What Is a Stochastic Differential Equation?
The machinery is now complete. The Itô integral, its process version, the Itô formula, and
the product rule were all built for one purpose, and this page begins to use them for it.
We return to the model that opened the
construction of the integral, an ordinary
differential equation perturbed by rapidly fluctuating noise,
\[
\frac{dX_t}{dt} = b(t, X_t) + \sigma(t, X_t)\, W_t ,
\]
where \(b\) is a systematic drift, \(\sigma\) modulates the strength of the noise, and
\(W_t\) is the hoped-for noise at time \(t\). We saw there that no genuine stochastic
process satisfies the axioms such a noise should obey, and that the equation survives only
after being reshaped, with the noise entering through Brownian increments rather than
pointwise values. The reshaped object is the integral equation
\[
X_t = X_0 + \int_0^t b(s, X_s)\, ds + \int_0^t \sigma(s, X_s)\, dw_s .
\]
This page studies such equations. Following custom, we write them in differential
shorthand as
\[
dX_t = b(t, X_t)\, dt + \sigma(t, X_t)\, dw_t ,
\]
a stochastic differential equation, and we emphasize once more that the
shorthand means the integral equation and nothing else. Differentials of Brownian paths do
not exist. Integrals against them do.
What It Means to Solve One
The shape of the integral equation is familiar. Its right-hand side is exactly the form of
an Itô process
with coefficients \(u(s, \omega) = b(s, X_s(\omega))\) and
\(v(s, \omega) = \sigma(s, X_s(\omega))\). The new feature is self-reference. In the
preceding pages the coefficients \(u\) and \(v\) were given data. Here they are built from
the unknown process itself, which is what makes the identity an equation. For this
page, a solution means an explicitly exhibited Itô process \(X_t\) for which
the integral equation holds for every \(t \geq 0\), almost surely. Part of the
requirement is that the substituted coefficients \(b(s, X_s)\) and \(\sigma(s, X_s)\)
satisfy the membership conditions of that definition, so that both integrals exist.
Verifying a proposed solution is therefore a computation
with the
Itô formula.
We differentiate the candidate and check that its differential reproduces the equation.
Two questions now separate. First, how does one find solutions, and what do they look like
for the equations that arise in practice? Second, do solutions exist for a general pair
\((b, \sigma)\), in what sense are they unique, and what data pins the solution down?
This page takes up the first question through a family of worked equations, each chosen because
its closed-form solution repays a promise made earlier in the track. The second question
has a substantial general answer, resting on a fixed-point iteration and a comparison
inequality, and it is the business of a dedicated page further along the track. Until
then we solve honestly but locally, verifying each candidate against the integral
equation while uniqueness, and with it the right to say the solution, waits.
The Reading Is Part of the Equation
One subtlety deserves its own paragraph before any equation is solved. The symbol string
\(dX_t = b\, dt + \sigma\, dw_t\) does not yet determine a mathematical object, because the
stochastic term admits more than one interpretation. Reading the integral as the Itô
integral, with left endpoint evaluation, gives one equation. Reading it as the
Stratonovich integral,
with midpoint evaluation, gives a genuinely different one, and the
conversion theorem
measures the gap between the two readings as a drift correction of size
\(\tfrac{1}{2}\, \frac{\partial \sigma}{\partial x}\, \sigma\). The choice of reading is
therefore part of the modeling decision, not a notational afterthought.
Our track is committed to the Itô reading, and every equation on this page is an Itô equation unless
stated otherwise. The commitment costs nothing, since the conversion dictionary transports
statements between the two calculi. It is not vacuous either. The first equation we solve
will make the difference concrete, producing two distinct closed-form solutions of the same
symbol string, one per reading, whose long-run mean behavior can differ dramatically.
