I would like to derive the HJB equation for the following stochastic optimal control problem:

$ \Phi(t,x)=\sup_{h} E \left[\exp \left\{\gamma \int_t^T g(X_s,h(s);\gamma)\ ds \right\} \right]$

where the state variable $X(t)$ is given

$dX(t)=\mu(X(t),h(t),t))\ dt + \sigma(X(t),h(t),t))\ d W(t) $

$g$ is a known function and $\gamma$ is just a constant different from $0$.

Honestly I never studied this topic formally and doing some research on the internet I read that the objective functional is usually written as an expectation of an integral over time. Splitting the integral in two, the Bellman principle can be readily applied and this yields the HJB equation.

In this case, however, we have the expectation of an exponential of an integral and quite frankly I am really not sure how to proceed.

Could you please help me? Thanks a lot :)

P.S. I will try to be more concrete: Let us consider the following problem

$ J(t_0,x_0)=\max_{u} E \left[ \int_{t_0}^T f(t,x,u)\ dt \right]$

subject to

$d x(t)=g(t,x(t),u(t))\ dt + \sigma (t,x(t),u(t))\ d B_t$.

Exploiting the linearity of the integral, we can write

$ J(t_0,x_0)=\max_{u} E \left( \int_{t_0}^{t_0+\Delta t} f(t,x,u)\ dt + \int_{t_0+\Delta t}^T f(t,x,u)\ dt \right)$.

Using the Bellman's principle and the law of iterated expectations, we can write

$J(t_0,x_0)=\max_{u, t_0 \leq t_0+\Delta} E \left[f(t,x,u) \Delta + J(t_0+\Delta,x_0+\Delta x) \right]$.

Applying Ito's lemma and taking the limit $\Delta t \to 0$, we derive the following HJB equation:

$-\frac{J (t,x)}{\partial t} = \max_{u} \left(f(t,x,u)+g(t,x,u) \frac{\partial J(t,x)}{\partial x} + \frac{1}{2} \sigma(t,x,u)^2 \frac{\partial^2 J}{\partial x^2}\right)$.

My question is the following: In order to derive the HJB equation for the exponential case, could we apply a similar reasoning? Since the exponential is not additive, it is not clear to me how to do it.

I found a paper where the author solves an exponential control problem claiming (but not proving) that the HJB equation is (up to a minus or plus sign)

$ \frac{\partial J}{\partial t} + sup (\frac{1}{2} \sigma^2 \nabla^2 J + \mu \nabla J -\frac{\gamma}{2} \sigma^2 (\nabla v)^2 - g)=0$.

Can someone explain to me where this equation comes from? As you can see, it has some terms in common with the "standard" one, but there is also some "new" term.