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

Abstract

We consider the portfolio optimization problem for the criterion of maximization of expected terminal log-utility. 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 log-utility. For power utilities a different approach is presented in the companion paper (Fujimoto et al. in Appl Math Optim 67(1):33–72, 2013).

Appendix

Proof of Lemma 4.1

Proof of statement (i). It follows from the two lemmas shown below.

Lemma 5.1

We have the following representation,

$$\begin{aligned} E\left[ f(\theta _t,h_t )|\mathcal{G }_t\right] \!=\!\displaystyle \sum _{k\ge 0}1_{]\tau _k, \tau _{k+1}]}(t) \frac{E\left[ f\left( \theta _t, \gamma \left( {\tilde{X}}_t \!-\! \tilde{X}_{\tau _k}, h_k\right) \right) 1_{\{ t\le \tau _{k+1} \} } |\mathcal{G }_k \right] }{ E\left[ 1_{\{ t\le \tau _{k+1} \} } |\mathcal{G }_k \right] }.\qquad \quad \end{aligned}$$

(5.1)

Proof

It suffices to prove that for any $\mathcal{G }_t-$adapted process $Z_t$

$$\begin{aligned}&E\left[ E\left[ f(\theta _t,h_t )|\mathcal{G }_t\right] Z_t\right] \nonumber \\&\quad {=}E\left[ \!\displaystyle \sum _{k\ge 0}1_{]}\tau _k, \tau _{k+1}](t) \frac{E\!\left[ \!f\left( \!\theta _t ,\gamma \left( \!{\tilde{X}}_t {-} \tilde{X}_{\tau _k}, h_k\!\right) \right) 1_{\{ t\le \tau _{k+1} \} } |\mathcal{G }_k \!\right] }{E\left[ 1_{\{ t\le \tau _{k+1} \} } |\mathcal{G }_k \right] }\, Z_t\!\!\right] \!.\nonumber \\ \end{aligned}$$

(5.2)

First notice that any $\mathcal{G }_t-$adapted process $Z_t$ has the representation (see Bremaud 1981)

$$\begin{aligned} Z_t = \displaystyle \sum _{k\ge 0}1_{]\tau _k, \tau _{k+1}]}(t) Z_k(t) + Z_{\infty }1_{]\tau _\infty , \infty [ }(t) , \end{aligned}$$

(5.3)

with the process $Z_k(t)$ being $\mathcal{G }_k \otimes \mathcal{B }(\mathbb{R }_+)$-measurable. Furthermore, under our assumptions, for all $t>0, \lim _{n\rightarrow \infty }1_{ \{ \tau _n < t\} } =0$ and thus

$$\begin{aligned} Z_t = \displaystyle \sum _{k\ge 0}1_{]\tau _k, \tau _{k+1}]}(t) Z_k(t). \end{aligned}$$

(5.4)

Note, finally, that $E\left[ 1_{\{\tau _k< t\le \tau _{k+1} \}} |\mathcal{G }_k \right] =1_{]\tau _k, \infty )} (t) E[1_{\{ t\le \tau _{k+1} \}}|\mathcal{G }_k ]]$. We then have

$$\begin{aligned} E\left[ E\left[ f(\theta _t,h_t )|\mathcal{G }_t\right] Z_t\right]&= E[f(\theta _t,h_t )\displaystyle \sum _{k\ge 0}1_{]\tau _k, \tau _{k+1}]}(t) Z_k(t)] \\&= \displaystyle \sum _{k\ge 0}E\left[ E\left[ f(\theta _t,h_t )1_{\{ t\le \tau _{k+1} \} }|\mathcal{G }_k\right] 1_{\{ \tau _k <t \} } Z_k(t)\right] \\&= \displaystyle \sum _{k\ge 0}E\Bigg [\!\frac{E\left[ f(\theta _t,h_t )1_{\{ t\le \tau _{k+1} \} }|\mathcal{G }_k\right] }{ E[1_{\{ t\le \tau _{k+1} \} } |\mathcal{G }_k ]} E\Bigg [\! 1_{]\tau _k, \tau _{k+1}]}(t) Z_k(t)|\mathcal{G }_k\Bigg ]\Bigg ] \\&= \displaystyle E[\sum _{k\ge 0} 1_{]\tau _k, \tau _{k+1}]}(t)\frac{E\left[ f(\theta _t,h_t ) 1_{\{ t\le \tau _{k+1} \} }|\mathcal{G }_k\right] }{ E[1_{\{ t\le \tau _{k+1} \} } |\mathcal{G }_k ]} Z_t], \end{aligned}$$

