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A left-continuous adapted process is a predictable process. Define the elementary predictable approximationAt every $s>0$, its sample time approaches $s$ from the left, including when $s$ is itself a grid point. Thus $H_s^{(n)}\to H_s$ pathwise by left continuity.

Use the continuous semimartingale decomposition $X=X_0+M+A$, with $M$ a continuous local martingale and $A$ a continuous finite-variation process. Localize so that on a fixed horizon $T$, the integrands are bounded by a deterministic constant, $[M{]}_T$ is bounded, and the total-variation process of $A$ at $T$ is bounded. This is possible using local boundedness and the finite pathwise clock and variation.

For the finite-variation integral, the dominated convergence theorem givesalmost surely. For the martingale integral, the Itô isometry and the Doob L2 maximal inequality giveHere the localized bounds justify dominated convergence also in expectation. Remove localization to conclude that the full elementary integrals converge in uniform convergence on compacts in probability to $\int HdX$.

The sum of completed intervals differs from the full elementary integral at time $t$ byIts maximum on $[0,T]$ tends to zero almost surely, by the local bound on $H$ and uniform continuity of $X$. ThereforeThis proves left-endpoint approximation of a continuous semimartingale integral, including the unfinished interval convention.

Use the local convention for the martingale problem, which is appropriate for arbitrary measurable coefficients. A continuous adapted process $X$ solves $\mathbf{M}(a,b)$ if the coefficient integrals are locally finite and, for every $f\in C_c^{\infty }(\mathbb{R})$,is a local martingale. Here locally finite means ${\int }_0^T(|a(X_s)|+|b(X_s)|)ds<\infty$ almost surely for each $T<\infty$. Suitable localization upgrades the identities to true martingale identities. With bounded coefficients and bounded test derivatives, this distinction disappears on finite horizons.

A Markov diffusion generated by $L$ is a time-homogeneous Markov process with continuous paths whose infinitesimal generator on the stated test domain is $L$. Its transition operators satisfy the generator limit, and Dynkin formula gives precisely the preceding martingale problem identities. Some terminology uses L-diffusion for the test-function identities alone; the Markov property is an additional assertion unless uniqueness supplies it.

Suppose now $a={\sigma }^2>0$. Use smooth compactly supported functions agreeing with $x$ and $x^2$ on $[-R,R]$, and stop before exiting this interval. Removing these stops shows thatis a continuous local martingale. Consequently $X$ is a continuous semimartingale. The martingale problem applied to the second cutoff function says thatis a local martingale. The Itô formula for $X^2$ also says thatis a local martingale. Subtracting, the continuous finite-variation process $[N{]}_t-{\int }_0^{t}a(X_s)ds$ is a local martingale and starts at zero. The theorem that a continuous finite-variation local martingale is constant givesDefineThe integrand is predictable and is stochastically integrable, since its integral against the bracket is exactly $t$. Thus $B$ is a continuous local martingale with $B_0=0$ and $[B{]}_t=t$. The Lévy characterization of Brownian motion states that these conditions make $B$ a Brownian motion relative to the given filtration. Associativity of stochastic integrals recovers $N=\int \sigma (X)dB$, and thereforeThe original process is a weak solution of a stochastic differential equation, with its driving Brownian motion constructed on the original space.

Conversely, the Itô formula applied to any weak solution with locally integrable coefficients gives the martingale problem identities. A Markov diffusion with generator $L$ gives them by Dynkin formula. A well-posed martingale problem, with the usual measurable path-space setup, identifies a unique family of diffusion laws and yields the Strong Markov property. Without uniqueness, the test-function identities alone do not assert that every chosen solution is Markov. This specifies both the equivalence with weak equations and the role of uniqueness in identifying a diffusion.

Use the predictable sigma-algebra generated by $A\times \{0\}$, $A\in {\mathcal{F}}_0$, and $A\times (s,t]$, $A\in {\mathcal{F}}_s$. A previsible process is a measurable function for this sigma-algebra. Set $J_t=1_{\{S<t\le T\}}$, with $J_0=0$. The events $\{S<t\}$ and $\{T<t\}$ are in ${\mathcal{F}}_t$, because they are countable unions of events $\{S\le q\}$ and $\{T\le q\}$ for rational $q<t$. Thus $J$ is an adapted process. Each path is left-continuous, including at the two endpoints, and is zero if $S=T$.

