Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2010 / A (Stochastic Differential Equations)

Write the stochastic differential equation in integral form and replace the unknown integrand by the preceding Picard iterate. With the deterministic initial vector $a$, the recursion isEach iterate is a continuous adapted process. The Itô isometry inductively gives its square integrability on every bounded time interval, so all these Itô integrals are well defined.

Use $s$ for the dummy integration variable in the printed integral. The Itô isometry givesThe 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.

Set $K={\sum }_i\Vert A^i{\Vert }_{\mathrm{op}}^2$, where the norms are operator norms, and $D_m=y^{(m+1)}-y^{(m)}$. Independence of the Brownian coordinates and the vector Itô isometry giveInduction, as in part (b), yields the Factorial Picard estimate for a linear stochastic differential equationFor $t\le T$, the triangle inequality in $L^2$ therefore bounds any tail of the iterates by the corresponding tail ofThis numerical series converges by the ratio test: successive terms have ratio $\sqrt{KT/(m+2)}\to 0$. Thus the iterates are Cauchy in $L^2$, uniformly in the deterministic time parameter on each bounded interval. Completeness of Lp spaces gives a limit $y_t$. Once its continuous adapted version is chosen in part (d), the Itô isometry permits passage to the limit in the integral equation: the squared norm of each integral difference is bounded by $K{\int }_0^T\mathbb{E}|y_s^{(m)}-y_s{|}^{2}ds\to 0$. Consequently the Picard limit solves the stochastic differential equation.

For every $m$, $D_m$ is a continuous square-integrable vector martingale. Its norm is a submartingale, so the Doob L2 maximal inequality and the estimate from part (c) giveThe Cauchy-Schwarz inequality now showsBy Tonelli's theorem, the random sum of these suprema is finite almost surely. Therefore the telescoping series $a+{\sum }_m{D}_m(t)$ converges absolutely and uniformly on $[0,T]$ outside one null event. Its limit is continuous and agrees at each fixed time with the $L^2$ limit from part (c). Taking the intersection of the probability-one events for integer $T$ provides a single version on the whole half-line. HenceThis also gives the requested pointwise almost-sure convergence of the full sequence.

Standard Brownian motion is centered, so $m_t=0$. If $s\le t$, write $B_t=B_s+(B_t-B_s)$; the final increment is independent of $B_s$ and has mean zero. ThusThis is the Brownian covariance kernel. Together with joint Gaussianity and continuous paths, it gives the Gaussian-process characterization of Brownian motion.

For positive times the transformed process is centered and jointly Gaussian, andIts paths are continuous away from zero. To verify continuity at zero, we need $B_u/u\to 0$ almost surely as $u\to \infty$. The strong law of large numbers applied to the independent increments $B_{n+1}-B_n$ gives $B_n/n\to 0$. On the unit interval after $n$, the Doob L2 maximal inequality givesThe Borel-Cantelli first lemma, first for each positive rational $\epsilon$, shows that these between-integer fluctuations divided by $n$ tend to zero. Hence $B_u/u\to 0$ for all real $u\to \infty$, and $t{B}_{1/t}\to 0$ as $t\downarrow 0$.

The Gaussian-process characterization of Brownian motion now proves $X$ is standard Brownian motion, with respect to its natural filtration. This is time inversion of Brownian motion; the transformed process need not be adapted to the original forward filtration.

The Itô formula for $f(x)=x^4$ givesThe stochastic integral has expectation zero: its square integrability on a finite horizon follows from the finite sixth moments of a normal distribution. Since $\mathbb{E}{B}_s^2=s$, integration and expectation yield

Put $Q_t=⟨M,M{⟩}_t$ and $F(x,q)=e^{\sigma x-{\sigma }^{2}q/2}$. In the two-variable Itô formula, $Q$ has finite variation, while the only nonzero quadratic term is $d⟨M⟩=dQ$. ThereforeThis proves the stochastic exponential solves the equation. Its initial value is $e^{\sigma M_0}$; no normalization $M_0=0$ was stipulated.

The resulting process is initially a continuous local martingale. Since $Q_t\ge 0$ and $|M_t|<C$,A bounded local martingale is a true martingale: apply the martingale identity to a localizing sequence and use dominated convergence in conditional expectations. The deterministic bound works on every finite horizon, proving $Z$ is a martingale on every $[0,T]$. The bound on $Q$ also gives a positive deterministic lower bound $e^{-|\sigma |C-{\sigma }^{2}C/2}$, although the upper bound already proves the martingale assertion.