Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 2 / c

Fix $t<\infty$. Part (b)(i) makes $d{\mathbb{Q}}_t=Z_{t}d\mathbb{P}$ a probability measure on ${\mathcal{F}}_t$, equivalent to $\mathbb{P}$ since $Z_t>0$. Its density process on $[0,t]$ is $Z_s=\mathbb{E}[Z_t\mid {\mathcal{F}}_s]$. The Girsanov theorem, applied to the stochastic exponential with integrand $-\beta B_s$, says thatis standard Brownian motion under ${\mathbb{Q}}_t$. The required local square-integrability follows from path continuity, and the true martingale property was proved in part (b)(i).

By part (b)(ii), under ${\mathbb{Q}}_t$ the process $B$ is therefore the zero-started Ornstein-Uhlenbeck process:This identifies its full law on continuous path space, rather than just its one-time marginal. It is a centered Gaussian process, since it consists of deterministic Itô integrals against Brownian motion. For $r,s\in [0,t]$, the Itô isometry gives its covariance function:In particular, its Ornstein-Uhlenbeck transition distribution at the terminal time is

Put $c=\beta /2-\alpha$. The change of measure and part (a)(iii) yieldFor a centered normal random variable $G$ of variance $v>0$, a Gaussian integral gives the Gaussian quadratic exponential momentHere $c\le \beta /2$ and $\beta v_t<1/2$ for finite positive $t$, so the required positivity holds, including when $c>0$. ThusExpanding the two exponentials in terms of hyperbolic functions givesAt $t=0$ this is one. As $\beta \downarrow 0$, the denominator tends to $1+2\alpha t$, recovering part (a)(i). At $\alpha =0$ it gives the Laplace transform of the integrated square of Brownian motion. The joint terminal-and-integral transform is the Joint Laplace transform of Brownian square and its time integral.