For a direct measurability argument, letEach summand is measurable for the predictable sigma-algebra, since its coefficient is ${\mathcal{F}}_{k{2}^{-n}}$-measurable. At every $t>0$ the sampling time approaches $t$ strictly from below, so left-continuity gives $J_t^{(n)}\to J_t$. ThereforeThe original PDF has a right-closed interval. The open right endpoint in the converted TeX is a transcription error; replacing $(S,T]$ by $(S,T)$ is not valid for arbitrary stopping times.

Fix $t>0$ and partition $[0,t]$ into $m$ equal intervals, with $r_k=kt/m$. Round $t\wedge T$ upward within this interval. The corresponding sampled value isEvery time in this sum is at most $t$. The stopping time property makes each indicator function ${\mathcal{F}}_t$-measurable, while the adapted process property makes each sampled value ${\mathcal{F}}_t$-measurable. Consequently $Y_m$ is ${\mathcal{F}}_t$-measurable.

The rounded times approach $t\wedge T$ from the right, so right continuity gives $Y_m\to X_{t\wedge T}$ for the chosen pathwise càdlàg version. If path regularity is instead stated only almost surely, the usual complete filtration handles the exceptional null set; on an incomplete filtration one should formulate that case as existence of an adapted version. At $t=0$, the value is simply $X_0$. Thus $X^T$ is an adapted process, by adaptedness of a stopped right-continuous process. The argument actually needs neither the martingale property nor boundedness of $T$.

A càdlàg martingale is an adapted process $X$ with càdlàg paths, integrable $X_t$ at each finite time, andThe càdlàg requirement means that the chosen path version is right-continuous and has a finite left limit at every positive time. With a complete filtration, it suffices that this holds on one common event of probability one, since the process can be redefined on the exceptional null set.

The usual hypotheses for a filtration are completeness and right continuity. Completeness means that ${\mathcal{F}}_0$ contains every subset of every null event of the ambient probability space; right continuity means

Fix the initial state $x$, or a prescribed initial distribution. A weak stochastic solution consists of a filtered probability space, a Brownian motion $W$ in that filtration, and a continuous adapted process $X$, with the prescribed initial law, such that the integrals exist andalmost surely for every $t$. The integrability conditions on each finite interval are ${\int }_0^t|b(X_s)|ds<\infty$ and ${\int }_0^t\sigma (X_s{)}^{2}ds<\infty$ almost surely. The probability space and driving Brownian motion are part of what may be chosen.

A strong stochastic solution is constructed on a space carrying a specified driving Brownian motion and specified initial variable. It satisfies the same equation and is adapted to the completed filtration generated by that initial variable and the Brownian motion. Equivalently, it is a nonanticipating measurable function of those data, requiring no additional randomness. For a random initial variable, it is independent of future Brownian increments, as required by the Brownian property of the enlarged filtration.

The scalar Lévy characterization of Brownian motion states that an adapted process $X$ starting at zero is a Brownian motion if and only if it is a continuous local martingale with

For necessity, the centered independent increments make a Brownian motion a martingale. Their conditional second moments show that $X_t^2-t$ is also a martingale. The defining uniqueness of the quadratic variation compensator gives $⟨X{⟩}_t=t$.

For sufficiency, fix $\theta \in \mathbb{R}$ and apply the Itô formula toThe time drift cancels the second-order Itô term, leaving $d{Z}_t=i\theta Z_{t}d{X}_t$. Its real and imaginary parts are local martingales. On any deterministic interval $[0,T]$, $|Z_t|=e^{{\theta }^{2}t/2}\le e^{{\theta }^{2}T/2}$, so the bounded local martingale criterion makes them true martingales. ThereforePart (b) proves the Brownian property. This also supplies a proof of Lévy's characterization of Brownian motion through conditional Fourier transforms.