and thus we obtain (5.2) since

$$\begin{aligned} f(\theta _t ,h_t)&= \displaystyle \sum _{k=0}^{\infty } 1_{[\tau _{k},\tau _{k+1})}(t) f(\theta _t ,\gamma (\tilde{X}_t - \tilde{X}_{\tau _k}, h_k)), \end{aligned}$$

which follows from (2.16). $\square$

Lemma 5.2

We have the following equation

$$\begin{aligned}&E\left[ \int \limits _t^T f(\theta _s,h_s )ds | \tau _0 = t, \pi _{\tau _0} = \pi \right] \nonumber \\&\quad =E\left[ \displaystyle \sum _{k\ge 0} \hat{C}(\tau _k,\pi _{\tau _k},h_k) 1_{ \{ \tau _k <T \} } | \tau _0 = t, \pi _{\tau _0} = \pi \right] \end{aligned}$$

(5.5)

with $\hat{C}(t,\pi ,h)$ defined by (4.6) in Definition 4.1.

Proof

For simplicity, in the following formula we shall use the notation

$$\begin{aligned} E^{t,\pi }[\cdot ]\equiv E[\cdot \;|\tau _0=t, \pi _{\tau _0}=\pi ] \end{aligned}$$

Using (5.1) we have similarly as above

$$\begin{aligned}&E^{t,\pi }\left[ \int \limits _t^T E\left[ f(\theta _s ,h_s)|\mathcal{G }_s\right] ds \right] \nonumber \\&\quad =E^{t,\pi }\left[ \int \limits _t^T \displaystyle \sum _{k\ge 0}1_{]\tau _k, \tau _{k+1}]}(s) \frac{E\left[ f(\theta _s ,\gamma (\tilde{X}_s - \tilde{X}_{\tau _k}, h_k))1_{\{ s< \tau _{k+1} \} } |\mathcal{G }_k \right] }{ E\left[ 1_{\{ s\le \tau _{k+1} \} } |\mathcal{G }_k \right] } ds \right] \nonumber \\&\quad =E^{t,\pi }\left[ \displaystyle \sum _{k\ge 0} \int \limits _t^T 1_{]\tau _k, \infty )} (s) E\left[ f(\theta _s ,\gamma (\tilde{X}_s - \tilde{X}_{\tau _k}, h_k)) 1_{\{ s< \tau _{k+1} \} } |\mathcal{G }_k \right] ds \right] \nonumber \\&\quad =E^{t,\pi }\left[ \displaystyle \sum _{k\ge 0} \int \nolimits _t^T 1_{]\tau _k, \infty )} (s) E\left[ e^{ -\int \nolimits _{\tau _k}^s n(\theta _u)du }f(\theta _s ,\gamma (\tilde{X}_s - \tilde{X}_{\tau _k}, h_k)) |\mathcal{G }_k \right] ds \right] \nonumber \\&\quad =E^{t,\pi }\left[ \displaystyle \sum _{k\ge 0} \int \limits _t^T 1_{]\tau _k, \infty )} (s) E\left[ E[e^{ -\int \nolimits _{\tau _k}^s n(\theta _u)du } f(\theta _s ,\gamma (\tilde{X}_s - \tilde{X}_{\tau _k}, h_k)) |\mathcal{G }_k \vee \sigma \{ \theta _{\tau _k} \} ]|\mathcal{G }_k \right] ds\right] \nonumber \\ \end{aligned}$$

