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For asset dynamics $dS=\mathrm{diag}(S)(\mu dt+\sigma dW)$, let $\lambda ={\sigma }^{-1}\mu$ and $Z=\mathcal{E}(-\int {\lambda }^{T}dW)$. The Itô product rule cancels the drift of each $Z{S}^i$, leaving the displayed stochastic integral. When $\mu$ is bounded and $\sigma {\sigma }^T\ge \epsilon I$, $\lambda$ is bounded and the Novikov condition makes $Z$ a true martingale on every finite horizon. If $\sigma$ is also bounded, each $Z{S}^i/S_0^i$ satisfies the Novikov condition as a stochastic exponential with bounded integrand ${\sigma }_i-\lambda$, so it too is a true martingale.

The conditional expectation of $\sin B_T$ at time $t$ is $e^{-(T-t)/2}\sin B_t$, by independent Brownian increment. Itô's formula cancels its time derivative against its second spatial derivative and leaves the displayed stochastic integral. The integrand is bounded, so Itô isometry verifies its square integrability.

For independent Brownian motions $W,B$, the strong law for Brownian motion makes ${\int }_0^{\infty }{e}^{2W_s-s}ds$ finite almost surely, so bracket localization gives an almost sure limit of the stochastic integral. The arctangent transform of a two-noise affine diffusion and endpoint convergence of a bounded angle diffusion identify its distribution function as $1/2+\arctan (x)/\pi$. Conditional Gaussianity rules out atoms.

If the entire integrand path is independent of the integrating Brownian motion, conditioning on that path gives a deterministic square-integrable integrand. The Itô isometry and Gaussian approximation by simple integrands show that its stochastic integral is conditionally centered normal with the stated variance. A strictly positive finite conditional variance implies that the unconditional distribution has no atoms.

Planar Brownian motion started away from zero almost surely avoids zero. On every finite time interval its continuous path therefore has positive minimum modulus, making the logarithmic stochastic integral well-defined. Since complex quadratic covariation satisfies $d[Z,Z]=0$, analytic Itô calculus gives the displayed exponential identity without a drift correction. Its real part is the change in logarithmic radius and its imaginary part is the continuous accumulated argument, including every winding. This is a pathwise logarithmic lift, not a single-valued logarithm on the entire punctured plane.

A self-financing investor with consumption satisfies $X=H\cdot P$ and $dX=H\cdot dP-cdt$. The Itô product rule and $d[Y,X]=H\cdot d[Y,P]$ give the displayed equation. For a positive local martingale deflator $Y$, the first term is a stochastic integral against the vector of deflated asset-price local martingales, not their current levels.

An elementary predictable process may use finitely many ordered stopping times ${\tau }_j$, with bounded ${\mathcal{F}}_{{\tau }_j}$-measurable coefficients $h_j$ on $({\tau }_j,{\tau }_{j+1}]$. Its elementary stochastic integral is ${\sum }_j{h}_j(X_{t\wedge {\tau }_{j+1}}-X_{t\wedge {\tau }_j})$. The stopped indicators are left-continuous and adapted, so the process is predictable; common refinement shows the sum is independent of its representation. Deterministic endpoints recover the usual simple predictable process. These integrands give the precise good-integrator test in the Bichteler-Dellacherie theorem.

Suppose a finite-horizon market has a strictly positive numéraire $B$ and an equivalent measure under which all prices $S^i/B$ are local martingales. The discounted wealth of a self-financing strategy is a stochastic integral against those prices. If it is bounded below by a deterministic constant, that wealth is a supermartingale: add the lower-bound constant to obtain a nonnegative local martingale, then apply conditional Fatou to a localizing sequence. A zero-cost admissible portfolio therefore has expected discounted terminal wealth at most zero. Nonnegative terminal wealth with positive probability of a strict gain would have strictly positive expectation under the equivalent measure, a contradiction. Thus there is no admissible arbitrage. True martingales are not needed for this implication.

The solution ofisIt has variance $(1-e^{-2\lambda t})/(2\lambda )$. Starting from $N(0,(2\lambda {)}^{-1})$ makes it stationary with covariance $e^{-\lambda |t-s|}/(2\lambda )$.

