indistinguishable | 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.

There is a natural definition of ‘pathwise’ stochastic integrals of a certain type of ‘simple’ process with respect to cadlag non-decreasing processes. It can be a shown that a function is of finite variation iff it can be expressed as the difference of two such functions. Hence, these finite variation processes can be used as variable of integration via an obvious linear extension. One direction of this result is obvious; the other is fiddly. To proceed, we show that the total valuation process is cadlag (and, obviously, increasing), and then check that $a'=\frac12(v+a),a''=\frac12(v-a)$ are processes satisfying the conditions of the result.

Our overall aim is to define integrals with respect to Brownian Motion since that is (in a sense to be made precise through the Dubins-Schwarz theorem later) the canonical non-trivial stochastic process with non-zero quadratic variation. The result we demonstrate shows that it is not possible to define the integral with respect to BM through pathwise finite variation integrals.

Theorem: $M\in\mathcal{M}_{c,loc},M_0=0$ a.s. is of finite variation. Then M is indistinguishable from 0.

We will show this for M a bounded martingale with bounded variation. Why does this suffice? In general, set $S_n:=\inf\{t,V_t\leq n\}$ , noting that V is continuous adapted non-decreasing. If $M^{S_n}\equiv 0\,\forall n$ , then we are done, as the $S_n$ s are increasing. But this is a bounded martingale with bounded variation.

To prove this, we make use of the orthogonality relation which is a key trick for this sort of result: If M is a martingale, with $M_s,M_t\in L^2$ , for s<t, then just by multiplying out:

$\mathbb{E}[(M_t-M_s)^2|\mathcal{F}_s]=\mathbb{E}[M_t^2-M_s^2|\mathcal{F}_s]$ a.s.

Now, for this particular result, we decompose

$\mathbb{E}[M_t^2]=\mathbb{E}\left[\sum_{k=0}^{2^n-1}(M_{(k+1)2^{-n}t}^2-M_{k2^{-n}t}^2)\right]=\mathbb{E}[\sum (M_{(k+1)2^{-n}t}-M_{k2^{-n}t})^2]$

and then we bound this last term as

$\leq \mathbb{E}\left[\sup_k [M_+-M_-]\sum_k |M_+-M_-|\right]$

Now, as $n\uparrow\infty$ , we have $\sum_k |M_+-M_-|\uparrow V_t\leq N$ by the boundedness assumption. Furthermore, M is almost surely continuous on [0,t] and so it is in fact uniformly continuous, which allows us to conclude that

$\sup_k |M_+-M_-|\downarrow 0$

By bounded convergence, this limit applies equally under the expectation, and so conclude that $\mathbb{E}M_t^2=0$ for each time t, and so for each time t the martingale is almost surely equal to 0. In the usual, can lift this to rational points by countability, then to all points by continuity.

Eventually Almost Everywhere

A blog about probability and olympiads by Dominic Yeo

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SDEs and L-Diffusions

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