Use a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t),\mathbb{P})$, and put ${\mathcal{F}}_{0-}={\mathcal{F}}_0$. The predictable sigma-algebra on $\Omega \times [0,\infty )$ isEquivalently it is generated by left-continuous adapted processes. A real-valued previsible process is a function $(\omega ,t)\mapsto H_t(\omega )$ that is measurable for $\mathcal{P}$ and the Borel sigma-algebra of $\mathbb{R}$.

A simple predictable process has the formwhere $h_{-1}$ is bounded and ${\mathcal{F}}_0$-measurable, and each $h_i$ is bounded and ${\mathcal{F}}_{t_i}$-measurable. Set $H=0$ after $t_n$; allowing a final infinite interval gives an equivalent dense class. The values may be restricted to finitely many values if desired, by approximating bounded measurable functions by simple functions. The initial-time term matters for arbitrary finite measures on $\mathcal{P}$: a measure can charge time zero, even though the stochastic integral against a continuous martingale does not.

A continuous martingale is an adapted process with almost surely continuous paths, $\mathbb{E}|M_t|<\infty$ for every $t$, andA continuous local martingale is continuous and adapted and has a localizing sequence of stopping times $T_n\uparrow \infty$ almost surely such that every $M_t^{T_n}=M_{t\wedge T_n}$ is a true martingale. The initial value is normally integrable; the frequently used class ${\mathcal{M}}_{c,\mathrm{loc}}$ is normalized to start at zero.

The usual conditions for a filtration are completeness and right continuity. Completeness means that ${\mathcal{F}}_0$ contains every subset of each $\mathbb{P}$-null set in the ambient $\mathcal{F}$; right continuity means ${\mathcal{F}}_t={\bigcap }_{u>t}{\mathcal{F}}_u$. These conditions concern the filtered probability space, whereas continuity of a martingale concerns its sample paths.

Let the infinitesimal generator be the second-order operatorwhere $a(x)$ is symmetric positive semidefinite and $b(x)$ is a vector. In the standard bounded-coefficient setting take $a,b$ bounded and measurable. An L-diffusion is a continuous adapted process $X$ for which, for every $g\in C_b^2({\mathbb{R}}^d)$,Here $C_b^2$ includes boundedness of the function and its derivatives through order two. This is the true-martingale version of the diffusion martingale problem. The matrix $a$ is the diffusivity, and $b$ is the drift. An Itô diffusion with coefficient $\sigma$ has $a=\sigma {\sigma }^{\mathsf{T}}$.

A more general local martingale problem may permit unbounded coefficients and require only a local martingale. That convention is weaker and must not silently replace the true martingale property used here and in part (b). The bounded-coefficient convention guarantees all finite-horizon integrability needed below; a sufficient more general replacement is $\mathbb{E}{\int }_0^T({\sum }_i|b_i(X_s)|+{\sum }_i{a}_{ii}(X_s))ds<\infty$ for every $T$.

Work on a filtered probability space with the usual conditions for a filtration. A continuous martingale belongs to the class of L2-bounded continuous martingales whenThis means a uniform bound over the entire time interval, not merely being square-integrable at each individual time. The L2 martingale convergence theorem gives a terminal variable $M_{\infty }$ and almost sure convergence and convergence in the Lebesgue space $L^2$; moreover $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$ by conditional expectation.

The predictable sigma-algebra $\mathcal{P}$ on $\Omega \times [0,\infty )$ is the smallest sigma-algebra making every left-continuous adapted process measurable. Equivalently, it is generated bywhere the sample coordinate comes first. A previsible process is precisely a process measurable for this predictable sigma-algebra. Values at time zero matter for its definition, although they make no contribution to integration against a continuous quadratic variation starting at zero.

