Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 24 (Stochastic Calculus and Applications)

Write the predictable sigma-algebra on positive times asIt is equivalently the smallest sigma-algebra making all left-continuous adapted processes measurable. Indeed, the indicator of each generating rectangle is left-continuous and adapted, and approximating any left-continuous adapted process by its values at the preceding points of deterministic partitions gives the reverse inclusion. Previsibility records information available immediately before observation. If time zero is included, add $B\times \{0\}$ with $B\in {\mathcal{F}}_0$. The displayed description in part (b) uses positive times, as indicated by its product space.

Observing $X_s$ reveals exactly whether $\omega <s$. Thus the natural filtration isEvery generating set either contains the entire tail $[t,\infty )$ or avoids it, so this tail remains one indistinguishable atom. Conversely, these initial intervals generate the full relative Borel sigma-algebra on $(0,t)$: use rational endpoints below $t$, together with $(0,t)$ itself. Therefore every Borel set contained in $(0,t)$, and its complement, belongs to ${\mathcal{F}}_t$. We obtainThe strict endpoint is essential: the value at $\omega =t$ is still $X_t=1$, so it cannot be distinguished from a later lifetime.

Put $D=\{(\omega ,t):\omega \ge t\}$ and let $\mathcal{C}$ denote the proposed family of sets. This is a sigma-algebra: complements replace $T$ and $A$ by their complements, and countable unions replace them by their unions. For a generating predictable rectangle $B\times (s,u]$, part (a) says either $B\subseteq (0,s)$ or $B^c\subseteq (0,s)$. In the first case its trace on $D$ is empty; in the second it is exactly $D\cap (\Omega \times (s,u])$. Its trace on $D^c$ is always product-Borel. Hence $\mathcal{P}\subseteq \mathcal{C}$.

For the reverse inclusion, $X$ is left-continuous and adapted, so $D=\{X=1\}$ belongs to the predictable sigma-algebra. Every deterministic Borel set in time also belongs to that sigma-algebra, hence $D\cap (\Omega \times T)$ is predictable. For a product rectangle $E\times T$,Each term is predictable because $E\cap (0,s)\in {\mathcal{F}}_s$. Every pair with $\omega <t$ lies in such a term by choosing $\omega <s<t$. The class of product-Borel sets $A$ for which $A\cap D^c$ is predictable is a sigma-algebra containing the product rectangles, so it contains $\mathcal{F}\otimes \mathcal{F}$. ThusThis survival-observation predictable sigma-algebra expresses a sharp distinction: before and at the lifetime the only observable coordinate is time, whereas strictly after it the lifetime is known.

For a simple predictable process, define its stochastic integral byCommon refinement of the deterministic partitions makes this definition independent of the representation. Each summand is a martingale increment multiplied by information already available at its left endpoint, so the resulting process is a continuous square-integrable martingale starting at zero. Its terminal value is the same sum with $t$ replaced by $t_n$.

Write ${\Delta }_{k}M=M_{t_{k+1}}-M_{t_k}$. If $j<k$, then $Z_{t_j}{\Delta }_{j}M{Z}_{t_k}$ is ${\mathcal{F}}_{t_k}$-measurable. By the conditional expectation identity for increments of a martingale,The conditional bracket isometry for stopped martingale increments givesFor completeness, this identity follows by applying the optional stopping theorem to $M$ and $M^2-[M]$ and expanding the square. The latter process is a true uniformly integrable martingale here: the Doob L2 maximal inequality bounds $\mathbb{E}{\sup }_t|M_t{|}^2$, stopped square identities and Fatou's lemma give $\mathbb{E}[M{]}_{\infty }<\infty$, and its absolute supremum is bounded by ${\sup }_t|M_t{|}^2+[M{]}_{\infty }$. Consequently all the stopped identities pass to the limit.

Multiplying the conditional identity by $Z_{t_k}^2$, taking expectations, and using the vanished cross terms proves the Itô isometry:

To extend the stochastic integral, use the finite quadratic-variation measure ${\nu }_M(C)=\mathbb{E}\int 1_{C}d[M]$ on the predictable sigma-algebra. The density of simple predictable processes for finite measures gives approximations to every $H\in L^2({\nu }_M)$. The Itô isometry makes the corresponding terminal integrals Cauchy in $L^2$, and the Doob L2 maximal inequality makes the whole processes Cauchy in expected squared supremum. Their limit defines a continuous martingale, and the isometry persists. For a locally bounded predictable process, choose increasing stopping times tending to infinity on which it is bounded, also restrict the time horizon, and perform this construction after stopping. The isometry shows that the stopped definitions agree on overlaps, so they patch into a uniquely defined continuous local martingale $H\cdot M$.

