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Suppose $\tilde{\mathbb{P}}$ is a probability measure with $d\tilde{\mathbb{P}}=Zd\mathbb{P}$, so $Z\ge 0$ and $\mathbb{E}Z=1$. For a sub-sigma-algebra $\mathcal{G}$, write $Z_{\mathcal{G}}=\mathbb{E}[Z\mid \mathcal{G}]$. If $X$ is integrable under $\tilde{\mathbb{P}}$, then on $\{Z_{\mathcal{G}}>0\}$,This is Bayes formula for conditional expectation. It changes the underlying probability measure as well as conditioning on the available information. The set where $Z_{\mathcal{G}}=0$ has zero $\tilde{\mathbb{P}}$-probability, so a version there can be chosen arbitrarily.

To verify the formula, let $Y$ denote the ratio. For $A\in \mathcal{G}$, measurability of $Y$ and the tower property give ${\int }_{A}Yd\tilde{\mathbb{P}}=\mathbb{E}[1_{A}ZY]=\mathbb{E}[1_A{Z}_{\mathcal{G}}Y]=\mathbb{E}[1_{A}ZX]={\int }_{A}Xd\tilde{\mathbb{P}}$. These are exactly the defining conditional-expectation identities. For a filtration, choosing $\mathcal{G}={\mathcal{F}}_t$ gives the usual time-indexed form. The denominator is the likelihood weight after reducing the full information to $\mathcal{G}$, not the original pointwise weight $Z$.

Let $({\mathcal{F}}_n)$ be a filtration. For events $B_n\in {\mathcal{F}}_n$, put $p_n=\mathbb{P}(B_n\mid {\mathcal{F}}_{n-1})$. Then ${\sum }_n{1}_{B_n}<\infty$ almost surely on $\{{\sum }_n{p}_n<\infty \}$, while ${\sum }_n{1}_{B_n}=\infty$ almost surely on $\{{\sum }_n{p}_n=\infty \}$. The proof applies the martingale convergence theorem to compensated sums ${\sum }_{j\le n}(1_{B_j}-p_j)$ and uses Kronecker lemma in the divergent case.

On $(0,1{]}^d$ with Lebesgue probability measure, the dyadic filtration is generated by the partitions into half-open cubes of side $2^{-n}$. The conditional expectation of an integrable function is its average on each atom. The increasing partitions generate the Borel sigma-algebra. Step functions on these partitions are dense in $L^1$; approximation and the Doob maximal inequality show that their conditional averages converge in $L^1$ and almost everywhere to the original function.

A filtration is an increasing family of sigma-algebras: ${\mathcal{F}}_s\subseteq {\mathcal{F}}_t$ whenever $s\le t$. It represents the information available as time passes.

For a Markov jump process with off-diagonal rate kernel $q(x,dy)$, the jump measure records the jump time and the destination state. Relative to a filtration for which the process has the Markov jump property, its predictable compensator is the displayed rate measure, up to explosion. The corresponding generator is $Qf(x)=\int [f(y)-f(x)]q(x,dy)$. An enlarged filtration revealing future jumps need not preserve this compensator, so the filtration condition is essential.

A Markov jump process is a nonexplosive, piecewise-constant continuous-time Markov process. Transition rates $q(x,y)$ give exponential holding time of rate ${\sum }_{y\ne x}q(x,y)$ and next-state probabilities proportional to these rates. Being Markov relative to a specified filtration requires the conditional future law given that filtration to depend only on the present state.

// Target: probability-and-statistics.bigb

Let $X$ be an adapted Càdlàg process on a finite state space, and let $Q$ be a conservative transition-rate matrix. It solves the martingale problem for $Q$ when, for every real function $f$ on the state space,is a martingale for the specified filtration. On a finite state space all such test functions and their generator values are bounded.

A Continuous-time Markov chain with generator $Q$ has this property by Dynkin's formula. Conversely, fix a terminal time $t$ and put $u(s,x)=(e^{(t-s)Q}f)(x)$ for $0\le s\le t$. The backward equation gives ${\partial }_{s}u+Qu=0$. Applying the martingale identities to its finitely many time-dependent coefficients shows $u(s,X_s)$ is a martingale. ThusTesting state indicators recovers the entire conditional transition law, proving the Markov property and identifying its semigroup. The càdlàg and filtration assumptions make this a path-process characterization, rather than a statement only about separately fixed-time versions.

A precise continuous Optional time-change theorem is the following. Let $M$ be a continuous local martingale with $M_0=0$ for a filtration satisfying the usual conditions. Let $({\tau }_s{)}_{s\ge 0}$ be a family of random times: every ${\tau }_s$ is a finite Stopping time, and its sample paths in $s$ are continuous and nondecreasing, with ${\tau }_0=0$. Thenis a continuous local martingale for the time-changed filtration ${\mathcal{G}}_s={\mathcal{F}}_{{\tau }_s}$. The stopping-time sigma-algebra, rather than the deterministic-time sigma-algebra at $s$, describes the information available after the clock has advanced to ${\tau }_s$.

For a proof, localize $M$ by increasing stopping times $R_n$ so that $M^{R_n}$ is bounded. Optional sampling makes $M_{{\tau }_s}^{R_n}$ a bounded martingale for $\mathcal{G}$. The new-clock stopping times $S_n=\inf \{s:{\tau }_s\ge R_n\}$ tend to infinity, because each ${\tau }_s$ is finite and $R_n\to \infty$. continuity of the clock identifies the process stopped at $S_n$ with the corresponding old-clock Stopped process. These are localizers for $N$.