Throughout, \(w_t\) is the standard one-dimensional Brownian motion of the preceding
pages, with Brownian filtration \(\{\mathcal{F}_t\}_{t \geq 0}\) containing all null sets,
and every Itô integral denotes its continuous version. The coefficients \(b\) and
\(\sigma\) are measurable functions on \([0, \infty) \times \mathbb{R}\), specified
concretely in each example. The initial value \(X_0\) is
\(\mathcal{F}_0\)-measurable, and in the examples of this page and the next it will be a constant;
genuinely random initial values return with the general theory. Everything on this page
is one-dimensional. The
systems version of the theory, with vector states and several driving motions, reuses the
multidimensional toolkit already established and is not needed for the equations below.
Geometric Brownian Motion
The first equation models growth whose rate is noisy. A population of size
\(N_t\) grows at a relative rate that fluctuates around a constant \(r\), so the growth
rate itself is \(r + \alpha W_t\) with \(\alpha\) a constant measuring the size of the
fluctuation. Under the Itô reading, to which the previous section committed us, the model becomes
\[
dN_t = r N_t\, dt + \alpha N_t\, dw_t ,
\quad N_0 \gt 0 \text{ a given constant} .
\]
Both coefficients are proportional to the unknown. The noise does not push the population
around by fixed amounts; it perturbs the population proportionally, which is the
natural regime for growth, for chemical concentrations, and for prices; in finance this
equation is a standard model of a risky asset.
Finding the Candidate
The following calculation is a heuristic whose only job is to produce a candidate
solution. Divide the equation by \(N_t\), so that the left side becomes the relative
increment \(dN_t / N_t\). For a positive process, ordinary calculus would read
\(dN_t / N_t = d(\ln N_t)\) and integrate at once. Itô calculus charges a correction. The
second-order term of the Itô formula, applied informally to the logarithm with the
bookkeeping rule \((dw_t)^2 = dt\), gives
\[
\begin{align*}
d(\ln N_t)
&= \frac{dN_t}{N_t} - \frac{1}{2\, N_t^2}\, (dN_t)^2 \\\\
&= r\, dt + \alpha\, dw_t - \tfrac{1}{2} \alpha^2\, dt .
\end{align*}
\]
Integrating from \(0\) to \(t\) suggests
\(\ln N_t - \ln N_0 = \bigl( r - \tfrac{1}{2} \alpha^2 \bigr) t + \alpha w_t\), an
explicit candidate that the theorem below states and verifies.
Every step so far presumed things we have not justified, positivity of \(N\) above all.
None of that matters now, because the candidate is explicit and can be verified directly
against the integral equation, and the heuristic can be discarded. One preliminary identity
makes both the verification and the moment computation short. For a constant
\(c \in \mathbb{R}\), the function \(h \equiv c\) is a deterministic step function, so
its
exponential martingale
\(Z^{(c)}_t = \exp \bigl( c\, w_t - \tfrac{1}{2} c^2 t \bigr)\) falls under the
exponential martingale identity,
which yields \(\mathbb{E}\bigl[ Z^{(c)}_t \bigr] = 1\), that is,
\[
\mathbb{E}\bigl[ e^{c\, w_t} \bigr] = e^{\frac{1}{2} c^2 t}
\quad \text{for every } c \in \mathbb{R},\ t \geq 0 .
\]
Theorem: Geometric Brownian Motion
Let \(r, \alpha \in \mathbb{R}\) and let \(N_0 \gt 0\) be a constant. The process
\[
N_t = N_0 \exp \Bigl( \bigl( r - \tfrac{1}{2} \alpha^2 \bigr) t + \alpha w_t \Bigr) ,
\quad t \geq 0 ,
\]
called geometric Brownian motion, has the following properties.
(i)
\(N\) is an Itô process with strictly positive paths, and for every \(t \geq 0\),
almost surely,
\[
N_t = N_0 + \int_0^t r N_s\, ds + \int_0^t \alpha N_s\, dw_s ,
\]
so \(N\) solves \(dN_t = r N_t\, dt + \alpha N_t\, dw_t\).
(ii)
\(\mathbb{E}[N_t] = N_0\, e^{r t}\) for every \(t \geq 0\).
Proof.
Membership.
