Brownian motion is a phenomenon in which small particles suspended in a liquid tend to move in pseudo-random paths through the...
What is Standard Brownian Motion?In this expository note, we explain several different historical approaches to the construction of standard Brownian motion.Applied Probability and Stochastic Processesdoi:10.1007/978-981-15-5951-8_4Krishna B. Athreya...
for all . (Actually, it suffices to verify this constraint at time , as the heat equation (1) will then preserve this constraint.) Indeed, in this case, one can interpret as the probability distribution of a Brownian motion where is a stochastic process with initial probability distribution...
(i) (Real case) Let be a real Brownian motion with base point . Let be a bounded stopping time – a bounded random variable with the property that for any time , the event that is determined by the values of the trajectory for times up to (or more precisely, this event is measurable...
In this paper we revisit an open problem posed by Aldous on the max-entropy win-probability martingale: given two players of equal strength, such that the win-probability is a martingale diffusion, which of these processes has maximum entropy and hence gives the most excitement for the spectator...
The model predicts that the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. It incorporates the constant price variation of the stock, the time value of money, the option's strike price, and the time to the option's expiry when it's app...
We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the ...
To verify whether the two conductance states represent the structurally distinct conformation of the channel, with dilation taking place at the ϕ-clamp [56], several molecular probes known to modify PA63 channel conductance were tested [11]. First, partitioning of a water-soluble polymer, poly...
can anyone tell me what is the meaning of number e,i mean how it is discovered ? If I'm not mistaken, in the 17th century, banks became interested in the problem of how to compute continual (compound?) interest, and that pretty much drove the number "e" forth, since it is...
Remark 1 It is easy to verify that needs to be growing exponentially in in order for the above argument to work, which leads to tower-exponential bounds in the number of cells in the partition. It was shown by Gowers that a tower-exponential bound is actually necessary here. By varying ...