A process is locally bounded if there are stopping times ${\tau }_n\uparrow \infty$ and finite deterministic constants $K_n$ bounding it on the stopped intervals. A locally bounded predictable process is integrable against a semimartingale. Continuous adapted processes are locally bounded by stopping their magnitudes and time.

// Target: probability-and-statistics.bigb

A martingale representation theorem states conditions under which every martingale in a filtration can be represented as a stochastic integral with respect to one or more fundamental martingales.

For a fixed initial value $X_0=x_0$, the Ornstein-Uhlenbeck process has the normal distributionThe two parameters of $\mathcal{N}(m,v)$ are the expected value and variance, not the standard deviation. At $t=0$ the variance is zero and the value is fixed. More generally, its transition kernel over any interval of length $h>0$ has conditional expected value $\mu +(x-\mu )e^{-\theta h}$ and conditional variance ${\sigma }^2(1-e^{-2\theta h})/(2\theta )$ when the current value is $x$. The coefficients depend on elapsed time; the initial probability distribution determines whether the resulting stochastic process is stationary.^[ref]

In the stipulated example $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$, $x_0=6$, the expected value at time two is $2+4e^{-2}\approx 2.5413$ and the variance is $1-e^{-4}\approx 0.9817$. The initial distance from two has fallen by a factor $e^{-2}$, but the standard deviation has grown to about $0.9908$. Approaching the reference level in expected value is compatible with growing uncertainty from a perfectly known initial value.

The solid probability density functions use times $0.25$, $1$ and $2$; the dashed curve is the limiting normal distribution with expected value two and variance one. The curves are exact model predictions. There is no ordinary density for the fixed value at time zero.

A short derivation uses the integrating factor $e^{\theta t}$. The Itô product rule gives $d[e^{\theta t}(X_t-\mu )]=\sigma e^{\theta t}d{W}_t$, henceThe stochastic integral has zero expected value. The Itô isometry gives its variance as ${\sigma }^2{\int }_0^t{e}^{-2\theta (t-u)}du$. Its deterministic integrand makes it a Gaussian random variable, yielding the displayed normal distribution. Shifting the integration interval gives the same transition kernel from any later time.

Let ${\Delta }_t$ be the number of stock units and ${\beta }_t$ the number of bank-account units, with predictable holdings satisfying the integrability conditions for the price integrals. The portfolio value is $V_t={\Delta }_t{S}_t+{\beta }_t{B}_t$. It is self-financing ifThus changes in value come only from gains on the assets held. Rebalancing transfers money between the stock and bank account; it does not inject or withdraw cash. The definition is an identity of stochastic integrals and automatically includes the bookkeeping needed when holdings themselves vary randomly.

For deterministic square-integrable volatility, apply Itô's formula to the exponential to obtainThe stochastic integral is centered Gaussian with variance ${\int }_0^t\sigma (u{)}^{2}du$.

Under the risk-neutral measure the drift is $r$. Conditional on $S_t=s$, put $\tau =T-t$ and $v={\int }_t^T\sigma (u{)}^{2}du$. ThenIntegrating the discounted call option against this normal distribution yieldsThus the total remaining variance replaces ${\sigma }^2\tau$ throughout the ordinary formula; the corresponding effective constant volatility is $\sqrt{v/\tau }$. If $v=0$, use the limiting deterministic value $(s-K{e}^{-r\tau }{)}_{+}$.

Write $\tau =T-t$ and $a=e^{-\alpha \tau }$. The stochastic integral in b has a deterministic integrand, so it is a Gaussian random variable independent of the information available at $t$. Its variance, by the Itô isometry, isThus $x_T$ conditionally has a normal distribution with expected value $m=a{x}_t+(1-a)\tilde{\mu }$ and variance $v$. The moment-generating function of a normal distribution gives ${\mathbb{E}}_t[e^{x_T}]=e^{m+v/2}$, so the conditional mean of a log-mean-reverting price isThe correction $v/2$ is essential: exponentiating the conditional mean of the logarithm alone would omit the dispersion of the lognormal distribution.