(5.6)

Since $\left( \theta _t, \tilde{X}_t\right)$ is a time homogeneous Markov process,

$$\begin{aligned}&E\left[ e^{ -\int \nolimits _{\tau _k}^s n(\theta _u)du } f\left( \theta _s ,\gamma \left( \tilde{X}_s - \tilde{X}_{\tau _k}, h_k\right) \right) |\mathcal{G }_k \vee \sigma \{ \theta _{\tau _k} \} \right] \nonumber \\&\quad =E\!\left[ \!e^{ -\int \nolimits _{0}^t n(\theta _u)du } f\left( \theta _t ,\gamma \left( \tilde{X}_t - x, h\right) \right) \bigg | \theta _0 =\theta , \tilde{X}_0=x\right] \bigg |_{t=s-\tau _k, \theta =\theta _k,x=\tilde{X}_{\tau _k},h=h_k }\nonumber \\ \end{aligned}$$

(5.7)

We now have, recalling the definition of $r_{ji}(t,z)$ in (3.8),

$$\begin{aligned}&E\left[ e^{ -\int \nolimits _{0}^t n(\theta _s)ds)} f\left( \theta _t ,\gamma \left( \tilde{X}_t - x, h\right) \right) | \theta _0 =\theta , \tilde{X}_0=x\right] \nonumber \\&\quad =E\left[ e^{ -\int \nolimits _{0}^t n(\theta _s)ds ) }E\left[ f(\theta _t ,\gamma (\tilde{X}_t - x, h)) \bigg | \mathcal{F }^{\theta }_t\vee \{\tilde{X}_0=x\} \right] \bigg | \theta _0 =\theta , \tilde{X}_0=x\right] \nonumber \\&\quad =E\left[ e^{ -\int \nolimits _{0}^t n(\theta _s)ds ) }\int \limits _{\mathbb{R }^m}f(\theta _t ,\gamma ( z , h)) \rho ^{\theta }_{0,t} (z) dz \bigg |\theta _0 =\theta , \tilde{X}_0=x\right] \nonumber \\&\quad =E\left[ \,\,\,\int \limits _{\mathbb{R }^m} \sum \limits _{ij} 1_{ \{ \theta _t = e_i , \theta _0 = e_j \} } f(e_i ,\gamma ( z , h))\right. \nonumber \\&\quad \quad \left. \times E\left[ e^{ -\int \limits _{0}^t n(\theta _s)ds } \rho ^{\theta }_{0,t} (z) | \theta _t = e_i , \theta _0 = e_j \right] dz \bigg |\theta _0 =\theta , \tilde{X}_0=x\right] \nonumber \\&\quad =E\left[ \,\,\,\int \limits _{\mathbb{R }^m} \sum \limits _{ij} 1_{ \{ \theta _t = e_i , \theta _0 = e_j \} } f(e_i ,\gamma ( z , h)) r_{ji}(t,z) dz \bigg |\theta _0 =\theta , \tilde{X}_0=x\right] \nonumber \\&\quad =\int \limits _{\mathbb{R }^m} \sum \limits _{ij} f(e_i ,\gamma ( z , h)) r_{ji}(t,z) p_{ji}(t) 1_{ \{ \theta = e_j \} }dz. \end{aligned}$$

(5.8)

