Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 36 / 4

A weak solution of a stochastic differential equation consists of a choice of probability space, filtration, driving Brownian motion and adapted solution with the specified initial law. The driving noise and the space are part of what may be chosen.

A strong solution of a stochastic differential equation is constructed for a prescribed Brownian motion and initial value, as a nonanticipating measurable functional of that data; for deterministic initial value it is adapted to the usual augmentation of the Brownian filtration. No further independent randomness is required.

Uniqueness in law means that any two weak solutions of a stochastic differential equation with the same initial distribution have the same law as solution processes. Pathwise uniqueness means that two solutions on the same filtered space, driven by the same Brownian motion and with the same initial value almost surely, are indistinguishable: $\mathbb{P}(X_t={\tilde{X}}_t\text{ for every }t)=1$. The latter concerns equality of paths on a common space, whereas uniqueness in law concerns equality of probability distributions.

Put $S=\Vert \sigma {\Vert }_{\infty }$, $K=\Vert b{\Vert }_{\infty }$, and write $X_t-x_0=M_t+A_t$, where $M_t={\int }_0^t\sigma (X_s)d{B}_s$ and $A_t={\int }_0^{t}b(X_s)ds$. The Itô isometry and boundedness giveContinuity and boundedness of $b$ then imply $\mathbb{E}b(X_s)\to b(x_0)$. Since $M$ has mean zero,For the variance, $\mathbb{E}{M}_t^2={\int }_0^t\mathbb{E}\sigma (X_s{)}^{2}ds$. The remaining terms satisfyContinuity of $\sigma$ and the initial $L^2$ convergence give $\mathbb{E}\sigma (X_s{)}^2\to \sigma (x_0{)}^2$. Expanding the variance and dividing by $t$ proves the initial mean and variance derivatives of an Itô diffusion:The derivatives at the initial time are right derivatives because the process is indexed by $t\ge 0$.

Apply the Itô formula separately to cosine and sine, using $[B{]}_t=t$:In vector standard form $d{Z}_t=\sigma (Z_t)d{B}_t+b(Z_t)dt$, with one-dimensional driving noise and two-dimensional state, the coefficient functions areThe original trigonometric process starts at $(1,0)$, but these coefficient functions define a stochastic differential equation on all of ${\mathbb{R}}^2$.

Both coefficient functions are linear, hence globally Lipschitz. Their norms are bounded by a constant times $|(x,y)|$, so they satisfy the linear growth condition for an SDE. The existence and pathwise uniqueness theorem for a stochastic differential equation gives a unique global strong solution of a stochastic differential equation for every deterministic starting point $(x_0,y_0)\in {\mathbb{R}}^2$. Linear growth prevents finite-time explosion.

The theorem gives pathwise uniqueness, and strong existence with pathwise uniqueness gives uniqueness in law. Thus every initial value is permitted, and both forms of uniqueness hold. The coefficients need not be bounded for this theorem; their global Lipschitz and linear-growth properties are sufficient.

For any initial point, rotate its vector through the Brownian angle:This is a nonanticipating function of the prescribed Brownian path and has the required initial value. Equivalently, $Z_t=X_t+i{Y}_t=(x_0+i{y}_0)e^{i{B}_t}$. The Itô formula gives $d{Z}_t=i{Z}_{t}d{B}_t-Z_{t}dt/2$, whose real and imaginary parts are exactly the equations in part c. Therefore the displayed formula is the strong solution of a stochastic differential equation, and uniqueness from part d makes it the only one.

The Brownian rotation in the plane preserves radius: $X_t^2+Y_t^2=x_0^2+y_0^2$. Nonzero starting points move around their initial circle, and the origin remains fixed.