The Itô formula decomposes the smooth transform into a continuous local martingale and a finite-variation process:If $f(M)$ has finite variation, then $N$ also has finite variation. A continuous finite-variation local martingale is constant, so $N=0$ andIt remains to show that the second-order term vanishes; the preceding observation alone does not establish this.

Let $Z=\{a:f^{\prime }(a)=0\}$. By the occupation-times formula for the local time of a semimartingale,Therefore $L_t^a(M)=0$ for almost every $a\notin Z$. Also $f^{\prime \prime }(a)=0$ for almost every $a\in Z$: if $f^{\prime }(a)=0$ and $f^{\prime \prime }(a)\ne 0$, continuity of $f^{\prime \prime }$ makes $f^{\prime }$ strictly monotone in a neighborhood of $a$, so that zero is isolated. Such isolated zeros form a countable set. Localizing to bounded paths and finite quadratic variation, then using the occupation-times formula once more, givesBoth terms in the Itô formula now vanish. Apply this on all rational times and use continuity to obtain the simultaneous conclusionThis is the finite-variation smooth transform of a continuous local martingale property.

There is an initial-value qualification in the printed conclusion. As written, nothing requires $A_0=0$. For example, $M\equiv 0$ and $A\equiv 1$ satisfy the hypotheses, because $M^2-A\equiv -1$ is a local martingale, but $M_t$ is not $N(0,1)$. The correct variance is $A_t-A_0$; the printed formula holds under the usual normalization $A_0=0$.

Indeed, $M^2-[M]$ is a continuous local martingale. Subtracting it from $M^2-A$ shows that $[M]-A$ is a continuous local martingale with finite variation, hence is its initial constant $-A_0$. ThusIn particular, $C$ is deterministic, continuous, nonnegative and increasing, regardless of the apparent allowance of a general finite-variation process in the premise.

For a real parameter $\theta$, the Itô formula gives the complex local martingaleOn every deterministic interval $[0,T]$, its modulus is at most $e^{{\theta }^2{C}_T/2}$, so its real and imaginary parts are bounded local martingales, hence true martingales. Taking expectations and using $Z_0=1$ yieldsBy the uniqueness theorem for characteristic functions,A zero variance means the point mass at zero. This proves the requested normalized version by a direct characteristic function calculation.

First stop the solution on leaving $[-n,n]$, at ${\tau }_n$, and set $F_n(t)=\mathbb{E}{\sup }_{u\le t}|X_{u\wedge {\tau }_n}{|}^2$. The stochastic integral representation, the Doob L2 maximal inequality and the Cauchy-Schwarz inequality give, for $0\le t\le 1$,Here stopping ensures square integrability before the calculation, and $8{\sigma }^2+2t{b}^2\le 8({\sigma }^2+b^2)$ explains the constant. The Gronwall inequality gives $F_n(t)\le e^{8At}-1$. For globally Lipschitz functions as coefficients, the global existence theorem for stochastic differential equations with Lipschitz coefficients supplies a nonexplosive solution. Letting $n$ increase and using Fatou's lemma proves the slightly stronger maximal second-moment bound under linear growth:

For coefficients that are only locally Lipschitz functions, let ${\pi }_n(x)=\max (-n,\min (x,n))$ and replace the coefficients by $\sigma ({\pi }_n(x))$ and $b({\pi }_n(x))$. These are globally Lipschitz functions, agree with the originals on $[-n,n]$, and obey the same linear growth condition for an SDE, since $|{\pi }_n(x)|\le |x|$. The globally defined solutions agree until their common exits from $[-n,n]$ by pathwise uniqueness. The uniform preceding estimate and Markov's inequality implyPatch the solutions before their increasing exit times. Their limiting lifetime exceeds $1$ almost surely by this bound. Thus a pathwise unique strong solution exists throughout $[0,1]$ even for locally Lipschitz coefficients.

Separate $L$ into a diffusion generator and the constant killing term $-1/2$. The appropriate stochastic differential equation and its explicit geometric Brownian motion solution areThe second-order term in the Itô formula cancels the drift of $\log |X|$ when $x\ne 0$; if $x=0$, the solution stays zero.