We finally have

$$\begin{aligned}&E^{t,\pi }\left[ \int \limits _t^T f(\theta _s,h_s )ds \right] =E^{t,\pi }\left[ \int \limits _t^T E[f(\theta _s ,h_s)|\mathcal{G }_s] ds \right] \nonumber \\&\quad =E^{t,\pi }\left[ \displaystyle \sum _{k\ge 0} \int \limits _t^T 1_{]\tau _k, \infty )} (s) E[E[e^{ -\int \limits _{\tau _k}^s n(\theta _u)du } f(\theta _s ,\gamma (\tilde{X}_s - \tilde{X}_{\tau _k}, h_k)) |\mathcal{G }_k \vee \sigma \{ \theta _{\tau _k} \} ]|\mathcal{G }_k ] ds \right] \nonumber \\&\quad =E^{t,\pi }\left[ \displaystyle \sum _{k\ge 0} 1_{ \{ \tau _k <T \} } \int \limits _{\tau _k}^T \int \limits _{\mathbb{R }^m} \sum \limits _{ij} f(e_i , \gamma (z ,h_k)) r_{ji}(s-\tau _k, z) p_{ji}(s-\tau _k) \pi _{\tau _k}^j dzds \right] \nonumber \\&\quad =E^{t,\pi }\left[ \displaystyle \sum _{k\ge 0} \hat{C}(\tau _k,\pi _{\tau _k},h_k) 1_{ \{ \tau _k <T \} } \right] . \end{aligned}$$

(5.9)

$\square$

Proof of statement (ii) of Lemma 4.1.

We start by proving that $\hat{C}(t,\pi , h)$ is Lipschitz continuous with respect to $t$.

$$\begin{aligned} \hat{C}(t,\pi , h)&= \int \limits _{ t}^{T} \int \limits _{\mathbb{R }^m}\sum \limits _{i,j} f(e_i ,\gamma (x, h))r_{ji}(s-t,x)p_{ji}(s-t)\pi ^j dxds \nonumber \\&= \int \limits _{ 0}^{T-t} \int \limits _{\mathbb{R }^m}\sum \limits _{i,j} f(e_i ,\gamma (x, h))r_{ji}(s,x)p_{ji}(s)\pi ^j dxds . \end{aligned}$$

(5.10)

Thus

$$\begin{aligned} \left| \hat{C}(t,\pi , h)-\hat{C}(\bar{t},\pi , h)\right|&= \left| \,\,\int \limits _{T - \bar{t}}^{T - t} \int \limits _{\mathbb{R }^m}\sum \limits _{i,j} f(e_i ,\gamma (x, h))r_{ji}(s,x)p_{ji}(s)\pi ^j dxds \right| \nonumber \\&\le \Vert f \Vert |t-\bar{t}| , \end{aligned}$$

(5.11)

where $\Vert f \Vert := \sup _{e\in E, h\in \bar{H}_m}\Vert f(e, h) \Vert$. Next, let us prove that $C(t,\pi , h)$ is Lipschitz continuous with respect to $\pi$ [in the metric introduced in (3.21)].

$$\begin{aligned} \left| \hat{C}(t,\pi , h){-}\hat{C}(t,\bar{\pi }, h)\right|&= \left| \int \limits _{0}^{T - t} \int \limits _{\mathbb{R }^m}\sum \limits _{i,j} f(e_i ,\gamma (x, h))r_{ji}(s,x)p_{ji}(s)(\pi ^j {-}\bar{\pi }^j )dxds \right| \nonumber \\&\le \Vert f \Vert T |\pi -\bar{\pi }|= \Vert f \Vert T \sum _{i=1}^N |\pi (e_i)-\bar{\pi }(e_i)| \nonumber \\&\le \Vert f \Vert T \Vert \pi -\bar{\pi } \Vert _{TV} \le \Vert f \Vert T \frac{2}{\log 3}d_H(\pi , \bar{\pi }) , \end{aligned}$$

(5.12)

where we have used (3.20).

