Keywords: Portfolio selection, utility function, Inada condition, square-integrable, Ito-process.
1 The wealth process
Consider a market with $l + 1$ assets, and an m-dimensional Brownian motion $W$ defined on the filtered probability space $\Big(\Omega, \mathcal{F}, \mathbf{P}; (\mathcal{F}_t)_{t > 0}\Big)$. One of the asset is risk-free, which means the return of this asset is $\mathcal{F}_{t-1}$ measurable. The dynamic of this asset can be written as:
$dS^0(t) = r(t)S^0(t)dt$
, where $r(t)$ can be random but always $\mathcal{F}_{t-1}$ measurable. The dynamic of other assets can be written as:
$dS(t) = \mu(t) S(t) dt + S(t) \sigma(t) dW_t$
, where $\mu(t)$ is a diagonal matrix, and $\sigma\sigma^T$ is the covariance matrix.
At time t, the number of the $i$th asset is $N^{i}(t)$; the dollar amount is $\pi^{i}(t)$. Let’s assume the agent’s trading is self-financing. Therefore, the dynamic of the wealth process can be written as:
$dx(t) = \sum_{i=0}\limits^l N^{i}(t)d S^{i}(t) = N \mu S_tdt + N \sigma S_t d W_t = \mu S_t N dt + \sigma S_t N dW_t$
$dx(t) = \mu \pi(t) dt + \sigma \pi(t) dW(t) = r(t) \sum(\pi(t)) dt + (\mu - r(t)) \pi(t) dt + \sigma \pi(t) dW(t)$
Note that by the definition of self-financing, $\sum(\pi(t)) = x(t)$.
Therefore, the wealth process can be written as:
$dx_t = [r(t)x(t) + B^{\prime}(t)\pi (t)]dt + \sigma (t) \pi (t) dW_t$
,where $B(t) = (\mu^{1}(t) - r(t), \dots, \mu^{l}(t) - r(t))$ (the excess of the return)
$\sigma(t) = (\sigma^{ij}(t))_{l \times m}$.
Note that we need to add extra constraints (square integrable) to the portfolio process $\pi_{t}$, i.e. $\mathbb{E}(\int_{0}^T \pi(t)^T\pi(t)dt) < +\infty$. This constraint has economic sense since we can only use limited leverage in the investment. The portfolio follows the above constraints is called admissible portfolio.
2 Consumption Decision
An admissible consumption plan should also be non-negative and square-integrable in $[0, T]$. Note that although we have $l + 1$ assets in the market, due to the self-financing and consumption plan, the freedom degree in the asset allocation is $l$. Therefore, let’s just choose the risky asset.
Another interesting setting in Merton problem is quite implicit. The system is generated by the filtration generated by the Brownian motion, which means all the uncertainties comes from the randomness of the Brownian motion. However, the drifts and diffusions are $\mathcal{F}_t-$adapted. This property indicate the implementation details, we can use the current information to determine the drift and diffusion.
3 Utility function
A concave utility function indicate the risk reversion condition.
1) U is defined on $\mathbb{R}^{+}$ and $U(0) = 0$.
2) $\dot{U}(0) = +\infty$
3) $\dot{U}(+\infty) = 0$
Utility function works as the pivot role in the portfolio selection. Utility function works as a paneity of the variance (or risk), which make the investment problem to a second order issue.
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