a constant vector
so that
, we define another
-dimensional Brownian motion
,
by
![](https://www.scirp.org/html/8-1490151\5027c05e-f1e6-42d3-a243-a1473806f2d7.jpg)
which is correlated with
as
![](https://www.scirp.org/html/8-1490151\dbecbf2e-0519-43c9-b509-cc1b65808848.jpg)
where
denotes the quadratic covariation, and
denotes Kronecker’s delta. We consider the price processes
, described by the following system of SDEs:
(2.1)
with the initial values
,
and
. Here, we denote by
the totality of
-dimensional, real, symmetric matrices, and
. Furthermore, for
, we define
(2.2)
where
,
, and
. Also, we assume that
is full rank, that is,
(2.3)
We also assume that
, and that
satisfies
(2.4)
Condition (2.4) ensures
almost everywhere on
, which was established by Mayerhofer et al. (2011) using a generalized form of SDE, including a jump martingale part. The
-valued process
is a stochastic factor process, which linearly depends on the covariance structure of
as
(2.5)
as well as on the interest rate
(2.6)
and on the so-called risk premium of
,
(2.7)
Remark 2.1 From (2.6), we see that the interest rate process
is included in the so-called affine class:
is an affine function of
and the process
, whose infinitesimal generator is given by
(2.8)
where
and
(2.9)
is indeed an affine diffusion. To review affine processes and their financial applications, see, for example, [21], [22], and the references therein.
Remark 2.2 The condition (2.7) on the structure of the risk-premium vector is rewritten as
![](https://www.scirp.org/html/8-1490151\fca6e682-8d5d-4df9-a23b-9471fdf12ba2.jpg)
So we interpret that the so-called mean-variance term in portfolio optimization theory is assumed to be constant.
The following are concrete examples of setting up (2.1) and (2.2).
Example 2.1 (Stochastic Covariance) Let
![](https://www.scirp.org/html/8-1490151\1f10f7ac-2efe-454b-a23d-f2d2f7cfafd8.jpg)
Concretely, we have
![](https://www.scirp.org/html/8-1490151\80316d78-5926-43c3-8250-679cc0ab1db6.jpg)
with the third equation in (2.1).
describes the infinitesimal covariance and the risk premium of
as
![](https://www.scirp.org/html/8-1490151\300e03fb-e75f-4c6d-be82-33f573673d84.jpg)
and
![](https://www.scirp.org/html/8-1490151\15e36c3e-2975-48d8-99f8-5e96c53a7bb0.jpg)
This is exactly the model employed in Section 1 of [18] to study expected CRRA-utility maximization of terminal wealth.
Example 2.2 (Stochastic Covariance and Interest Rate) We present a slight generalization of Example 2.1 to include stochasticity of interest rates. Let
![](https://www.scirp.org/html/8-1490151\64fc5129-48f7-4343-866a-8acf0e34929b.jpg)
where we set
if
. Then, letting
![](https://www.scirp.org/html/8-1490151\dac16cdb-aae8-40ec-8835-47d6f4b5db1a.jpg)
and
, we see that
![](https://www.scirp.org/html/8-1490151\999b5f43-a964-4daf-a10e-74d5e0e2ffbe.jpg)
is the risk-free interest rate with the latent factor
and that
![](https://www.scirp.org/html/8-1490151\9dc461aa-5a4c-45ec-a5cb-1a30d3849105.jpg)
where
describes the infinitesimal covariance and the risk premium of
as
![](https://www.scirp.org/html/8-1490151\cb98fc63-8e9a-4e97-b042-f513c7880c59.jpg)
and
![](https://www.scirp.org/html/8-1490151\ad51d42b-e4bd-467b-ad22-85dc255fe7f7.jpg)
Example 2.3 (Cox-Ingersoll-Ross Interest Rate Factor) Let
![](https://www.scirp.org/html/8-1490151\fcaa7969-4fd7-45cc-ac91-e0118964117a.jpg)
Then, we see
![](https://www.scirp.org/html/8-1490151\870ef0a5-1d64-4644-b749-dff7690a9770.jpg)
This financial market model with Cox-Ingersoll-Ross’s interest rate
is treated in [23] to study (1.3).
Under the financial market model comprising (2.1) and (2.2) with the assumptions (2.3) and (2.4), we are interested in treating the risk-sensitive asset management problem (1.3).
