15 research outputs found
Minimization Solutions to Conservation Laws with Non-smooth and Non-strictly Convex Flux
Conservation laws are usually studied in the context of sufficient regularity
conditions imposed on the flux function, usually and uniform convexity.
Some results are proven with the aid of variational methods and a unique
minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these
classical results can be extended to a flux function that is not necessarily
smooth or uniformly or strictly convex. Although uniqueness a.e. of the
minimizer will generally no longer hold, by considering the greatest (or
supremum, where applicable) of all possible minimizers, we can successfully
extend the results. One specific nonlinear case is that of a piecewise linear
flux function, for which we prove existence and uniqueness results. We also
approximate it by a smoothed, superlinearized version parameterized by
and consider the characterization of the minimizers for the
smooth version and limiting behavior as to that of the
sharp, polygonal problem. In proving a key result for the solution in terms of
the value of the initial condition, we provide a stepping stone to analyzing
the system under stochastic processes, which will be explored further in a
future paper.Comment: 27 pages, 5 figure
A Minimization Approach to Conservation Laws With Random Initial Conditions and Non-smooth, Non-strictly Convex Flux
We obtain solutions to conservation laws under any random initial conditions
that are described by Gaussian stochastic processes (in some cases
discretized). We analyze the generalization of Burgers' equation for a smooth
flux function for
under random initial data. We then consider a piecewise linear, non-smooth and
non-convex flux function paired with general discretized Gaussian stochastic
process initial data. By partitioning the real line into a finite number of
points, we obtain an exact expression for the solution of this problem. From
this we can also find exact and approximate formulae for the density of shocks
in the solution profile at a given time and spatial coordinate . We
discuss the simplification of these results in specific cases, including
Brownian motion and Brownian bridge, for which the inverse covariance matrix
and corresponding eigenvalue spectrum have some special properties. We
calculate the transition probabilities between various cases and examine the
variance of the solution in both and . We also
describe how results may be obtained for a non-discretized version of a
Gaussian stochastic process by taking the continuum limit as the partition
becomes more fine.Comment: 36 pages, 5 figures, small update from published versio
Asset Price Volatility and Price Extrema
The relationship between price volatility and expected price market extremum is examined using a fundamental economics model of supply and demand. By examining randomness through a microeconomic setting, we obtain the implications of randomness in the supply and demand, rather than assuming that price has randomness on an empirical basis. Within a general setting of changing fundamentals, the volatility is maximum when expected prices are changing most rapidly, with the maximum of volatility reached prior to the maximum of expected price. A key issue is that randomness arises from the supply and demand, and the variance in the stochastic differential equation governing the logarithm of price must reflect this. Analogous results are obtained by further assuming that the supply and demand are dependent on the deviation from fundamental value of the asset
Analytical and Numerical Results on Escape of Brownian Particles
A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called "window" or "gate") in which it is absorbed. The main problems are to determine the first hitting time and spatial distribution. A closed formula for the mean first hitting time is given for a gate of any size. Also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed. Numerical simulations of the stochastic process with finite step size and sufficient amount of sample paths are compared with the exact solution to the Brownian motion (the limit of zero stepsize), providing an empirical formula for the divergence. Histograms of first hitting times are also generated
Establishing Cryptocurrency Equilibria Through Game Theory
We utilize optimization methods to determine equilibria of cryptocurrencies. A core group, the wealthy, fears the loss of assets that can be seized by a government. Volatility may be influenced by speculators. The wealthy must divide their assets between the home currency and the cryptocurrency, while the government decides the probability of seizing a fraction the assets of this group. We establish conditions for existence and uniqueness of Nash equilibria. Also examined is the separate timescale problem in which the government policy cannot be reversed, while the wealthy can adjust their allocation in reaction to the government’s designation of probability