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Expected Power-Utility Maximization Under Incomplete Information and with Cox-Process Observations

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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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Correspondence to Wolfgang J. Runggaldier.

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The opinions expressed are those of K. Fujimoto and not those of The Bank of Tokyo-Mitsubishi UFJ.

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

$$ E^{t,\pi} \Biggl[ \Biggl( \sum_{i=0}^m h^i \exp\bigl( X_T^i - X_t^i \bigr) \Biggr)^{\mu}\Biggr] \leq E^{t,\pi} \Biggl[ \sum _{i=0}^m h^i \exp\bigl( \mu \bigl(X_T^i - X_t^i \bigr) \bigr) \Biggr] . $$
(A.1)

For each i and t∈[0,T],

$$ \begin{aligned}[c] &\underline{m} (T-t) \leq\int _t^T m_i(\theta_s )ds \leq\bar{m} (T-t), \\ &\int_t^T \sigma_i^2( \theta_s )ds \leq\bar{\sigma}^2 (T-t). \end{aligned} $$
(A.2)

Thus, we have

(A.3)

and

(A.4)

Therefore, from (A.1) it follows that

(A.5)

To obtain the lower estimate, applying Jensen’s inequality yields

$$ \Biggl(E^{t,\pi} \Biggl[ \sum_{i=0}^m h^i \exp\bigl( X_T^i - X_t^i \bigr) \Biggr]\Biggr)^{\mu} \leq E^{t,\pi} \Biggl[ \Biggl( \sum _{i=0}^m h^i \exp\bigl( X_T^i - X_t^i \bigr) \Biggr)^{\mu}\Biggr] . $$
(A.6)

Since x μ is a decreasing function, we have

(A.7)

 □

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

(A.8)

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 ),

(A.9)

Notice furthermore that

$$ \bigl|D(h,x)-D(h,y)\bigr|\leq|x-y| $$
(A.10)

holds because

$$ \bigl|\nabla_x D(h,x)\bigr|\leq1. $$
(A.11)

Therefore,

(A.12)

Finally, we prove the continuity with respect to h. By the definition of D(h,x) and Jensen’s inequality,

$$ \exp\bigl( \mu D(h, x ) \bigr) = \Biggl( \sum _{i=0}^m h^i \exp\bigl( x^i \bigr) \Biggr)^{\mu} \leq\sum_{i=0}^m h^i \exp\bigl( \mu x^i \bigr) \leq\sum _{i=0}^m \exp\bigl( \mu x^i \bigr) . $$
(A.13)

Therefore, for m≥1

$$ \exp\bigl( \mu D(h,X_{T} - X_{t} ) \bigr) \leq\sum _{i=0}^m \exp\bigl( \mu\bigl( X^i_{T} - X^i_{t} \bigr)\bigr). $$
(A.14)

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

$$ E^{t,\pi}\bigl[ \exp\bigl( \mu\bigl( X^i_{T} - X^i_{t} \bigr) \bigr)\bigr] < \infty. $$
(A.15)

Then, applying the dominated convergence theorem, for \(\ {h_{j}}\subset \bar{H}_{m}\), s.t. \(\lim_{j\rightarrow\infty} h_{j} =h\in\bar{H}_{m}\)

(A.16)

 □

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

$$I_1=E^{t,\pi}\bigl[ \exp\bigl( \mu D(h,X_{\tau_1} - X_{t} ) + \mu g(\tau_1, \pi_1) \bigr) 1_{\{ \tau_1 \leq T \}}\bigr] $$

and

$$I_2=E^{t,\pi}\bigl[ \exp\bigl( \mu D(h,X_T-X_t) \bigr) 1_{\{ \tau_1 > T \}} \bigr]. $$

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

(A.17)

by using Jensen’s inequality. On the other hand, we obtain

(A.18)

again by using Jensen’s inequality and (A.2).

(ii) (estimate (3.19)). By using Jensen’s inequality, we have

(A.19)

from (A.2), since the function x μ is decreasing. On the other hand, by using Jensen’s inequality, we have

(A.20)

because of (A.2).

(iii) (estimate (3.20)). Since

$$ {\hat{\xi}} (t,\pi,h;g) =\frac{1}{\mu}\log(I_1+I_2). $$
(A.21)

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

$$ \exp\bigl( \mu D(h,X_{T\wedge\tau_1} - X_{t} ) + \mu g( \tau_1,\pi_1 ) \bigr) \leq\sum _{i=0}^m \exp\bigl( \mu\bigl(X^i_{T\wedge\tau_1} - X^i_{t}\bigr) + \mu g(\tau_1, \pi_1 ) \bigr). $$
(A.22)

Similarly to (A.15), we have for each i

$$ E^{t,\pi}\bigl[ \exp\bigl( \mu\bigl( X^i_{T\wedge\tau_1} - X^i_{t}\bigr) + \mu g(\tau_1, \pi_1 ) \bigr) \bigr] < \infty. $$
(A.23)

Applying the dominated convergence theorem, for \(\ {h_{n}}\subset\bar{H}_{m}\), s.t. \(\lim_{n\rightarrow \infty} h_{n} =h\in\bar{H}_{m}\)

(A.24)

We next prove that, for \(g\in{\mathcal{G}}_{1}\), the relation (3.21) holds. For this purpose, recalling Corollary 2.2, we rewrite

