Abstract
We consider the problem of maximization of expected terminal power utility (risk sensitive criterion). The underlying market model is a regime-switching diffusion model where the regime is determined by an unobservable factor process forming a finite state Markov process. The main novelty is due to the fact that prices are observed and the portfolio is rebalanced only at random times corresponding to a Cox process where the intensity is driven by the unobserved Markovian factor process as well. This leads to a more realistic modeling for many practical situations, like in markets with liquidity restrictions; on the other hand it considerably complicates the problem to the point that traditional methodologies cannot be directly applied. The approach presented here is specific to the power-utility. For log-utilities a different approach is presented in Fujimoto et al. (Preprint, 2012).
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Appendix
Appendix
Proof of Proposition 3.1
The proof is contained in the following two lemmas.
Lemma A.1
For t∈[0,T], \(h \in\bar{H}_{m} \), we have the estimate (3.10).
Proof
Since x μ is convex, Jensen’s inequality applies and we obtain
For each i and t∈[0,T],
Thus, we have
and
Therefore, from (A.1) it follows that
To obtain the lower estimate, applying Jensen’s inequality yields
Since x μ is a decreasing function, we have
□
Lemma A.2
\(\bar{W}^{0}(t,\pi, h) \) in Definition 3.1 (see (3.9)) is a continuous function on \([0,T]\times{\mathcal{S}}_{N} \times\bar{H}_{m}\) and the estimates (3.12), (3.13) hold.
Proof
Let us first prove the continuity of \(\bar{W}^{0} (t,\pi,h)\) with respect to π. Owing to (2.4) and recalling p ji (t) that was defined in (2.28), we have
Next, we show the continuity of \(\bar{W}^{0} (t,\pi,h)\) with respect to t. First notice that, due to the time homogeneity of the process (X t ,θ t ),
Notice furthermore that
holds because
Therefore,
Finally, we prove the continuity with respect to h. By the definition of D(h,x) and Jensen’s inequality,
Therefore, for m≥1
Since \(X_{T}-X_{t}=\{X_{T}^{i} - X_{t}^{i}\}_{i=1,\ldots,m}\) is, conditionally on \({\mathcal{F}}^{\theta}\), Gaussian with mean \(\{\int_{t}^{T} m_{i}(\theta_{s} )ds\}_{i=1,\ldots,m}\) and covariance \(\{\int_{t}^{T} (\sigma\sigma^{*})^{ij}(\theta_{s} )ds\}_{i,j=1,\ldots,m}\) we have
Then, applying the dominated convergence theorem, for \(\ {h_{j}}\subset \bar{H}_{m}\), s.t. \(\lim_{j\rightarrow\infty} h_{j} =h\in\bar{H}_{m}\)
□
Proof of Proposition 3.2
Again, the proof is contained in the following two lemmas.
Lemma A.3
For each \(g \in{\mathcal{G}}\), we have the three estimates (3.18), (3.19) and (3.20).
Proof
Let us first set
and
Recall also that n(θ t ) is the intensity of the Cox process describing the observations and that the dynamics of the filter process π t was given in Corollary 2.2 in terms of the function M(t,x,π).
(i) (estimate (3.18)). Since \(g \in{\mathcal{G}}\), from the definition of \(\rho_{t,T}^{\theta}(z)\) in (2.12) and from (A.2) we obtain
by using Jensen’s inequality. On the other hand, we obtain
again by using Jensen’s inequality and (A.2).
(ii) (estimate (3.19)). By using Jensen’s inequality, we have
from (A.2), since the function x μ is decreasing. On the other hand, by using Jensen’s inequality, we have
because of (A.2).
(iii) (estimate (3.20)). Since
The estimate (3.20) follows from (i) and (ii). □
Lemma A.4
For all \(g \in{\mathcal{G}}\), the function \(\exp(\mu\hat{\xi}(t,\pi,h;g) )\) is continuous with respect to h. Furthermore, for each \(g \in {\mathcal{G}}_{1}\) the relation (3.21) holds and for each \(g \in {\mathcal{G}}_{2}\) the relation (3.22) holds.