Under the specified risk-neutral measure, the Itô product rule and $d(B_t^{-1})=-r_t{B}_t^{-1}dt$ giveThus discounted traded zero-coupon bonds have zero drift. Assume the usual integrability and admissibility conditions that make discounted replicating values true martingales, not merely local martingales. If $X$ is an attainable financial payoff with ${\mathbb{E}}_Q|B_T^{-1}X|<\infty$, its self-financing portfolio value $V_t$ has terminal value $V_T=X$ and satisfiesMultiplying by $B_t$ yieldsThis proves the requested conditional pricing formula. In a Brownian filtration, the Martingale representation theorem realizes the discounted conditional expectation as a stochastic integral. If an available bond has nonzero Itô diffusion exposure, its holdings can match that integral, with the remaining value in the bank account, giving replication under the usual square-integrability conditions.

These attainability or market completeness hypotheses matter for a completely arbitrary contingent claim: existence of a risk-neutral measure alone need not imply a unique price for every unspanned financial payoff, and a zero-drift local martingale alone need not satisfy the displayed terminal expectation identity. In the general path-dependent Heath-Jarrow-Morton model, write the value as $V_t$; notation $V(t,r_t)$ is justified only when the current short rate is a sufficient state for the financial payoff and dynamics.

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 $H\cdot M$.

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.

Here is a finite-horizon Girsanov theorem statement with the sign fixed explicitly. Let $B$ be a Brownian motion under $P$, and let $\theta$ be predictable with ${\int }_0^T{\theta }_s^{2}ds<\infty$ almost surely. Ifis a true martingale with $\mathbb{E}{L}_T=1$, then the measure $Q$ defined by $dQ=L_{T}dP$ makes $B_t-{\int }_0^t{\theta }_{s}ds$ a Brownian motion up to $T$. The sufficient Novikov condition is $\mathbb{E}\exp (\frac{1}{2}{\int }_0^T{\theta }_s^{2}ds)<\infty$. Merely being a local martingale is not a sufficient density condition.

For a deterministic absolutely continuous function $h$ with $h(0)=0$ and derivative $g\in L^2[0,T]$, take $\theta =g$. Novikov's condition holds because its exponential is deterministic and finite. Under $Q$, the canonical process has the law of a Brownian motion plus $h$. Thus for every bounded measurable path functional $F$,This derives the Cameron-Martin theorem density formula from the Girsanov theorem, rather than assuming the two measures have the same drift convention. Replacing $h$ by $-h$ changes the sign of the stochastic integral.

The question's path space is the whole half-line, not just one finite horizon. When $g\in L^2(0,\infty )$, the density martingale satisfiesIt has a uniformly integrable limit $L_{\infty }=\exp ({\int }_0^{\infty }{g}_{s}d{B}_s-\Vert g{\Vert }_2^2/2)$ of mean one, strictly positive almost surely. Integrals here converge in $L^2$ and almost surely. The finite-time identities then identify $L_{\infty }$ as the density on the full path sigma-algebra, since finite-time cylinder events generate it. This is Cameron-Martin shifts on infinite Wiener path space. Finite-horizon absolute continuity of measures alone does not imply infinite-horizon absolute continuity of measures.

For continuous semimartingales $U,V$, the semimartingale integration by parts formula isIndeed each partition increment satisfiesSumming telescopes the left side. The first two sums converge to the stochastic integrals by left-endpoint approximation, and the last sum converges to the quadratic covariation by polarization of the quadratic-increment formula. All limits are in uniform convergence on compacts in probability. Hence the equality holds at every time outside one null set, using continuity. In differential notation it is the Itô product rule $d(UV)=UdV+VdU+d[U,V]$.

For a vector of continuous semimartingales $X=(X^1,\dots ,X^d)$ and $F\in C^{1,2}$, the multidimensional Itô formula isOne route from the product formula is to apply it repeatedly to powers and products of the coordinates, obtaining this expression for polynomials; polarization identifies the mixed second derivatives. Time is another coordinate of finite variation, so it has no quadratic-covariation contribution. Localize to compact ranges, approximate a $C^{1,2}$ function and its indicated derivatives by smooth polynomial approximations, and pass to the limit using stochastic-integral continuity and the variation bounds on the finite-variation terms. Equivalently, second-order Taylor expansion along partitions gives the same limit, with integration by parts accounting for every quadratic term.