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Consider the problem of maximizing expected lifetime utility

$$ V(a_t) \equiv \max_c\mathrm{E}_t \int_t^\infty e^{\rho (s - t)}u(c_t)\mathrm{d}t $$ subject to a state process $\mathrm{d}a_t$ which is stochastic only because of jumps which arrive with probability $\lambda$ and have random size $\theta_t$. I have two questions and reference requests about such a problem.

  1. If $\theta_t$ was deterministic, and to make things easy, constant, the HJB for this problem would be

$$ \rho V = \max_c \{u(c) + V'(a) \mu + \lambda(\tilde{V} - V)\}, $$ where $\mu$ is the deterministic part of the state process and $\tilde{V}$ is the value function conditional on the jump. Now if $\theta_t$ is random and distributed as $F$, I think the HJB would become

$$ \rho V = \max_c \{u(c) + V'(a) \mu + \lambda\int(\tilde{V} - V)\mathrm{d}F\}. $$

Is this correct? Are there any references that prove this? One would need something analogous to an Ito's Lemma with random coefficients. Sennewald (2005) discusses the case with a deterministic $\theta_t$.

  1. Is there a stochastic maximum principle formulation of the problem? What would it look like? Any references here would be greatly appreciated.

Thanks!

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