Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 202 / 6

Interpret the Stratonovich integral in its standard extension to continuous semimartingale integrands:This extension is needed because $f^{\prime }(X)$ is generally a semimartingale rather than a local martingale. Also the introductory prose in the PDF reverses integrand and integrator relative to its displayed definition; the displayed definition fixes the convention used here.

Apply the Itô formula to $f^{\prime }(X)$, using $f\in C^3$:The continuous finite-variation process in the second term has zero quadratic covariation with $X$, while the quadratic covariation rule for an Itô integral givesA second application of the Itô formula, now to $f(X)$, therefore yieldsAll unbounded coefficients are handled by stopping $X$ on compact spatial intervals, then removing the localizing sequence.

Take $X=Y=B$, a standard Brownian motion with $B_0=0$. The Itô formula and $⟨B{⟩}_t=t$ giveIf this nonnegative stochastic process, which starts from zero, were a local martingale, question 4(i) would make it a supermartingale and force its expectation to be at most zero. But $\mathbb{E}(B_t^2/2)=t/2>0$ for $t>0$. Hence a Stratonovich integral is in general not a local martingale.

The limit is in probability, or more strongly in uniform convergence on compacts in probability; a pathwise limit for every continuous local martingale is not implicit. The original PDF uses $\lceil t{2}^n\rceil$, whereas the supplied TeX changes this to $\lfloor t{2}^n\rfloor$; both versions have the same limit by continuity.

Put $\delta =2^{-n}$ and $m=\lfloor t/\delta \rfloor$. For the completed intervals, the algebraic identityseparates a left-point sum from a quadratic covariation sum.

The results from stochastic calculus used are these: predictable left-step approximations to a continuous adapted integrand converge in their Itô integrals in uniform convergence on compacts in probability; and products of increments of continuous semimartingales on deterministic partitions of an interval with vanishing mesh converge to their quadratic covariation in the same sense. The first follows after localization from the Itô isometry and the Doob L2 maximal inequality, since the integrand approximations converge uniformly on compact intervals; the second is the deterministic-partition characterization of quadratic covariation.

Thus the two sums converge to ${\int }_0^t{Y}_{s}d{X}_s$ and $\frac{1}{2}⟨Y,X{⟩}_t$. For the ceiling convention, at most one additional full interval extends past $t$; on $[0,T+1]$ its absolute contribution is at most ${\sup }_{s\le T+1}|Y_s|$ times the modulus of continuity of $X$ at scale $\delta$, which tends to zero almost surely. Therefore

The space-time Hermite polynomials obtained by scaling the Probabilists' Hermite polynomials are genuine polynomials in both variables, including at $t=0$:This expansion follows by differentiating the defining Gaussian exponential, and shows that $H_n(x,0)=x^n$ extends smoothly through $t=0$. Consequently the Itô formula is applicable from time zero. The given backward heat equation cancels the drift, and the derivative identity givesThe integrand is continuous and adapted, hence previsible, and stopping the Brownian motion and time makes it bounded. Thus $H_n(B_t,t)$ is a local martingale for every $n\ge 0$. The $n=0$ case is the constant $H_0=1$; in fact Gaussian moments make the displayed Itô integral square-integrable on every finite horizon, so these are true martingales as well.

The first three Probabilists' Hermite polynomials are $h_1(x)=x$, $h_2(x)=x^2-1$ and $h_3(x)=x^3-3x$. Hence