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Stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford

Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford

2010

{c}

Use $s$ for the dummy integration variable in the printed integral. The <Itô isometry> gives
$$
\mathbb E|z_t^{(n+1)}|^2=C\int_0^t\mathbb E|z_s^{(n)}|^2\,ds
\leq\frac{C^{n+1}}{n!}\int_0^t s^n\,ds
=\boxed{\frac{C^{n+1}t^{n+1}}{(n+1)!}.}
$$
The assumed second-moment bound also proves the integrability needed to apply the <Itô isometry>. The extra time integration is the source of the factorial improvement.

Solution

Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2010 / A / 1 / b / Solution

Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2010 / A / 1 / b / Solution

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Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2010 / A / 1 / b / Solution

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