Specify an initial value or initial probability distribution. A strong solution of a stochastic differential equation uses a prescribed Brownian motion on a prescribed filtered probability space. It is continuous, adapted to the completed filtration generated by that Brownian motion and the initial value, and satisfiesalmost surely for every $t$, with ${\int }_0^t(|b(X_s)|+\sigma (X_s{)}^2)ds<\infty$ almost surely. The given initial variable is independent of future Brownian motion increments. Some courses allow a larger prescribed filtration in their definition of a strong solution of a stochastic differential equation; the Brownian-generated convention states explicitly what absence of extra randomness means.

A weak solution of a stochastic differential equation consists of a filtered probability space, a Brownian motion relative to its filtration, and a continuous adapted process $X$ satisfying the same integrability and integral equation. Here the space and noise are part of the unknown; the filtration may contain randomness beyond the initial variable and the driving path.

Pathwise uniqueness means that on any common filtered probability space, any two solutions with the same driving Brownian motion and the same initial value are indistinguishable. Uniqueness in law means that all weak solutions with the same initial probability distribution have the same law as path-valued random variables, even on different spaces. Pathwise uniqueness compares shared-noise paths; uniqueness in law compares path distributions. Neither uniqueness notion by itself asserts existence.

A locally defined stochastic process is a pair $(X,T)$, where $T$ is a lifetime stopping time and $X_t(\omega )$ is defined for $0\le t<T(\omega )$, that is, on a stochastic interval. For continuous local problems one uses stopping times $T_n\uparrow T$ with $T_n<T$ on $\{T>0\}$, so the process stopped at each $T_n$ has an ordinary continuous adapted process version. This is an announcing sequence for a stopping time; no value at $T$ or after $T$ is implicit in the pair.

A local solution of a stochastic differential equation on an open domain $U$ takes its values in $U$ before $T$ and, on every such stopped interval, satisfieswith the local integrability needed for the ordinary and Itô integrals. A maximal local solution of a stochastic differential equation has no extension to a strictly later lifetime that agrees with it before $T$. For locally Lipschitz continuous coefficients on $U$, take a nested compact exhaustion $K_n$ of $U$ and the successive exit times: their increasing limit is the maximal lifetime, and on $\{T<\infty \}$ the solution eventually leaves every compact subset of $U$. To obtain the announcing sequence with finite stops even when the path never exits a compact set, cap these exit times by $n$; the capped times still increase to the maximal lifetime. Thus a finite maximal lifetime means exit from the domain or explosion; it need not mean divergence to infinity. For $U=(0,\infty )$, reaching the boundary zero terminates the local solution even if an absorbing extension could be defined for a different domain.

A standard one-dimensional Brownian motion is a real-valued stochastic process $(B_t{)}_{t\ge 0}$ satisfying $B_0=0$ almost surely, having continuous paths almost surely, and having independent increments withHere $N(0,t-s)$ is the normal distribution with zero expected value and variance $t-s$. Thus the increments are also stationary increments.

If Brownian motion is defined relative to a specified filtration $({\mathcal{F}}_t)$, additionally require that it is an adapted process and that $B_t-B_s$ is independent of ${\mathcal{F}}_s$. For its natural filtration, this follows from independent increments. These are the defining properties of standard Brownian motion.

Let $\mathcal{P}$ be the predictable sigma-algebra on $\Omega \times [0,\infty )$, generated by $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,u]$ with $A\in {\mathcal{F}}_s$. A stochastic process $H$ is previsible precisely when $(\omega ,t)\mapsto H_t(\omega )$ is $\mathcal{P}$-measurable.

Every left-continuous adapted process is previsible. Indeed, on intervals $(k{2}^{-n},(k+1)2^{-n}]$, put $H_t^{(n)}=H_{k{2}^{-n}}$, and put $H_0^{(n)}=H_0$. Each approximation is a predictable process, since its coefficient is ${\mathcal{F}}_{k{2}^{-n}}$-measurable; left continuity gives $H_t^{(n)}\to H_t$ for every $t$, so measurability passes to the limit.

