Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 29K (Stochastic Financial Models)

A Brownian motion starts at zero, has continuous paths, and has independent increments with $W_t-W_s\sim N(0,t-s)$. The stated covariance makes every finite collection jointly Gaussian; disjoint increments have zero covariance and hence are independent. Their variances are interval lengths, proving all defining properties.

The transformed process is centred Gaussian andThe Brownian strong law makes $t{W}_{1/t}\to 0$ almost surely as $t\downarrow 0$, giving continuity at zero; elsewhere continuity is immediate. Part (a) now applies.

The joint Gaussian density of the independent increments $\Delta W_i$ changes under shifts $c\Delta t_i$ byMultiplying the density and integrating the bounded cylinder function $g$ proves the identity.

Apply Cameron-Martin theorem to the absolutely continuous path $h(t)=xt/T$ up to $T$ and $h(t)=x$ thereafter. Its derivative is $x/T$ on $[0,T]$ and zero later, so the Radon-Nikodym factor is $\exp ((x/T)W_T-x^2/(2T))$, yielding the formula for every bounded measurable functional of the path after $T$.