Next, let us prove that $C(t,\pi , h)$ is continuous with respect to $h$ [always in the metric introduced in (3.21)]. The function $f(e_i,h)$ is bounded and continuous with respect to $h$ for all $i$. Furthermore, $\gamma (x,h)$ is continuous with respect to $h$ for all $x\in \mathbb{R }^m$. Applying the dominated convergence theorem, for $\ {h_n}\subset \bar{H}_m$, s.t.$\displaystyle \lim _{n\rightarrow \infty } h_n =h\in \bar{H}_m$

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \hat{C}(t,\pi , h_n)&= \int \limits _{0}^{T - t} \int \limits _{\mathbb{R }^m}\sum \limits _{i,j} \displaystyle \lim _{n\rightarrow \infty } f(e_i , \gamma (x, h_n))r_{ji}(s,x)p_{ji}(s)\pi ^j dxds \nonumber \\&= \int \limits _{0}^{T - t} \int \limits _{\mathbb{R }^m}\sum \limits _{i,j} f(e_i ,\gamma (x, h))r_{ji}(s,x)p_{ji}(s)\pi ^j dxds \nonumber \\&= \hat{C}(t,\pi , h). \end{aligned}$$

(5.13)

$\hat{C}(t,\pi , h)$ is thus continuous with respect to each of the variables $t,\pi ,h$. However, continuity in $t,\pi$ is independent of the other variable. Hence, $\hat{C}(t,\pi , h)$ is a continuous function on $[0,T]\times \mathcal{S }_N \times \bar{H}_m$.

Proof of Lemma 4.2

Fix $n\ge 0$. Recall the definition of $h_{n}^i$ given in Sect. 2.3. Since $S_t$ is continuous and $V_t$ satisfies the self-financing condition, we obtain

$$\begin{aligned} h_{\tau _{n}-}^i=\frac{N_{n-1}^iS_{\tau _{n}-}^i}{V_{\tau _{n}-}} = \frac{N_{n-1}^iS_{\tau _{n}}^i}{V_{\tau _{n}}} =\frac{N_{n-1}^iS_{\tau _{n}}^i}{\sum _{i=0}^mN_{n}^iS_{\tau _{n}}^i}. \end{aligned}$$

Using (2.16), (2.18), for $all \ k\ge 1,h \in \mathcal{A }^{n},t\in [\tau _{n+k},T]$, one furthermore has

$$\begin{aligned} h^i_t= \gamma ^i(\tilde{X}_t - \tilde{X}_{\tau _{n+k}}, h_{n+k}) =\gamma ^i(\tilde{X}_t - \tilde{X}_{\tau _n}, h_{n}) . \end{aligned}$$

Therefore, using lemma 4.1(i) for $\ h\in \mathcal{A }^n$

$$\begin{aligned}&W(t,\pi , h.) = E\left[ \displaystyle \sum _{k=0}^{n -1} \int \limits _{\tau _{k}}^{T\wedge \tau _{k+1}} f(\theta _s ,\gamma (\tilde{X}_s - \tilde{X}_{\tau _k}, h_k)) ds 1_{\{\tau _{k}<T\}}\right. \nonumber \\&\quad \left. + \ \displaystyle \sum _{k=n}^{\infty } \int \limits _{\tau _{k}}^{T\wedge \tau _{k+1}} f(\theta _s , \gamma (\tilde{X}_s - \tilde{X}_{\tau _k}, h_k)) ds 1_{\{\tau _{k}<T\}} | \tau _0=t,\;\pi _{\tau _0}=\pi \right] \nonumber \\&= E\left[ \displaystyle \sum _{k=0}^{n-1 } \hat{C}(\tau _k,\pi _{\tau _k}, h_k)1_{\{\tau _{k}<T\}}+ \ \int \limits _{\tau _{n}}^{T} f\left( \theta _s ,\gamma \left( \tilde{X}_s - \tilde{X}_{\tau _n}, h_n\right) \right) ds1_{\{\tau _{n}<T\}} | \tau _0=t,\;\pi _{\tau _0}=\pi \right] .\nonumber \\ \end{aligned}$$

(5.14)

