Lévy's stochastic area

Let ${\displaystyle W=(W_{s}^{(1)},W_{s}^{(2)})_{s\geq 0}}$ be a two-dimensional Brownian motion in ${\displaystyle \mathbb {R} ^{2}}$ then Lévy's stochastic area is the process

${\displaystyle S(t,W)={\frac {1}{2}}\int _{0}^{t}\left(W_{s}^{(1)}dW_{s}^{(2)}-W_{s}^{(2)}dW_{s}^{(1)}\right),}$

where the Itō integral is used.^[2]

Define the 1-Form ${\displaystyle \vartheta ={\tfrac {1}{2}}(x^{1}dx^{2}-x^{2}dx^{1})}$ then ${\displaystyle S(t,W)}$ is the stochastic integral of ${\displaystyle \vartheta }$ along the curve ${\displaystyle \varphi$ :[0,t]\to \mathbb {R} ^{2},s\mapsto (W_{s}^{(1)},W_{s}^{(2)})}

${\displaystyle S(t,W)=\int _{W[0,t]}\vartheta .}$^[6]

Let ${\displaystyle x=(x_{1},x_{2})\in \mathbb {R} ^{2}}$, ${\displaystyle a\in \mathbb {R} }$, ${\displaystyle b=at/2}$ and ${\displaystyle S_{t}=S(t,W)}$ then Lévy computed

${\displaystyle \mathbb {E} [\exp(iaS_{t})]={\frac {1}{\cosh(b)}}}$

and

${\displaystyle \mathbb {E} [\exp(iaS_{t})\mid W_{t}=x]={\frac {b}{\sinh(b)}}\exp \left({\frac {\|x\|_{2}}{2t}}\left(1-b\coth \left(b\right)\right)\right),}$

where ${\displaystyle \|x\|_{2}}$ is the Euclidean norm.^{[2]: 172–173}

• In 1980 Yor found a short probabilistic proof.^[7]
• In 1983 Helmes and Schwane found a higher-dimensional formula.^[8]