# Stochastic control of jumps of random size

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!