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      All Questions

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      Stochastic Control: Markovian restriction

      Consider a stochastic control problem, $$v^C(0,x) = \mathbb{E} \Big[\int_0^\tau f(X_t,C_t) d t + (T-\tau)|X_\tau|\Big] $$ where $X_t$ is a weak solution to the SDE $$dX_t = C_t dB_t, \quad X_0 = x \...
      0
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      0answers
      41 views

      Reference: Stochastic Optimal Control with cost functional

      There are a variety of control problems for controlled diffusions $X_t^u$, with the terminal cost given by $$ J(u)\triangleq \mathbb{E}\left[g(X_T,u)+\int_0^t h(X_t,u_t)ds\right], $$ function $g$ and ...
      1
      vote
      1answer
      225 views

      When do supremum and expectation commute?

      This is an alternative form of the question in When do maximum and expectation commute? When we looking for conditions on $G(t,x(t))$ such that $$ \sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{...
      1
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      0answers
      38 views

      Smoothness of an optimal control problem for a point process

      Let $\theta \in \{0,1\}$ be an unknown state of the world. Let $P_0 := Prob(\theta = 1)$ at time $0$. Let $G_t$ and $B_t$ be two Poisson processes with stochastic intensity $\lambda_g e_t \theta$ ...
      3
      votes
      0answers
      148 views

      Optimal control of SDEs

      I've set up a system of stochastic differential equations that I'd like to control. I'm new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I'm not certain if I've set this up ...
      1
      vote
      0answers
      53 views

      Stochastic Control with Stochastic Cost-functional

      Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also? That is, let $X_t^u$ is the solution to a controlled SDE $$ dX_t = \mu(t,u_t,X_t^u)dt ...
      3
      votes
      1answer
      169 views

      Superlinear Convergence of a Markov Chain

      Suppose that we have a Markov process $\{Z_t\}_{t=0}^\infty$, where $Z_t \geq 0$ for any $t$. Assume that, conditioning on $Z_t = z_t$, we have $ \mathbb{E}\{Z_{t+1}|Z_t = z_t\} \leq \kappa z_t^2 $. ...
      6
      votes
      1answer
      346 views

      Supremum of a stochastic process

      Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard ...
      1
      vote
      2answers
      535 views

      Deriving the HJB equation for exponential utility

      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 ...
      4
      votes
      0answers
      182 views

      Stochastic subgradient descent almost sure convergence

      I was reading up on stochastic subgradient descent, and most sources i could find via google search give quick proofs on convergence in expectation and probability, and say that proofs of almost sure ...
      3
      votes
      1answer
      324 views

      Conditions for supremum and conditional Expectation to commute

      I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and have (for simplicity) two stopping times $\tau_1$ and $\tau_2$ such that $\tau_2 \leq \tau_1$ and $U:\Bbb R\...
      3
      votes
      1answer
      157 views

      Optimal control / Portoflio optimization: Maximize expected utility of total consumption

      I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is ...
      0
      votes
      0answers
      74 views

      Law of motion when initial condition is perturbed

      We know how to find the law of motion (Ito process) of the value function: $$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$ such that $$dX_t=\mu(t,X_t)...
      2
      votes
      1answer
      391 views

      Stochastic differential equation associated with an optimal control problem

      We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time $\...
      1
      vote
      1answer
      532 views

      Supremum in a Markov chain model

      A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

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