2.2. Deriving the HJB Equation
To tackle (1.3), we employ a dynamic programming approach: Recall that wealth process (1.1) of a selffinancing investor, combined with (2.1), is rewritten as
![](https://www.scirp.org/html/8-1490151\c1e26f76-63ae-4fc6-9882-234a3c1b0ffb.jpg)
So, we see
![](https://www.scirp.org/html/8-1490151\1a760f35-d8ac-4958-aa00-4266bd80715a.jpg)
where we set
(2.10)
Hence, we have
![](https://www.scirp.org/html/8-1490151\63c05aff-9675-44e7-9fa2-92d953f5aee5.jpg)
where we define
(2.11)
Let
![](https://www.scirp.org/html/8-1490151\0e1c3fb8-1a2d-4681-9473-81a7c7c4d034.jpg)
For
, we define the probability measure
on
by the formula
![](https://www.scirp.org/html/8-1490151\a9983ee8-d964-4ff6-9245-d8d9dea74b04.jpg)
By Cameron-Martin-Maruyama-Girsanov’s theorem, we see that the
-valued process
, defined by
![](https://www.scirp.org/html/8-1490151\ce92133d-bd00-48e7-9053-c39d3d2bbba2.jpg)
is a
-Brownian motion. Moreover, we see that
has the
-dynamics
![](https://www.scirp.org/html/8-1490151\7b0f31f0-68d1-42af-a45a-357c0f1a411a.jpg)
Recall that, for
, we have
(2.12)
where
denotes the expectation with respect to
. We now consider, for
,
![](https://www.scirp.org/html/8-1490151\1ade596c-7fa9-4704-8d08-17d48e3af78e.jpg)
where
![](https://www.scirp.org/html/8-1490151\15f7a90c-bc21-40be-8eef-6831a62a3b35.jpg)
The associated HJB equation is written as
(2.13)
By direct calculation, we can see the following.
Lemma 2.1 1) If
and
, then HJB Equation (2.13) is rewritten as
(2.14)
where we define
(2.15)
The maximizer for (2.13) is given by
![](https://www.scirp.org/html/8-1490151\3fc884e9-63e7-4a7e-b44c-0c03f2181f59.jpg)
2) If
and
, then HJB equation (2.13) is rewritten as
(2.16)
where
and
are given by (2.15). The maximizer for (2.13) is given by
![](https://www.scirp.org/html/8-1490151\238af13e-0ef6-4b2d-b3bc-3340ab504625.jpg)
3. Results
With the help of Lemma 2.1, it is straightforward to see the following.
Proposition 3.1 (Solution to the HJB equation) 1) If
, then
(3.1)
solves (2.13), or equivalently (2.14). Here,
and
solve the following system of ordinary differential equations:
(3.2)
2) If
, then
(3.3)
solves (2.13), or equivalently (2.16). Here,
and
solve the following system of ordinary differential equations:
(3.4)
Using this proposition, we obtain the following.
Theorem 3.1 (Verification and optimal strategy)
Define the filtration
by
. Let
(3.5)
and consider (1.3) with
. Then, the following assertions hold.
1) If
, then
, defined by
(3.6)
is optimal for (1.3). It holds that
(3.7)
2) If
, then
, defined by
(3.8)
is optimal for (1.3). The relation (3.7) holds.
The proof of the above theorem is given in Subsection 4.2 after preparing lemmas in Subsection 4.1.
4. Proofs
4.1. Lemmas for Exponential Martingale
We prepare the following two lemmas.
Lemma 4.1 Let
be
-progressively measurable so that
almost surely for all
.
Define
![](https://www.scirp.org/html/8-1490151\51f57306-ad14-47a7-b486-343fe8d5067b.jpg)
![](https://www.scirp.org/html/8-1490151\bfc77a06-e12f-4edd-8d7f-72d1eee7b86a.jpg)
Then,
is an
-martingale if and only if
is an
-martingale.
Proof. Denote
. For
, we have
![](https://www.scirp.org/html/8-1490151\c103caae-7f31-41f3-9bc5-efbccb24c401.jpg)
Lemma 4.2 Let
satisfy the following: for each
,
is
-measurable, and
![](https://www.scirp.org/html/8-1490151\6fb7f8b5-4089-4fa7-9084-92bcf9ce51db.jpg)
with some bounded
, where we write
for
. Then, the process
, defined by
![](https://www.scirp.org/html/8-1490151\c18e01bb-9699-4fc9-bbb1-22103ec639e0.jpg)
is a martingale.
