27 | May | 2012 | Eventually Almost Everywhere

The motivation for the development of differential calculus by Newton et al. was to enable us to deduce concrete properties of, say, particle motion defined implicitly through ODEs. And we proceed similarly for the stochastic differential. Having defined all the terms through Ito’s formula, and concluded that BM is in some sense the canonical stochastic process, we seek to solve so-called stochastic differential equations of the form:

$dX_t=b(X_t)dt+\sigma(X_t)dB_t$

While there is no reason not to consider processes in $\mathbb{R}^d$ , it is reasonable interesting to consider processes in one dimension. As with normal ODEs and PDEs, we have some intuitive notion if we specify some initial conditions, we should be able to set the differential equation up and ‘let it go’ like a functional clockwork mouse. Of course, we are conscious of the potential problems with uniqueness of solutions, stability of solutions, and general mathematical awkwardness that derives from the fact that we can’t treat all DEs as physical systems, with all the luxuries of definiteness that the physical world automatically affords. To establish some concreteness, we set up some definitions.

For a solution to the SDE, $E(\sigma,b)$ , we require a nice filtration $\mathcal{F}$ and a BM adapted to that filtration to drive the process $X_t$ , which satisfies $X_t=X_0+\int_0^t\sigma(X_s)dB_s+\int_0^tb(X_s)ds$ , and we require this for each $x_0\in\mathbb{R}^d$ s.t. $X_0=x_0$ a.s.
Uniqueness in law: all solutions to $E(\sigma,b)$ starting from each x have the same law. Obviously, this places no restriction on the underlying probability space and filtration.
A stronger condition is Pathwise uniqueness: Given the filtration, solutions are almost surely indistinguishable (that is, paths are equal everywhere).
We have not specified any conditions on the filtration $\mathcal{F}$ . It would be natural to consider only the minimal such filtration that works. If we can take $\mathcal{F}=\mathcal{F}^B$ , the natural filtration of the driving BM, we say the solutions is strong. If every solution is strong, then we have pathwise uniqueness, otherwise we would have a solution where we could choose which path to follow independently of the BM.

The key theorem here is Yamada-Watanabe: If there exist solutions and we have pathwise uniqueness, then we have uniqueness in law. Then for every $(\mathcal{F}_t)$ , and $\mathcal{F}_t$ -BM, the unique solution is strong.

Existence of solutions is particularly tractable when $\sigma,b$ are Lipschitz, as this opens the way for implicit constructions as the fixed points of contracting mappings. We make particular use of Gronwall’s lemma, which confirms an intuitive thought that differential inequalities have solutions bounded by solutions to the corresponding ODE. Concretely, for $latex f\geq 0,\,f(t)\leq a+b\int_0^tf(s)ds,\quad 0\leq t\leq T$, the lemma states that $f(t)\leq a\exp(bt)$ . The case a=0 is obviously of particular interest for demonstrating convergence results. We deploy this method to show that when $\sigma,b$ are Lipschitz, the SDE $dX_t=\sigma(X_t)dB_t+b(X_t)dt$ has pathwise uniqueness and for any triple of filtration $(\mathcal{F}_t)$ , $\mathcal{F}_t$ -adapted BM, and starting point x, there is a strong solution. Uniqueness in law then follows by Yamada-Watanabe, but we knew this anyway by composing measurable maps.

Now, given L, an operator on $C^2$ functions by:

$Lf(x)=\frac12\sum_{i,j}a_{i,j}(x)\frac{\partial^2 f}{\partial x^i\partial x^j}+\sum_i b_i(x)\frac{\partial f}{\partial x^i}$

We define X to be an L-diffusion if X’s local behaviour is specified (in distribution) by L(X). The first sum in the expression for L corresponds to diffusivity, while the second corresponds to (deterministic) drift. Formally, for a, b, bounded $X_t$ a L-diffusion is $\forall f\in C_b^2$ :

$M_t^f:=f(X_t)-f(X_0)-\int_0^t Lf(X_s)ds$ is a martingale.

Alternatively, can relax boundedness condition, and require $M_t^f\in\mathcal{M}_{c,loc}$ . To make a link to SDEs, define $a=\sigma\sigma^T$ (so in one dimension $a=\sqrt{\sigma}$ ), then solutions to $dX_t=\sigma(X_t)dB_t+b(X_t)dt$ are L-diffusions if boundedness conditions are met. Remember bounded implies Lipschitz implies solutions to SDEs. The result then follows directly from Ito’s formula.