Write \(g(t, x) = N_0 \exp \bigl( (r - \tfrac{1}{2} \alpha^2) t + \alpha x \bigr)\),
so that \(N_t = g(t, w_t)\) and \(g \in C^2([0, \infty) \times \mathbb{R})\). Brownian
motion itself is an Itô process with coefficients \(u = 0\) and \(v = 1\), so the
Itô formula
applies once we check its membership condition for the integrand
\(v \cdot \frac{\partial g}{\partial x}(s, w_s) = \alpha\, g(s, w_s) = \alpha N_s\).
The process \(\alpha N_s\) is adapted, being a continuous function of
\(w_s\), and has continuous paths, so it is progressively measurable by the
continuity criterion.
For square integrability, fix \(T \gt 0\). The integrand below is nonnegative and
jointly measurable, so
Tonelli's theorem
exchanges expectation and time integral, and the exponential moment identity with
\(c = 2\alpha\) gives
\[
\begin{align*}
\mathbb{E} \int_0^T \alpha^2 N_s^2\, ds
&= \alpha^2 N_0^2 \int_0^T e^{(2r - \alpha^2) s}\,
\mathbb{E}\bigl[ e^{2 \alpha w_s} \bigr]\, ds \\\\
&= \alpha^2 N_0^2 \int_0^T e^{(2r - \alpha^2) s}\, e^{2 \alpha^2 s}\, ds \\\\
&= \alpha^2 N_0^2 \int_0^T e^{(2r + \alpha^2) s}\, ds
\lt \infty .
\end{align*}
\]
Hence \(\alpha N \in \mathcal{V}(0, T)\) for every \(T \gt 0\).
The differential.
The partial derivatives of \(g\) are proportional to \(g\) itself,
\[
\frac{\partial g}{\partial t} = \bigl( r - \tfrac{1}{2} \alpha^2 \bigr)\, g ,
\quad
\frac{\partial g}{\partial x} = \alpha\, g ,
\quad
\frac{\partial^2 g}{\partial x^2} = \alpha^2\, g .
\]
With \(u = 0\) and \(v = 1\), the Itô formula gives, for every \(t \geq 0\), almost
surely,
\[
\begin{align*}
N_t
&= g(0, 0) + \int_0^t \Bigl(
\bigl( r - \tfrac{1}{2} \alpha^2 \bigr)
+ \tfrac{1}{2} \alpha^2
\Bigr)\, g(s, w_s)\, ds
+ \int_0^t \alpha\, g(s, w_s)\, dw_s \\\\
&= N_0 + \int_0^t r N_s\, ds + \int_0^t \alpha N_s\, dw_s .
\end{align*}
\]
The drift coefficient \(r N_s\) has continuous paths, so it is pathwise integrable on
bounded intervals and progressively measurable by the same continuity criterion, and
the diffusion coefficient \(\alpha N_s\) lies in \(\mathcal{V}(0, T)\) by the
membership step. The right-hand side therefore exhibits \(N\) as an Itô process whose
coefficients are exactly \(b(t, x) = r x\) and \(\sigma(t, x) = \alpha x\) evaluated
along the solution, which is claim (i). Positivity of every path is immediate from the
exponential form.
The mean.
The solution factorizes through the exponential martingale with \(c = \alpha\),
\[
N_t
= N_0\, e^{r t} \exp \bigl( \alpha w_t - \tfrac{1}{2} \alpha^2 t \bigr)
= N_0\, e^{r t}\, Z^{(\alpha)}_t ,
\]
and taking expectations, \(\mathbb{E}\bigl[ Z^{(\alpha)}_t \bigr] = 1\) gives
\(\mathbb{E}[N_t] = N_0\, e^{r t}\), which is claim (ii).
The factorization in the last step is worth keeping in view. Geometric Brownian motion is
the deterministic growth curve \(N_0 e^{rt}\) multiplied by a mean-one martingale. The
exponential martingale of the previous page is thereby promoted from a technical density
tool to the fluctuation factor of a population. The mean of the solution is exactly the solution of
the noiseless equation \(dN/dt = rN\). Under the Itô reading, mean-zero noise in the
growth rate leaves the mean population untouched, and all the influence of \(\alpha\) is
pushed into the fluctuations around that mean. The strict positivity in (i) also settles a
debt. The heuristic divided by \(N_t\) and took its logarithm, and the verified solution
shows that for this equation those operations were legitimate after all.
