The joint distribution of a geometric Brownian motion and its time-integral was derived in seminal paper by Yor (1992) using Lamperti’s transformation, leading to explicit solutions terms modified Bessel functions. In this paper, we revisit classic result the simple Laplace transform approach connection Heun differential equation. We extend methodology with affine drift show that process can be...

An American option grants the holder the right to select the time at which to exercise the option, so pricing an American option entails solving an optimal stopping problem. Difficulties in applying standard numerical methods to complex pricing problems have motivated the development of techniques that combine Monte Carlo simulation with dynamic programming. One class of methods approximates th...

An American option grants the holder the right to select the time at which to exercise the option, so pricing an American option entails solving an optimal stopping problem. Difficulties in applying standard numerical methods to complex pricing problems have motivated the development of techniques that combine Monte Carlo simulation with dynamic programming. One class of methods approximates th...

An American option grants the holder the right to select the time at which to exercise the option, so pricing an American option entails solving an optimal stopping problem. Difficulties in applying standard numerical methods to complex pricing problems have motivated the development of techniques that combine Monte Carlo simulation with dynamic programming. One class of methods approximates th...

This note contains two main results. (1) (Discrete time) Suppose S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose the initial option prices agree with the Cox–Ross–Rubinstein binomial tree model.) Then S is a geometric simple random walk. (2) (Continuous time) Suppose S = S0eσX−σ 〈X〉/2 is a cont...

Several challenges in the engineering or financial world can be resolved with a proper handle on data. Amongst other applications in engineering, system identification and parameter estimation are widely used in developing control strategies for automation. In this domain, there would be requirements to design an adaptive control system. In order to design an adaptive control system, an adaptiv...

Jump diffusion processes are often used as an alternative to geometric Brownian motion within continuous–time dynamic financial time series models. The advantages of the jump diffusion process are that it can not only account for discrete jumps in the path of the process, but it also provides a simple way of replacing the Gaussian return distributions that arise in geometric Brownian motion mod...