In developing the stochastic integral, much of our motivation has come from considering integrals with respect to Brownian Motion. In this section, we develop some results which justify that Brownian Motion is the canonical stochastic process with non-zero quadratic variation (which is related, but not directly equivalent to the property of infinite total variation). In particular, we shall observe the Dubins-Schwarz theorem, which shows that martingales with unbounded (as time $\rightarrow\infty$ ) quadratic variation ARE Brownian Motion, up to a (stochastic) time change.

Recall Levy’s characterisation of a d-dimensional BM, which allows us to avoid considering independent normal increments. Given $X^1,\ldots,X^d\in\mathcal{M}_{c,loc}$ :

$X=(X^1,\ldots,X^d)$ a BM iff $[X^i,X^j]_t=\delta_{ij}t$

Obviously, one direction has been shown as part of the construction and properties of quadratic variation. For the other direction,, because laws are precisely defined by characteristic functions, it suffices to show that

$\mathbb{E}\left[\exp(i\langle \theta,X_t-X_s\rangle)|\mathcal{F}_s\right]=\exp(-\frac12||\theta||^2(t-s))$

We set $Y_t:=\langle \theta,X_t\rangle$ , and deduce $[Y]=t||\theta||^2$ and $Z=\mathcal{E}(iY)=\exp(iY_t+\frac12[Y]_t)\in\mathcal{M}_{c,loc}$ , and furthermore is bounded on compact [0,t], hence is a true martingale. So $\mathbb{E}\left(\frac{Z_t}{Z_s}|\mathcal{F}_s\right)=1$ which is pretty much what was required.

Now, Dubins-Schwarz states

Theorem: Given $M\in\mathcal{M}_{c,loc}, M_0=0, [M]_\infty=\infty$ almost surely, if we set $\tau_s:=\inf\{t:[M]_t>s\}$ , then $B_s:=M_{\tau_s}$ is a $(\mathcal{F}_{\tau_s})$ -BM, with $M_t=B_{[M]_t}$ .

This final result is clear if $[M]_t$ is almost surely strictly increasing in t: just take $s=[M]_t$ in the definition.

We know B is cadlag: we first show B as defined is almost surely continuous. It remains to show $B_{s-}=B_s\,\forall s>0\iff M_{\tau_{s-}}=M_{\tau_s}$ , noting that $\tau_{s-}=\inf\{t\geq 0:[M]_t=s\}$ (by continuity) is a stopping time also.

The only interesting case is if $\tau_{s-}<\tau_s$ , for which need to show [M] is constant. This is intuitively obvious, but formally, we must appeal to $(M^2-[M])^{\tau_s}$ which is UI, since $\mathbb{E}[M^{\tau_s}]_\infty<\infty$ . Now may apply OST to obtain $\mathbb{E}[M_{\tau_s}^2-M_{\tau_{s-}}^2|\mathcal{F}_{\tau_{s-}}]=\mathbb{E}[(M_{\tau_s}-M_{\tau_{s-}})^2|\mathcal{F}_{\tau_{s-}}]=0$ which implies M is almost surely constant on $[\tau_{s-},\tau_s]$ . We need to lift this to the case where it holds for all s simultaneously almost surely. Note that cadlag almost surely plus almost surely continuous at each point does not implies almost surely continuous everywhere (eg consider H(U(0,1)) with H the Heaviside function and U a uniform distribution). Instead, we record intervals of constancy of both $M_t,[M]_t$ . That is, we set

$T_r=\inf\{t>r:M_t\neq M_r\},\quad S_r=\inf\{t>r:[M]_t\neq [M]_r\}$

Then these are cadlag, and by above $T_r=S_r\,\forall r\in\mathbb{Q}^+$ a.s. therefore $T_r=S_r\,\forall r$ almost surely. Thus M, [M] are constant on the same intervals.

We also check B is adapted to $\mathcal{G}_t=\mathcal{F}_{\tau_t}$ . STP $X_T1_{\{T<\infty\}}$ is $\mathcal{F}_T$ -measurable for X cadlag adapted. Approximating T discretely from above gives the result, exploiting that the result is clear if T has countable support. Now, obtain $M^{\tau_s}\in\mathcal{M}_c^2$ , so $M_{t\wedge \tau_s}$ UI by Doob, so by OST, get $\mathbb{E}[M_{\tau_s}|\mathcal{F}_{\tau_s}]=M_{\tau_r}$ , to get B a martingale. The finally: