Risk-Sensitive Asset Management under a Wishart Autoregressive Factor Model

The risk-sensitive asset management problem with a finite horizon is studied under a financial market model having a Wishart autoregressive stochastic factor, which is positive-definite symmetric matrix-valued. This financial market model has the following interesting features: 1) it describes the stochasticity of the market covariance structure, interest rates, and the risk premium of the risky assets; and 2) it admits the explicit representations of the solution to the risk-sensitive asset management problem.


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 n   , where is the initial wealth of the investor, and , is the dynamic invest-  which we call the risk-sensitive asset management problem. Here, T is a space of admissible investment strategies and is a subset of 2, , the totality of -dimensional -progressively measurable processes

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][2][3][4][5][6][7]. In those works, the price processes   0 , S S are given by the solutions to the following system of SDEs: Here, is the solution to a matrix differential R ti lin the cumulant expansi For the purpose, we employ a Wishart aut ocess 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 (cova ance) under the risk-neutral probability: see [11][12][13][14][15][16].  Modeling of multivariate asset price process und physical probability with stochastic covariance and mean-return rates: see [14,17,18].  Modeling of (term structure of) interest chastic intensity for credit risk: see [14,17,19,20]. Our market model defined by (2.1)- (2.4) in Sectio 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 se ith a Wishart autoregressive factor and present preliminary calculations of the associated Hamilton-Jacobi-Bellman (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.

Marke Factor
  denotes the quadratic covariation, and ij  denotes onecker's delta. We consider the price processes Kr   0 , S S , described by the following system of SDEs: with the initial values and where , and . Also, we assume We also assume that as well as on the interest rate and on the so-called risk premium of S , is indeed an affine diffusion. To re and their financial applications, see, for example, [21], as So we interpret that the so-called mean-v riance term in portfolio optimization theory is assumed to be const 2). view affine processes [22], and the references therein.
Remark 2.2 The condition (2.7) on the structure of the risk-premium vector is rewritten  , , , ,

Deriving the HJB Equation
To tackle (1.3), we employ a dynamic programming approach: Reca For , we define the probability measure denotes the expectation with respect to The associated HJB equation is written as By direct calculation, we can see the following.
The maximizer for (2.13) is given by

Results
With the help of Lemma 2.1, it is straightforward to see the following. uivalently (2.14). Here, the following system of ordinary differential equations: Using this proposition, we obtain the following.

Theorem 3.1 (Verification and optimal strategy)
Define the filtration     0, and consider (1.3) wit . Then, the following assertions hold.

Lemmas for Exponential Martingal
We prepare the following two lemmas. , ,