The Stratonovich Reading of the Same String
Now read the same symbol string \(dN_t = r N_t\, dt + \alpha N_t\, dw_t\) in the
Stratonovich sense. The
conversion theorem
expresses the midpoint integral as the Itô integral plus half the integral of
\(\frac{\partial \sigma}{\partial x} \cdot v\). For \(\sigma(t, x) = \alpha x\), a solution has
diffusion coefficient \(v = \alpha N\), so the correction integrand is
\(\tfrac{1}{2} \alpha \cdot \alpha N_s = \tfrac{1}{2} \alpha^2 N_s\). The theorem was
proved under simplifying hypotheses on the coefficients, and we use it here as the
transport principle it expresses. At the level of equations it converts the Stratonovich
string into the Itô equation
\[
d\bar{N}_t = \bigl( r + \tfrac{1}{2} \alpha^2 \bigr) \bar{N}_t\, dt
+ \alpha \bar{N}_t\, dw_t ,
\]
the same equation with the drift shifted upward by half the squared noise strength. No new
verification is needed. The theorem above with \(r\) replaced by
\(r + \tfrac{1}{2} \alpha^2\) solves this equation in closed form, and the shift cancels
the Itô correction in the exponent,
\[
\bar{N}_t = N_0 \exp \bigl( r t + \alpha w_t \bigr) ,
\quad
\mathbb{E}\bigl[ \bar{N}_t \bigr] = N_0\, e^{ ( r + \frac{1}{2} \alpha^2 ) t } .
\]
The Stratonovich reading thus produces exactly what ordinary calculus would have guessed
at the pathwise level, the exponential of the integrated growth rate, and this is no
accident. Midpoint evaluation is the choice that preserves the classical chain rule, so
the classical formula survives.
The price appears in the statistics. The two solutions
differ in the exponent by precisely the correction term \(\tfrac{1}{2} \alpha^2 t\), and
their means separate exponentially fast, \(N_0 e^{rt}\) against
\(N_0 e^{(r + \frac{1}{2}\alpha^2) t}\). For any \(r\) with
\(-\tfrac{1}{2} \alpha^2 \lt r \lt 0\), the mean of the Itô solution decays to zero while
the mean of the Stratonovich solution grows without bound. One symbol string, two
readings, opposite long-run fates for the mean population. Which reading is the right
model depends on the physical origin of the noise, not on mathematics, and the
mathematics has now made the stakes of that choice exact.
The exponential martingale did double duty in this section, organizing both the
verification and the mean. In the next equation it is promoted again. There it will not
merely describe the solution but produce it, acting as an integrating factor that
absorbs the stochastic term entirely.
A Stochastic Integrating Factor
The second equation keeps the multiplicative noise of geometric Brownian motion but
decouples the drift from the unknown,
\[
dY_t = r\, dt + \alpha Y_t\, dw_t ,
\quad Y_0 \in \mathbb{R} \text{ a given constant} ,
\]
with \(r, \alpha \in \mathbb{R}\). The model describes a quantity that receives a steady
deterministic inflow at rate \(r\) while its current level is shaken proportionally by
noise. The trick of the previous section fails immediately. Dividing by \(Y_t\) no longer
closes the equation, because the drift term becomes \(r\, dt / Y_t\) and the unknown
survives on the right-hand side.
A different classical idea transfers instead. For the
ordinary equation \(dy/dt = r + a y\), one multiplies by the integrating factor
\(e^{-at}\), chosen so that the factor's own derivative cancels the term proportional to
the unknown. Here the term proportional to the unknown is the noise term, so the
integrating factor must itself be random, and the natural candidate is the reciprocal of
the solution of the homogeneous equation \(dZ_t = \alpha Z_t\, dw_t\). That homogeneous
solution is already on the table. It is geometric Brownian motion with \(r = 0\), which is
precisely the exponential martingale \(Z_t = Z^{(\alpha)}_t\) of the previous section. We
therefore set
\[
F_t = \frac{1}{Z_t} = \exp \Bigl( - \alpha w_t + \tfrac{1}{2} \alpha^2 t \Bigr) .