The quadratic variation transforms by $⟨N{⟩}_s=⟨M{⟩}_{{\tau }_s}$. Applying the same argument to $M^2-⟨M⟩$ proves the identity. A Finite-variation process remains of finite variation on each compact new-time interval, since its variation is bounded by the old variation up to the finite time ${\tau }_s$.

For example, the deterministic clock ${\tau }_s=cs$, $c\ge 0$, turns standard Brownian motion into a Gaussian martingale of variance $cs$. General random time substitutions require the stopping-time and filtration conditions; path continuity alone establishes continuity, not the martingale claim.

Use a filtration with respect to which $B$ is Brownian. The process is adapted and continuous because it is a continuous function of $B_t$ and $t$. The moment-generating function givesFor $0\le s<t$, writeThe increment is independent of ${\mathcal{F}}_s$ and normal with variance $t-s$, so the conditional expectation of the second factor is one. ThusThis verifies the martingale property from the definition and proves square integrability at each finite time, as well as continuity. It does not assert uniform square integrability over an infinite time horizon.

Lévy's characterization of Brownian motion states that a real continuous local martingale $M$ with $M_0=0$ and quadratic variation $⟨M{⟩}_t=t$ is a standard Brownian motion with respect to its filtration. Equivalently, its increments are independent of the past and have centered normal distribution with variances equal to elapsed times.

Let $H_s=\mathrm{sgn}({\sup }_{r\le s}{B}_r-1)$. The running maximum is continuous and adapted, so $H$ is bounded and progressively measurable process. It has a predictable version for integration; changing its value at the single switching time does not change its stochastic integral. Thus $X=\int HdB$ is a continuous local martingale starting at zero. Since $H_s^2=1$,Lévy's characterization of Brownian motion yields $X$ is standard Brownian motion. The sign depends on the entire past through the running maximum, but the unit quadratic-variation rate is exactly what the characterization needs.

Fix a finite horizon $T$ and use the completed natural Brownian filtration. Martingale representation theorem states that every $F\in L^2({\mathcal{F}}_T)$ can be writtenwhere $H$ is predictable and is unique up to equality for $dsd\mathbb{P}$ almost every $(s,\omega )$.

The Martingale representation theorem states that every square-integrable martingale $M$ in this filtration has a continuous version of the formwith the same integrability and uniqueness condition. The completed natural filtration has only null-set randomness at time zero, so $M_0$ is almost surely constant. The first statement gives the second by representing $M_T$ and taking conditional expectations; the second gives the first by taking $M_t=\mathbb{E}(F\mid {\mathcal{F}}_t)$.

There is also the local version: every local martingale in the Brownian filtration has a continuous version representable by a predictable, locally square-integrable integrand. The filtration restriction matters; an enlarged filtration with independent additional martingales need not have representation using $B$ alone.

A finite-horizon form of Girsanov theorem is as follows. If $\theta$ is predictable, ${\int }_0^T{\theta }_s^{2}ds<\infty$ almost surely, and its stochastic exponential $M$ is a true martingale of mean one, then under the probability $d\mathbb{Q}=M_{T}d\mathbb{P}$ the processis Brownian motion. The same conclusion holds on the completed natural filtration, since a strictly positive density gives equivalent probabilities.

Part (b) establishes precisely the required density martingale and integrand ${\theta }_s={\partial }_x\log p(1-s,B_s,y)$. ThereforeIn particular it is the requested local martingale. Equivalently, under $\mathbb{Q}$, $B$ solves $d{B}_t=(y-B_t)(1-t{)}^{-1}dt+d{X}_t$, the Brownian bridge drift coefficient toward $y$ at time one. Keeping $T<1$ avoids the singular endpoint in this density change.

Let $T$ be a rate-one exponential random variable independent of a uniform random variable $U$ on $(0,1)$. In its natural filtration, this Markov jump process jumps once from zero to its mark and then stays there. Its destination rate kernel is $q(x,dy)=1_{\{x=0\}}{1}_{(0,1)}(y)dy$. For a Borel mark set $B\subseteq (0,1)$, the martingale $1_{\{T\le t,U\in B\}}-|B|(t\wedge T)$ follows directly from the memoryless residual lifetime. Information about the future mark or lifetime must not be inserted into the initial filtration.

A time-changed filtration records the information available after a new clock has reached an old time. Let $({\mathcal{F}}_t)$ be a filtration and let $({\tau }_s)$ be a nondecreasing family of finite stopping times. DefineThe second expression is the stopping-time sigma-algebra. If $s\le u$ and $A\in {\mathcal{F}}_{{\tau }_s}$, thenThus ${\mathcal{G}}_s\subseteq {\mathcal{G}}_u$, as required for a filtration.

For the deterministic clock ${\tau }_s=2s$, the new information is ${\mathcal{F}}_{2s}$, not ${\mathcal{F}}_s$. A continuous adapted process sampled at these stopping times is adapted to the new filtration. Preservation of the local martingale property requires further hypotheses such as those in the optional time-change theorem; merely substituting arbitrary random times does not establish it.