Proof
Let us first prove the continuity of \(\exp(\mu\hat{\xi}(t,\pi,h;g) )\). From (A.13), we have for m≥1
Similarly to (A.15), we have for each i
Applying the dominated convergence theorem, for \(\ {h_{n}}\subset\bar{H}_{m}\), s.t. \(\lim_{n\rightarrow \infty} h_{n} =h\in\bar{H}_{m}\)
We next prove that, for \(g\in{\mathcal{G}}_{1}\), the relation (3.21) holds. For this purpose, recalling Corollary 2.2, we rewrite
Then, recalling the Definition 3.3 of \(\hat{\xi}(\cdot)\), from (3.20) in Proposition 3.2 and (2.4) it follows that
Furthermore, by the definition the definition of \({\mathcal{G}}_{1}\) (see (3.15) in Definition 3.2), using also (3.10)
Therefore, we obtain
Finally, to prove that for \(g\in{\mathcal{G}}_{2}\) the relation (3.22) holds, we rewrite, using the time homogeneity of (X t ,θ t ),
Therefore, recalling that \(\bar{t}<t\),
Now we have
We also have, using (3.20),
Further, since |D(h,x)−D(h,y)|≤|x−y| holds from (A.10), we obtain
Since (X t ,θ t ) is a time homogeneous process, we have
where l is the function defined in (3.1). Hence, we obtain
Since \(g\in{\mathcal{G}}_{2}\), we have
by using (3.18) in Proposition 3.2.
Putting all the estimates together, we finally obtain
□
Proof of Proposition 3.5
The equality (3.46) is shown in Lemma A.5 below. This lemma is followed by Lemma A.6 that is preliminary to Lemma A.7, from which then (3.47) follows.
Lemma A.5
For each n≥0, the equality (3.46) holds.
Proof
By definition we have
Moreover, \({\bar{W}}^{0}(t,\pi)\in{\mathcal{G}}_{1}\cap{\mathcal{G}}_{2}\) because of Proposition 3.3. Therefore, in Corollary 3.1, we set \(g(t,\pi)={\bar{W}}^{0}(t,\pi)\) and obtain a Borel function \(\hat{h}^{(n)}(t,\pi)\) satisfying (3.31) for n≥0. Then,
We also have a Borel function \({\bar{h}}(t,\pi)\) such that
We define a strategy \(\bar{h}^{(n)}\in{\mathcal{A}}^{n}\) as follows.
First, to show that \(\bar{W}_{n}(t,\pi)\leq W_{n}(t,\pi)\), we rewrite \(\bar{W}^{n}\) as follows,
Noting that
we have
inductively. By Corollary 3.3 we then have
Next, we shall prove the converse inequality. By applying Lemma 3.4, we have for \(h \in{\mathcal{A}}^{n}\)
By the definition of \({\hat{\xi}}\) and \(\bar{W}^{1}\) we have
Therefore, for \(h\in\mathcal{A}^{n}\), we have inductively
□
Lemma A.6
For all \(\ h\in{\mathcal{A}}\), we have
Proof
We shall first give an estimate for I 1(n). From (3.11) in Proposition 3.1 it follows that for \(h \in\bar{H}_{m}\)
Therefore, we have
by using Proposition 3.2(i) because clearly \(\exp( ( \mu\underline{m} + \frac{( \mu\bar{\sigma})^{2}}{2} )(T-t) ) \in {\mathcal{G}}\). Thus, we obtain inductively
and therefore we see that
On the other hand, since \(1_{\{ \tau_{n} > T \}} = \sum_{j=0}^{n-1 }1_{\{ \tau_{j} < T \leq\tau_{j+1} \}} \), we have
Noting that {τ k <T}∩{T≤τ j+1}=∅ for all k≥j+1, we have
and that
Therefore, we obtain
having used Lemma 3.1. This completes the proof. □
Lemma A.7
The equality (3.47) holds.
Proof
By the definition of \({\mathcal{A}}^{n}\), the inclusions \({\mathcal {A}}^{n}\subset{\mathcal{A}}^{n+1} \subset{\mathcal{A}}\) hold for n≥0 and we have
From the definition of W n(t,π) and W(t,π) it follows that
Therefore, from Lemma A.5 we have
Thus, from Proposition 3.4 and (3.39), we obtain
On the other hand, for \(h \in{\mathcal{A}}\)
Letting n→∞ and applying Lemma A.6,
and hence, we obtain
□
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Fujimoto, K., Nagai, H. & Runggaldier, W.J. Expected Power-Utility Maximization Under Incomplete Information and with Cox-Process Observations. Appl Math Optim 67, 33–72 (2013). https://doi.org/10.1007/s00245-012-9180-2
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DOI: https://doi.org/10.1007/s00245-012-9180-2