Proof of Lemma 4.6

By the definition of $\mathcal{A }^n$, for $n\ge 0, \mathcal{A }^n\subset \mathcal{A }^{n+1} \subset \mathcal{A }$, hence,

$$\begin{aligned} \displaystyle \sup _{h \in \mathcal{A }^n} W(t,\pi ,h.) \le \displaystyle \sup _{h \in \mathcal{A }^{n+1}} W(t,\pi ,h.) \le \displaystyle \sup _{h \in \mathcal{A }} W(t,\pi ,h.) . \end{aligned}$$

(5.15)

By the definition of $W^{n}(t,\pi )$ and $W(t,\pi )$

$$\begin{aligned} W^{n}(t,\pi ) \le W^{n+1}(t,\pi ) \le W(t,\pi ) . \end{aligned}$$

(5.16)

Using Lemma 4.5, for $\ n,m\ge 0$

$$\begin{aligned} \bar{W}^{n}(t,\pi ) \le \bar{W}^{n+m}(t,\pi ) \le W(t,\pi ) . \end{aligned}$$

(5.17)

Letting $m\rightarrow \infty$

$$\begin{aligned} \bar{W}^{n}(t,\pi ) \le \bar{W}(t,\pi ) \le W(t,\pi ) . \end{aligned}$$

(5.18)

Proof of Lemma 4.7

For$\ h \in \mathcal{A }, W(t,\pi ,h)$ defined by (4.8) satisfies

$$\begin{aligned} W(t,\pi ,h.)&= E\left[ \displaystyle \sum _{k=0}^{n-1 } \hat{C}(\tau _k,\pi _{\tau _k} ,h_k)1_{\{\tau _{k}<T\}} |\tau _0=t,\pi _{\tau _0}=\pi \right] \nonumber \\&\quad + \ E \left[ \displaystyle \sum _{k=n}^{\infty } \hat{C}\left( \tau _k,\pi _{\tau _k}, h_k\right) 1_{\{\tau _{k}<T\}} |\tau _0=t,\pi _{\tau _0}=\pi \right] \nonumber \\&= E\left[ \displaystyle \sum _{k=0}^{n-1 } \hat{C}(\tau _k,\pi _{\tau _k}, h_k)1_{\{\tau _{k}<T\}}+ \int \limits _{\tau _{n}}^{T} f\left( \theta _s ,\gamma \left( \tilde{X}_s - \tilde{X}_{\tau _n}, h_n\right) \right) ds1_{\{\tau _{n}<T\}} \right. \nonumber \\&\quad \left. - \ \int \limits _{\tau _{n}}^{T} f(\theta _s ,\gamma (\tilde{X}_s - \tilde{X}_{\tau _n}, h_n)) ds1_{\{\tau _{n}<T\}} |\tau _0=t,\pi _{\tau _0}=\pi \right] \nonumber \\&\quad + \ E[W(\tau _n,\pi _{\tau _n}, h.)1_{\{\tau _{n}<T\}} |\tau _0=t,\pi _{\tau _0}=\pi ]. \nonumber \\&\le W^n(t,\pi )+\left| E \left[ \int \limits _{\tau _{n}}^{T} f\left( \theta _s ,\gamma \left( \tilde{X}_s - \tilde{X}_{\tau _n}, h_n\right) \right) ds1_{\{\tau _{n}<T\}} |\tau _0=t,\pi _{\tau _0}=\pi \right] \right| \nonumber \\&\quad + \ E\left[ W(\tau _n,\pi _{\tau _n}, h.)1_{\{\tau _{n}<T\}} |\tau _0=t,\pi _{\tau _0}=\pi \right] \nonumber \\&\le {\bar{W}}^n(t,\pi )+2\Vert f \Vert T P(\tau _{n}<T | \tau _{0}=t). \end{aligned}$$

(5.19)

because of the representation of $W^n(t,\pi )$ in Corollary 4.2 (Eq. 4.13) and Lemma 4.5. Thus, by letting $n \rightarrow \infty$, we obtain

$$\begin{aligned} W(t,\pi ,h.) \le \bar{W}(t,\pi ) \end{aligned}$$

(5.20)

for all $\ h \in \mathcal{A }$.