(A.25)

Then, recalling the Definition 3.3 of \(\hat{\xi}(\cdot)\), from (3.20) in Proposition 3.2 and (2.4) it follows that

(A.26)

Furthermore, by the definition the definition of \({\mathcal{G}}_{1}\) (see (3.15) in Definition 3.2), using also (3.10)

(A.27)

Therefore, we obtain

(A.28)

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 ),

(A.29)

Therefore, recalling that \(\bar{t}<t\),

(A.30)

Now we have

(A.31)

We also have, using (3.20),

$$ J_2\leq\exp\biggl( \biggl( \mu\underline{m} + \frac{( \mu\bar {\sigma})^2}{2} \biggr) (T-t) \biggr) \biggl(\frac{\bar{n}}{ \underline{n} } \biggr ) \bigl(e^{\underline{n} (t-\bar{t}) } -1 \bigr) . $$
(A.32)

Further, since |D(h,x)−D(h,y)|≤|xy| holds from (A.10), we obtain

(A.33)

Since (X t ,θ t ) is a time homogeneous process, we have

(A.34)

where l is the function defined in (3.1). Hence, we obtain

(A.35)

Since \(g\in{\mathcal{G}}_{2}\), we have

(A.36)

by using (3.18) in Proposition 3.2.

Putting all the estimates together, we finally obtain

(A.37)

 □

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

$$ \bar{W}^0(t,\pi)= W^0(t,\pi). $$
(A.38)

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

$${\bar{W}}^{0}(t,\pi)=\sup_h {\bar{W}}^0(t,\pi,h)={\bar{W}}^0\bigl(t,\pi,{\bar{h}}(t,\pi) \bigr). $$

We define a strategy \(\bar{h}^{(n)}\in{\mathcal{A}}^{n}\) as follows.

$$ \begin{aligned}[c] \bar{h}^{(n)}_k&=\hat{h}^{(n-1-k)}(\tau_{k}, \pi_k ),\quad k=0,\ldots, n-1, \\ \bar{h}^{(n)}_n&= {\bar{h}}(\tau_n, \pi_n), \\ \bar{h}^{(n)}_k&=\gamma\bigl(\tilde{X}_{\tau_k}- \tilde{X}_{\tau_{n}},\bar{h}^{(n)}_{n}\bigr),\quad k\geq n+1. \end{aligned} $$
(A.39)

First, to show that \(\bar{W}_{n}(t,\pi)\leq W_{n}(t,\pi)\), we rewrite \(\bar{W}^{n}\) as follows,

(A.40)

Noting that

(A.41)

we have

(A.42)

inductively. By Corollary 3.3 we then have

(A.43)

Next, we shall prove the converse inequality. By applying Lemma 3.4, we have for \(h \in{\mathcal{A}}^{n}\)

(A.44)

By the definition of \({\hat{\xi}}\) and \(\bar{W}^{1}\) we have

(A.45)

Therefore, for \(h\in\mathcal{A}^{n}\), we have inductively

(A.46)

 □

Lemma A.6

For all \(\ h\in{\mathcal{A}}\), we have

(A.47)

Proof

(A.48)

We shall first give an estimate for I 1(n). From (3.11) in Proposition 3.1 it follows that for \(h \in\bar{H}_{m}\)

$$ \exp\bigl( \mu W^0(t,\pi, h) \bigr) \leq\exp\biggl( \biggl( \mu \underline{m} + \frac{( \mu\bar{\sigma})^2}{2} \biggr) (T-t) \biggr). $$
(A.49)

Therefore, we have

(A.50)

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

(A.51)

and therefore we see that

$$\lim_{n\rightarrow\infty} I_1(n) = 0. $$

On the other hand, since \(1_{\{ \tau_{n} > T \}} = \sum_{j=0}^{n-1 }1_{\{ \tau_{j} < T \leq\tau_{j+1} \}} \), we have

(A.52)

Noting that {τ k <T}∩{Tτ j+1}=∅ for all kj+1, we have

(A.53)

and that

(A.54)

Therefore, we obtain

(A.55)

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

$$ \sup_{h \in{\mathcal{A}}^n} W(t,\pi,h.) \leq\sup_{h \in {\mathcal{A}}^{n+1}} W(t,\pi,h.) \leq \sup_{h \in{\mathcal{A}}} W(t,\pi,h.). $$
(A.56)

From the definition of W n(t,π) and W(t,π) it follows that

$$ W^{n}(t,\pi) \leq W^{n+1}(t,\pi) \leq W(t,\pi) . $$
(A.57)

Therefore, from Lemma A.5 we have

$$ \bar{W}^{n}(t,\pi) \leq\bar{W}^{n+1}(t,\pi) \leq W(t,\pi). $$
(A.58)

Thus, from Proposition 3.4 and (3.39), we obtain

$$ \bar{W}(t,\pi) \leq W(t,\pi). $$
(A.59)

On the other hand, for \(h \in{\mathcal{A}}\)

(A.60)

Letting n→∞ and applying Lemma A.6,

$$ \bar{W}(t,\pi)\geq W(t,\pi, h.) $$
(A.61)

and hence, we obtain

$$ \bar{W}(t,\pi) = W(t,\pi) . $$
(A.62)

 □

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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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