The Multifractal Nature of Volterra-L\'{e}vy Processes

We consider the regularity of sample paths of Volterra-L\'{e}vy processes. These processes are defined as stochastic integrals M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, where $X$ is a L\'{e}vy process and $F$ is a deterministic real-valued function. We derive the spectrum of singularities and a result on the 2-microlocal frontier of $\{M(t)\}_{t\in [0,1]}$, under regularity assumptions on the function $F$.Comment: 21 pages, Stochastic Processes and their Applications, 201

Is higher-order evidence evidence?

Suppose we learn that we have a poor track record in forming beliefs rationally, or that a brilliant colleague thinks that we believe P irrationally. Does such input require us to revise those beliefs whose rationality is in question? When we gain information suggesting that our beliefs are irrational, we are in one of two general cases. In the first case we made no error, and our beliefs are rational. In that case the input to the contrary is misleading. In the second case we indeed believe irrationally, and our original evidence already requires us to fix our mistake. In that case the input to that effect is normatively superfluous. Thus, we know that information suggesting that our beliefs are irrational is either misleading or superfluous. This, I submit, renders the input incapable of justifying belief revision, despite our not knowing which of the two kinds it is

The Miner's Dilemma

An open distributed system can be secured by requiring participants to present proof of work and rewarding them for participation. The Bitcoin digital currency introduced this mechanism, which is adopted by almost all contemporary digital currencies and related services. A natural process leads participants of such systems to form pools, where members aggregate their power and share the rewards. Experience with Bitcoin shows that the largest pools are often open, allowing anyone to join. It has long been known that a member can sabotage an open pool by seemingly joining it but never sharing its proofs of work. The pool shares its revenue with the attacker, and so each of its participants earns less. We define and analyze a game where pools use some of their participants to infiltrate other pools and perform such an attack. With any number of pools, no-pool-attacks is not a Nash equilibrium. With two pools, or any number of identical pools, there exists an equilibrium that constitutes a tragedy of the commons where the pools attack one another and all earn less than they would have if none had attacked. For two pools, the decision whether or not to attack is the miner's dilemma, an instance of the iterative prisoner's dilemma. The game is played daily by the active Bitcoin pools, which apparently choose not to attack. If this balance breaks, the revenue of open pools might diminish, making them unattractive to participants