Proof. The lemma follows from Lemma 4.1.5 of [24], an extension of Lemma 4.1.1 of [25]. Below, we reproduce the proof for self-containedness. Note that it suffices to show
(4.1)
Recall that
is progressively measurable. The proof of (4.1) consists of several steps.
First, writing
, we recall that
(4.2)
where
is defined by (2.8). From this, we can check that
![](https://www.scirp.org/html/8-1490151\2c051f71-38ee-468e-a195-b1a17f108b34.jpg)
for each
with some constant
. Also, we can check that
(4.3)
This follows from the relation
(4.4)
where
is arbitrary and the constant
is independent of
. Indeed, in (4.4), letting
and using Fatou’s lemma, (4.3) is deduced. To see (4.4), use (2.1) and Itô’s formula to deduce
![](https://www.scirp.org/html/8-1490151\a4e7066f-3ad4-4dbb-99f8-37360f4d1d06.jpg)
![](https://www.scirp.org/html/8-1490151\06e45b89-f94d-46f0-8112-30bc3571ab66.jpg)
where we use notation (2.8) and (2.9). From these, we see, from Itô’s formula,
![](https://www.scirp.org/html/8-1490151\73a46711-a023-4fa1-bc3e-f24381ffa6c2.jpg)
where
![](https://www.scirp.org/html/8-1490151\c1a33864-77f7-4902-8777-2f029ff19d78.jpg)
is the local-martingale part and
is the bounded-variation part, which satisfies
![](https://www.scirp.org/html/8-1490151\e33c2000-5430-402e-af0b-617e80147709.jpg)
with some constant
, independent of
. We can check that
; hence,
is a square-integrable martingale. Further, using (4.2) and recalling that
for conformable matrices
and
, we can check that
![](https://www.scirp.org/html/8-1490151\6f967045-f431-4090-abd7-152ad59b1e3c.jpg)
with some positive constant
, independent of
. So, taking the expectation, we deduce that
![](https://www.scirp.org/html/8-1490151\27a719d8-0458-4b47-9217-e74616af002f.jpg)
and that (4.4) follows from Gronwall’s inequality.
Next, use Itô’s formula for the following computation:
(4.5)
Here, we see that
![](https://www.scirp.org/html/8-1490151\fad5fac0-9913-4cc2-9272-6fa0b82754ad.jpg)
with some constant
; hence, the first term of the righthand side of (4.5) is a square-intergrable martingale. Also, we can deduce that
![](https://www.scirp.org/html/8-1490151\62dd0ced-d187-45b2-8b20-cf72d98e325b.jpg)
where
is a positive constant, independent of
. Taking the expectation, we see
![](https://www.scirp.org/html/8-1490151\51fad3b9-722c-492d-8315-3f3127b20f1b.jpg)
Letting
and using the dominated convergence theorem, we obtain (4.1).
4.2. Proof of Theorem 3.1
Let
be given by (3.1). Fix
and take
. Using these, define
(4.6)
where we use (2.10), (2.11), and the process
given by (2.1), and we set
. Using Itô’s formula, we see that
(4.7)
where we define the process
by
(see below)
Combining (4.6)-(4.8), we have, for
,
![](https://www.scirp.org/html/8-1490151\e0fecd79-eac5-4a23-a379-ad3539dec8b4.jpg)
Here, note that
is a martingale for any
by using Lemma 4.1 and 4.2 and that
almost everywhere on
since
solves HJB-equation
(2.13). So we deduce that
is a submartingale for each
. Taking the expectation, we see that
![](https://www.scirp.org/html/8-1490151\1d91da1b-4131-4fc2-bdee-04d6df3762e5.jpg)
Thus, we see that
(4.9)
for any
. Furthermore, if we define
by
![](https://www.scirp.org/html/8-1490151\cd8eb63c-9172-4fb9-9ce5-022a58a7eb25.jpg)
then, we deduce that
almost everywhere on
, from which we see that
is a martingale. Therefore, taking the expectation, we see that
![](https://www.scirp.org/html/8-1490151\6b7183c8-3863-4424-9e9c-459c8dc9b816.jpg)
that is,
(4.10)
Combining (4.9) and (4.10), we deduce that
(4.11)
Thus, letting
in (4.11), we have that
(4.8)and write
![](https://www.scirp.org/html/8-1490151\b360b736-9929-418d-ae66-ea0a40636876.jpg)
(3.7) follows from relation (2.12).