Fix a terminal time $t$. Apply the Itô formula to $e^{-s/2}u(t-s,X_s)$ for $0\le s\le t$. Its drift isAfter localization this is a local martingale, and it is bounded because $u$ is bounded. It is therefore a true martingale. Taking expectations at its endpoints gives the Feynman-Kac formula

For $t>0$ and $x\ne 0$, the change of variable $y=x{e}^z$ in the normal density produces the killed geometric Brownian heat kernel with respect to Lebesgue measure:For $xy>0$, an equivalent form isIn particular,The apparent missing $1/y$ in this last expression is absorbed into the completed square, together with the killing factor; it is not a transcription error. The total mass is $e^{-t/2}$, as expected for killing at rate $1/2$. At $x=0$ the fundamental kernel is instead the measure $e^{-t/2}{\delta }_0(dy)$; it has no Radon-Nikodym derivative with respect to Lebesgue measure. Thus $u(t,0)=e^{-t/2}g(0)$. At $t=0$ the kernel is ${\delta }_x$, and continuity of bounded $g$ gives the initial condition. This also specifies the fundamental solution at the degenerate point omitted by a density-only formula.

Use coordinates $1,2,3$ for paper, scissors and stone. Let $X^N$ take values in the lattice simplex $S_N=\{(n_1,n_2,n_3)/N:n_i\ge 0,{\sum }_i{n}_i=N\}$. A uniformly selected unordered pair consists of types $i$ and $j$, $i\ne j$, with probability $2n_i{n}_j/(N(N-1))$. Losing copies the winner, so the possible jumps of the continuous-time Markov chain areTies make no change. For arbitrary event rate $\lambda$, the respective transition rates are $2\lambda N{x}_1{x}_3/(N-1)$, $2\lambda N{x}_2{x}_1/(N-1)$ and $2\lambda N{x}_3{x}_2/(N-1)$. ChooseThen the rates are exactly $N{x}_1{x}_3$, $N{x}_2{x}_1$, $N{x}_3{x}_2$. The infinitesimal generator of this cyclic imitation chain iswhere $\mathcal{J}=\{e_1-e_3,e_2-e_1,e_3-e_2\}$ and the three $\beta$ functions are, in that order, $x_1{x}_3,x_2{x}_1,x_3{x}_2$. Its drift is precisely

Here is a precise fluid limit with its error control. Suppose $X_0^N\to x_0$ in probability, and let $x$ solve $\dot{x}=F(x)$, $x(0)=x_0$. For every finite $T$,Indeed, the martingale decomposition of a density-dependent jump process givesThis follows by compensating each transition count by its integrated rate; the compensated counts are square-integrable martingales, with variance equal to the expected compensator. Since $\Vert \ell {\Vert }^2=2$ and $x_1{x}_2+x_2{x}_3+x_3{x}_1\le 1/3$ on the simplex, the Doob L2 maximal inequality, applied coordinatewise, givesThe polynomial drift is Lipschitz on the compact simplex, with some finite constant $L$. It preserves the sum of the coordinates, and each coordinate solves an equation of the form ${\dot{x}}_i=x_i{h}_i(x)$, so nonnegativity is preserved. Thus the ordinary differential equation stays in the simplex and exists for all time. The Gronwall inequality giveswhich proves the claim by Markov's inequality. This proves the required version directly, rather than invoking an unspecified approximation theorem.

For an interior trajectory, differentiating the logarithm of the product of numbers givesThe same identity for the product itself holds on the boundary, so $x_1(t)x_2(t)x_3(t)$ is conserved. A strictly positive initial product keeps the deterministic trajectory away from the boundary. Except at $(1/3,1/3,1/3)$ these interior trajectories are periodic: the strictly concave function ${\sum }_i\log x_i$ has smooth compact level curves surrounding its unique maximum, and the vector field is nonzero and tangent to each such curve.

The finite Markov chain behaves differently at long times. For $P(x)=x_1{x}_2{x}_3$, the change in $P$ during the first jump isThe other two terms are cyclic permutations. Their first-order contributions cancel, leaving the exact product decay in a cyclic imitation chain identityFor $N\ge 3$, while all three counts are positive, their product is at least $N-2$, so $P(X_t^N)\ge (N-2)/N^3$. A missing type cannot reappear. If $\tau$ is the first loss of a type, thenOnce only two types remain, one always wins, so its count increases until an absorbing monochromatic vertex is reached, almost surely in finite time. The cases of one or two initially present types follow the same argument without the first stage. Every finite chain eventually becomes monochromatic, whereas an interior deterministic solution retains its positive product. The fluid limit holds on each fixed finite interval; it does not justify exchanging the limits $N\to \infty$ and $t\to \infty$.