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Kijima, C. Hara, and K. , World Scientiﬁc, Singapore, 2010, pp. 1– 41. [14] M. H. A. Davis and S. Lleo, Jump-Diﬀusion Risk-Sensitive Asset Management II: Jump-Diﬀusion Factors, preprint, Imperial College London, London, England, 2010. [15] W. H. Fleming, Optimal investment models and risk-sensitive stochastic control, in Mathematical Finance, IMA Vol. Math. Appl. 65, Springer, Berlin, 1995, pp. 75–88. [16] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, New York, 1975.

T ˜ k |1+μ is ˜ ∈ C 1,2 (QR ). 3: Proving that Φ QR bounded for μ > 0, which implies that η > n + 2. 12) ˜ k+1 + θg(t, x, hk ) Φ ˜k − Φ ˜ k+1 ˜ k − DΦ + f (x, hk )T D Φ for any admissible control h. e. in QR . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 1 of Fleming and Rishel [16], we see that there exists a Borel measurable function h∗ from (0, T ) × BR into J (in fact J ) such that ˜ + θg(t, x, h∗ )Φ ˜ = inf f (x, h∗ )T D Φ ˜ + θg(t, x, h)Φ ˜ f (x, h)T D Φ h∈J holds for almost all (t, x) ∈ (0, T ) × BR .

17). More2. 2). over, Φ 3. 14). Moreover, Φ is convex in its argument x. Proof. The proof is based on a series of results proved in sections 5–7. These combine to give us the following arguments. Existence of an optimal control. 12) admits a unique Borel measurable maximizer. 1). Thus, we can take this maximizer as our optimal asset allocation. ˜ is a C 1,2 ([0, T ] × Rn ) solution Existence of a classical (C 1,2 ) solution. 17). Uniqueness of the classical solution. The existence of zero beta policies enable us to deduce ˜ is bounded.