Kelly (1956) A new interpretation of information rate

November 18, 2020 paper

Shannon defined the information rate as the number of bits transmitted per second, which bit is a unit for entropy, i.e., probability is involved. We can relate the information rate with \(R=rH\), where \(r\) is the rate that message is delivered and \(H\) is the entropy of each message. So... [more]

Chatterjee (2015) Modeling credit risk

November 10, 2020 paper

This is a concise paper describing the Vasicek model. It serves the purpose of describing how banks should evaluate risk on loan assets, and provides a connection between the Black-Scholes option pricing model to pricing bonds or loans. [more]

Owen (1980) A table of normal integrals

November 8, 2020 paper

This is an artcle of my recent acquaintance. It is about various integrals involving the standard normal distribution \(\Phi(x)\) and its derivative. Besides it is a handy reference, it is interesting to see how the author organize the hundreds of integrals in a manner that is easy to lookup. [more]

A rough description of Radon-Nikodym derivative

November 8, 2020 blog math

Radon-Nikodym theorem suggests, in simplified terms, that if we have two measures \(\mu,\lambda\) of the same space with \(\mu\) absolutely continuous with respect to \(\lambda\) and a function \(\phi\) is \(\mu\)-integrable, then \(\int_A \phi d\mu = \int_A \phi \frac{d\mu}{d\lambda} d\lambda\) which the term \(d\mu/d\lambda\) is called the Radon-Nikodym derivative. [more]

Martingale and local martingale

October 28, 2020 blog math

Martingale is a stochastic process with the martingale property. If we have \(X_t\) as the stochastic process, the martingale property says that \(\mathbf{E}[X_t\mid\mathcal{F}_s] = X_s\), for \(s\lt t\). Closely related to this is the local martingale. However, the Wikipedia page does not have it clearly explained. Here is my narrative.... [more]