The exponential Brownian martingale

Given: AC, (H), the usual conditions, the ${\mathcal{F}}_0$-normalized everywhere-continuous adapted representative of the standard Brownian motion $B$, a real parameter $\theta$, and a finite horizon $T>0$.

Class structure. Under the usual conditions the continuity event is in ${\mathcal{F}}_0$, so setting $B$ to $0$ off it preserves adaptedness, all finite-dimensional laws, and the increment-independence hypothesis while making every path continuous. The normalized $B$ is therefore predictable and is a continuous Brownian Ito process with drift $0$ and diffusion coefficient $1$; the exponential process $Z$ is everywhere continuous and adapted, hence predictable, and locally bounded, hence locally square-integrable as an integrand. Continuous Brownian Ito processes Brownian motion Locally square-integrable predictable Brownian integrands

Ito formula. For $f\in C^{1,2}([0,\mathrm{\infty })\times \mathbb{R})$ the one-dimensional Ito formula holds for every continuous Brownian Ito process, so $f(t,B_t)=f(0,0)+{\int }_0^t({\mathrm{\partial }}_{t}f+\frac{1}{2}{\mathrm{\partial }}_x^{2}f)(s,B_s)ds+{\int }_0^t{\mathrm{\partial }}_{x}f(s,B_s)d{B}_s$ up to indistinguishability. One-dimensional Ito formula

Gaussian increments and exponential moment. For $0\le s<t$ the increment $B_t-B_s$ is independent of ${\mathcal{F}}_s$ with law $N(0,t-s)$; for $N$ with law $N(0,{\sigma }^2)$, $\sigma >0$, and real $\lambda$, $E{e}^{\lambda N}=e^{{\lambda }^2{\sigma }^2\mathrm{/}2}\mathrm{.}$ Indeed, substituting $x=\sigma y$ in the density $(2\pi {\sigma }^2{)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}(2{\sigma }^2)}$ and completing the square gives $e^{{\lambda }^2{\sigma }^2\mathrm{/}2}\int (2\pi {)}^{-1\mathrm{/}2}{e}^{-(y-\lambda \sigma {)}^2\mathrm{/}2}dy=e^{{\lambda }^2{\sigma }^2\mathrm{/}2}$ by the translation change of variables and The standard normal density has total mass one; the degenerate case $\sigma =0$ gives $E{e}^{\lambda N}=1$. Standard normal and normal laws Brownian covariance is equivalent to independent stationary normal increments Brownian motion A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions

Conditional expectation tools. If $X$ is integrable and independent of ${\mathcal{F}}_s$, then $E[X\mid {\mathcal{F}}_s]=EX$. If $Y$ is finite and ${\mathcal{F}}_s$-measurable and $X,YX\in L^1$, then $E[YX\mid {\mathcal{F}}_s]=YE[X\mid {\mathcal{F}}_s]$; the latter theorem also proves that $YE[X\mid {\mathcal{F}}_s]$ is integrable. Conditioning a known variable and an independent variable Taking out what is known Conditional expectation as an ae class Tower property of conditional expectation Continuous-time adapted processes and martingales

Integral interfaces. A finite-energy integral $\int HdB$ has a continuous version that is a square-integrable martingale with mean zero and isometry $E({\int }_0^{t}HdB{)}^2=E{\int }_0^t{H}^{2}ds$; the localized integral exists for locally square-integrable predictable integrands. Localized Ito integral The Ito integral process has a continuous martingale version Ito isometry and linearity in predictable L2 Elementary predictable Brownian integrands

AC bookkeeping. Choice is declared for the conditional-expectation interface. The Axiom of Choice