Zhu and He [(2018). A new closed-form formula for pricing European options under a skew Brownian motion. The European Journal of Finance 24(12): 1063–1074] provided an innovative closed-form solution by replacing the standard Brownian motion in the Black–Scholes framework using a particular skew Brownian motion. Their formula involves numerically integrating the product of the Guassian density and corresponding distribution function. Being different from their pricing formula, we derive a much simpler formula that only involves the Gaussian distribution function and Owen's $T$ function.

Simulated tempering is a popular method of allowing Markov chain Monte Carlo algorithms to move between modes of a multimodal target density $\pi$ . Tawn, Moores and Roberts (2021) introduces the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that, under appropriate assumptions, the ALPS algorithm mixes in time $O(d [\log d]^2)$ or O(d), depending on which version is used.

In this note we propose an exact simulation algorithm for the solution of (1)

\begin{equation} \label{eds-intro} {\rm d}X_t={\rm d}W_t+\bar{b}(X_t){\rm d}t,\quad X_0=x, \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}\mathrm{\right)}\mathrm{d}\mathit{t,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x,}$

\hbox{$\bar{b}$}b̅

\hbox{$\bar{b}(0+)\neq \bar{b}(0-)$}b̅(0 + ) ≠b̅(0 −)

(2)

\begin{equation} \label{edsbeta} {\rm d}X^\beta_t={\rm d}W_t+\bar{b}(X^\beta_t){\rm d}t + \beta {\rm d}L^0_t(X^\beta),\quad X_0=x \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{\right)}\mathrm{d}\mathit{t}\mathrm{+}\mathit{\beta }\mathrm{d}{\mathit{L}}_{\mathit{t}}^{\mathrm{0}}\mathrm{\left(}{\mathit{X}}^{\mathit{\beta }}\mathrm{\right)}\mathit{,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x}$

Nearly fifty years after the introduction of skew Brownian motion by Itô and McKean (1963), the first passage time distribution remains unknown. In this paper we first generalize results of Pitman and Yor (2011) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use these results to derive the first passage time distribution.