An adapted càdlàg process need not be previsible. For a concrete example, let $\xi$ take values $0,1$ with equal probabilities, let ${\mathcal{F}}_t$ be trivial for $t<1$ and ${\mathcal{F}}_t=\sigma (\xi )$ for $t\ge 1$, and set $H_t=\xi 1_{[1,\infty )}(t)$. This is an adapted càdlàg process, but $H_1=\xi$ is not ${\mathcal{F}}_{1-}$-measurable, where ${\mathcal{F}}_{1-}=\sigma ({\bigcup }_{s<1}{\mathcal{F}}_s)$. Sections at time $1$ of the generators of $\mathcal{P}$ all belong to ${\mathcal{F}}_{1-}$, so every predictable process has an ${\mathcal{F}}_{1-}$-measurable value there; this proves the counterexample directly.

Put $\Delta B_h=B_{t+h}-B_t$ and $R_h={\int }_t^{t+h}(H_s-H_t)d{B}_s$. The constant-in-time coefficient $H_t$ on $(t,t+h]$ is previsible, so the ratio minus $H_t$ is $R_h/\Delta B_h$. Since $|H|\le C$, continuity and the dominated convergence theorem implyThe Itô isometry gives $\mathbb{E}{R}_h^2\le h{\eta }_h$. For $\epsilon ,\delta >0$, use the Chebyshev inequality and the normal distribution of $\Delta B_h/\sqrt{h}$ to obtainFirst let $h\downarrow 0$, then $\delta \downarrow 0$; the normal distribution has no atom at zero. This provesNo independence between $R_h$ and the denominator is needed.

The first estimate suggested by the hint is valid: Brownian scaling and the Gamma function give $\mathbb{E}|\Delta B_h{|}^{-1/2}=h^{-1/4}{2}^{-1/4}\Gamma (1/4)/\sqrt{\pi }$. The second hint requires care about the placement of its expectation: with $V_h={\int }_t^{t+h}(H_s-H_t{)}^{2}ds$, the interpretation $\mathbb{E}|R_h{|}^{1/2}\le (\mathbb{E}{V}_h{)}^{1/4}$ is correct by the Jensen inequality and the Itô isometry, whereas the interpretation $\mathbb{E}|R_h{|}^{1/2}\le \mathbb{E}(V_h^{1/4})$ is false with constant $1$. For the latter interpretation, the Burkholder-Davis-Gundy inequality supplies a constant $C_{1/2}$ instead. One can see the obstruction with a Brownian motion stopped on exiting $(-1,1)$: if $\tau$ is that Brownian exit time, then $|B_{\tau }|=1$, $\mathbb{E}\tau =1$, and strict Jensen inequality gives $\mathbb{E}{\tau }^{1/4}<1=\mathbb{E}|B_{\tau }{|}^{1/2}$.

This obstruction also survives the continuity condition on $H$. On a long finite interval approximate $1_{\{s<\tau \}}$ by a continuous adapted process equal to $1$ until $\tau$ and decreasing linearly to $0$ during the next $ϵ$ units of time, and take its initial value to be $0$ by inserting a short continuous ramp before starting the exit experiment. The Itô isometry controls these approximations, and letting the interval length tend to infinity recovers the strict inequality above; hence some bounded continuous adapted example also violates the latter interpretation with constant $1$. The direct proof avoids relying on this ambiguous hint.

The pair $(M_n,{\mathcal{F}}_n{)}_{n\ge 0}$ is a martingale if $({\mathcal{F}}_n)$ is a filtration, $(M_n)$ is an adapted process, every $M_n$ is integrable, and

A semimartingale is an adapted process of the form $X=X_0+M+A$, where $M$ is a local martingale and $A$ is an adapted finite-variation process. A sequence of processes $X^n$ converges to $X$ in uniform convergence on compacts in probability, abbreviated ucp, when for every $T<\infty$ and $\delta >0$,

If two supermartingales satisfy $X_N^1\ge X_N^2$ on the event that the stopping time $N$ is finite, the displayed pasted adapted process is a supermartingale. On $\{N>n\}$ its next value is at most $X_{n+1}^1$, because a possible switch at $n+1$ only decreases it. On $\{N\le n\}$ it follows the second supermartingale. Conditioning these inequalities proves the assertion. Finite repeated pastings inherit the same property, allowing a multiplicative trading construction for upcrossings.