I am developing a Reinforcement learning algorithm for the optimal portfolio selection, using stochastic control approach. I have a market with risky and risk free assets. However becouse of some theoretical constraints, I need to assume that the drifts of both assets are the same, so I have

\begin{equation} dS_t = S_t(\alpha dt + \beta dB_t), \end{equation} with $S_0 = s >0,$ and

\begin{equation} dZ_t = \alpha Z_t dt, \end{equation} with $Z_0 = 1.$

For a self-financing portfoli $(\varphi_t) = (\varphi^Z_t,\varphi^S_t) \in \mathbb{R}^2,$ and $u_t = \varphi^S_t S_t$ being the amount of money being invested into risky asset at time $t,$ we have that wealth process satisfies

\begin{align}\label{eq:wealth} dX_t &= \varphi^Z_t dZ_t + \varphi^S_t dS_t \nonumber \nonumber\\ &= \alpha X_t dt + \beta u_t dB_t. \end{align}

I consider an infinite horizon framework where investor tries to maximize objective function given as
\begin{equation} J_u(x) = E \Big[ \int_0^\infty e^{-\rho t} \Big( u_t - \frac{(X_t)^2 + u_t^2}{2} \Big) dt \Big|X_0 = x \Big], \end{equation} for discount factor $\rho > 0.$ The particular form of this reward is again given by the theoretical constraints of my algorithm.

Now, my question is what could be some real life motivations behid such problem? Perhaps $Z$ and $S$ being same instruments where $S$ has some added variability (fixed and variable interest rate accounts)? For objective function my reasoning would be that investor tries to minimize its wealt (or exposure) while trying to maintain some level of its wealth in the risky asset. Am I making any sense? What would be some better explanaitions? Thank you in advance!


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