A random mathematical blog
Spitzer s formula is a remarkable result giving the precise joint distribution of the maximum and terminal value of a random walk in terms of the marginal distributions of the process. I have already covered the use of the reflection principle to describe the maximum of Brownian motion, and the same technique can be used for Continue reading Spitzer s Formula
In this post I look at the integral Xt = ∫0t 1{W≥0} dW for standard Brownian motion W. This is a particularly interesting example of stochastic integration with connections to local times, option pricing and hedging, and demonstrates behaviour not seen for deterministic integrals that can seem counter-intuitive. For a start, X is a martingale so has zero expectation. Continue reading On The…
In a 1986 article of The American Mathematical Monthly written by Richard Guy, the following question was asked, and attributed to Bogusłav Tomaszewski: Consider n real numbers a1, …, an such that Σiai2 = 1. Of the 2n expressions ±a1±⋯±an , can there be more with value 1 than with value ≤ 1? A cursory attempt to find such real numbers Continue reading Tomaszewski s Conjecture
Concentration inequalities place lower bounds on the probability of a random variable being close to a given value. Typically, they will state something along the lines that a variable Z is within a distance x of value μ with probability at least p, (1) Although such statements can be made in more general topological spaces, Continue reading Rademacher Concentration Inequalities
Probability and measure theory relies on the concept of measurable sets. On the real numbers ℝ, in particular, there are several different sigma-algebras which are commonly used, and a set is said to be measurable if it lies in the one under consideration. Probabilities and measures are only defined for events lying in a specific Continue reading Non-Measurable Sets
I continue the investigation of discrete barrier approximations started in an earlier post. The idea is to find good approximations to a continuous barrier condition, while only sampling the process at a discrete set of times. The difference now is that I will look at model independent methods which do not explicitly depend on properties Continue reading Model-Independent Discrete Barrier…
It is quite common to consider functions of real-time stochastic process which depend on whether or not it crosses a specified barrier level K. This can involve computing expectations involving a real-valued process X of the form (1) for a positive time T and function f: ℝ → ℝ. I am using the notation 𝔼[A;S] to denote the Continue reading Discrete Barrier Approximations
In stochastic calculus it is common to work with processes adapted to a filtered probability space . As with probability space extensions, It can sometimes be necessary to enlarge the underlying space to introduce additional events and processes. For example, many diffusions and local martingales can be expressed as an integral with respect to Brownian Continue reading Extending Filtered…
According to Kolmogorov s axioms, to define a probability space we start with a set Ω and an event space consisting of a sigma-algebra F on Ω. A probability measure ℙ on this gives the probability space (Ω, F , ℙ), on which we can define random variables as measurable maps from Ω to the reals or other measurable Continue reading Probability Space Extensions and Relative Products
Stochastic differential equations (SDEs) form a large and very important part of the theory of stochastic calculus. Much like ordinary differential equations (ODEs), they describe the behaviour of a dynamical system over infinitesimal time increments, and their solutions show how the system evolves over time. The difference with SDEs is that they include a source Continue reading Stochastic…