I am reading the paper Solution of the HJB Equations Involved in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu.

The authors consider the utility function $U: \mathbb{R} \to \mathbb{R}$, with

\begin{align} U(w) = -\exp{\left( - \gamma w \right)} \end{align}

and the dynamics of the risky asset and the dynamics of the auxiliary process as follows.

\begin{align} dS_{t} = S_{t}[\mu(Y_{t}) dt + \sigma(Y_{t})dW_{t}^{1} \\ dY_{t} = g(Y_{t})dt + \beta(Y_{t})[\rho W_{t}^{1} + \sqrt{1 - \rho^{2}} dW_{t}^{2}] \end{align} where $\rho$ is the correlation of the two noises.

According to the article, they want to compute a utility-based price option. For that purpose they make use of the dynamics of the wealth process

$dX_{t} = \alpha_{t}(\mu_{t}(Y_{t})dt + \sigma(Y_{t})dW_{t}^{1}), X_{0}=x$.

Where $\alpha_{t}$ is a $\mathcal{F}_{t}$-adapted process representing the amount of money invested in the risky asset at time $t $ such that

$E \int_{0}^{T} \alpha_{t}^{2} dt < \infty$

Question: Does anyone knows how the authors deduce the formula for the wealth process? I mean how can they deduce the formula without mentioning the riskless asset, and the interest rate? Why do they use $\alpha_{t}$ in the wealth process instead of $S_{t}$ that is the risky asset?

By the way, they make use of the formula

\begin{align} M_{t} = \exp{\left\lbrace \int_{0}^{t} \left[ -\gamma \alpha_{u} \sigma(Y_{u}) dW_{u}^{1} - \dfrac{1}{2} \gamma^{2} \alpha_{u}^{2} \sigma^{2}(Y_{u})du\right] \right\rbrace} \end{align} that is a martingale.

I would really appreciate any hint or reference about how to deduce this formula. Thanks in advance.


By investing the amount of $\alpha_t$ at time $t$ in the risky asset, the wealth is given by \begin{align*} X_t = \frac{\alpha_t}{S_t} S_t, \end{align*} where $\frac{\alpha_t}{S_t}$ is the units of the risky asset. For $\Delta$ sufficiently small, the wealth at time $t+\Delta$ becomes \begin{align*} X_{t+\Delta} = \frac{\alpha_t}{S_t} S_{t+\Delta}. \end{align*} Then, \begin{align*} X_{t+\Delta}-X_t &= \frac{\alpha_t}{S_t}\left(S_{t+\Delta} - S_t\right)\\ &\approx \frac{\alpha_t}{S_t} S_t \Big(\mu(Y_t) \Delta + \sigma(Y_t)\big(W_{t+\Delta}^1 - W_t^1\big) \Big). \end{align*} That is, \begin{align*} dX_t = \alpha_t\Big(\mu(Y_t) dt+ \sigma(Y_t)dW_t^1 \Big). \end{align*}

  • $\begingroup$ thanks for your answer. I made the same reasoning. However, if you apply Ito's formula you get $dX_{t} = \dfrac{\alpha_{t}}{S_{t}} dS_{t} + S_{t} d\dfrac{\alpha_{t}}{S_{t}} + d \langle S_{t} , \dfrac{\alpha_{t}}{S_{t}} \rangle$, and then you need to have $ S_{t} d\dfrac{\alpha_{t}}{S_{t}} + d \langle S_{t} , \dfrac{\alpha_{t}}{S_{t}} \rangle = 0$ (to have the dynamics of the wealth process) what seems very convenient and do not have a financial sense. $\endgroup$ – Ivan Aug 30 '18 at 0:53
  • $\begingroup$ Generally, you need another asset, such as the deposit account, to make the investment self-financing. But here we can only heuristically argue like that given that you have only the risky asset. $\endgroup$ – Gordon Aug 30 '18 at 12:58

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