How do we deal with Markov Decision Process solution with continuous state space?

Let there be a value function parameterized on \(\theta\):

\begin{equation} U_{\theta}(s) \end{equation}

Let us find the value-function policy of this utility:

\begin{equation} \pi(s) = \arg\max_{a} \qty(R(s,a) + \gamma \sum_{s’}^{} T(s’|s,a) U_{\theta}(s’)) \end{equation}

We now create a finite sampling of our state space, which maybe infinitely large (for instance, continuous):

\begin{equation} S \in \mathcal{S} \end{equation}

where, \(S\) is a set of discrete states \(\{s_1, \dots, s_{m}\}\).

Probabilistically Checkable Proofs

Every statement that has a polynomial time checkable proof has such a proof where the verifier only reads \(O(1)\) (constant) bits of the proof such hat…

• perfect completeness: correct statements will be accepted with probability 1
• soundness: false statements will be rejected with probability 0.99 (with epsilon as the reading constant increases)

PCP Theorem

For some constant \(\alpha > 0\), and for ever language \(L \in NP\), there exists a polynomial-time that makes every input \(x\) into a \(f(x)\) such that…

If we take entangled qubits, and separate them real far away, their behavior would be the same even despite it will take longer for light to travel.

Background

In the 60s, economists that the pricing of options were independent of pricing of underlying assets. Nowadays, we can see that, if the underlying assets were obeying of a Brownian Motion, there is no additional degree of freedom that options can bring: that knowing the stocks will tell you exactly through a DiffEQ how the option will evolve.

The idea, then, is that you can replicate options: by dynamically buying and selling pairs of securities in the same way as the option, your new portfolio can track the option exactly.