\]
Finding the Candidate
As before, a heuristic produces the candidate and is then discarded. Compute
\(d(F_t Y_t)\) informally by the product rule
\(d(FY) = F\, dY + Y\, dF + dF \cdot dY\), with the bookkeeping rules
\((dw_t)^2 = dt\) and \(dt \cdot dw_t = 0\). The Itô formula applied to \(F\) gives
\(dF_t = \alpha^2 F_t\, dt - \alpha F_t\, dw_t\), and substituting the equation for
\(dY_t\),
\[
\begin{align*}
d(F_t Y_t)
&= F_t \bigl( r\, dt + \alpha Y_t\, dw_t \bigr)
+ Y_t \bigl( \alpha^2 F_t\, dt - \alpha F_t\, dw_t \bigr)
+ \bigl( - \alpha F_t \bigr) \bigl( \alpha Y_t \bigr)\, dt \\\\
&= r F_t\, dt .
\end{align*}
\]
Every stochastic term cancels. The factor was built for exactly this purpose, and what
remains integrates pathwise, \(F_t Y_t = Y_0 + r \int_0^t F_s\, ds\). Multiplying back by
\(Z_t = F_t^{-1}\) yields the candidate, which the theorem below states in both compact
and expanded form.
Theorem: The Linear Equation with Multiplicative Noise
Let \(r, \alpha \in \mathbb{R}\), let \(Y_0 \in \mathbb{R}\) be a constant, and let
\(Z_t = \exp \bigl( \alpha w_t - \tfrac{1}{2} \alpha^2 t \bigr)\) and
\(F_t = Z_t^{-1}\). The process
\[
\begin{align*}
Y_t
&= Z_t \Bigl( Y_0 + r \int_0^t F_s\, ds \Bigr) \\\\
&= Y_0\, e^{\alpha w_t - \frac{1}{2} \alpha^2 t}
+ r \int_0^t e^{\alpha ( w_t - w_s ) - \frac{1}{2} \alpha^2 ( t - s )}\, ds
\end{align*}
\]
has the following properties.
(i)
\(Y\) is an Itô process, and for every \(t \geq 0\), almost surely,
\[
Y_t = Y_0 + r t + \int_0^t \alpha Y_s\, dw_s ,
\]
so \(Y\) solves \(dY_t = r\, dt + \alpha Y_t\, dw_t\).
(ii)
\(\mathbb{E}[Y_t] = Y_0 + r t\) for every \(t \geq 0\).
Proof.
The two factors as Itô processes.
Fix \(T \gt 0\) and take the constant step function \(h \equiv \alpha\) on
\([0, T]\). The
exponential martingale identity
states that \(\alpha Z_s \in \mathcal{V}(0, T)\) and that, for every
\(t \in [0, T]\), almost surely,
\(Z_t = 1 + \int_0^t \alpha Z_s\, dw_s\). Since \(T\) was arbitrary, \(Z\) is an Itô
process with coefficients \(u^Z = 0\) and \(v^Z = \alpha Z\). Next, the process
\(A_t = Y_0 + r \int_0^t F_s\, ds\) has continuous paths, and \(F\) is adapted with
continuous paths, so the integrand \(r F_s\) is pathwise integrable on bounded
intervals and progressively measurable by the
continuity criterion.
Hence \(A\) is an Itô process with \(u^A = r F\) and \(v^A = 0\), and \(Y = Z A\).
Membership.
The
stochastic product rule
requires \(A_s\, v^Z + Z_s\, v^A = \alpha Z_s A_s = \alpha Y_s\) to lie in
\(\mathcal{V}(0, T)\) for every \(T \gt 0\), which is exactly the membership our
working notion of solution demands of the diffusion coefficient \(\alpha Y_s\).
Adaptedness and path continuity of \(Y = Z A\) give progressive measurability by the
continuity criterion, so square integrability is the only open point. By (W1) of the
definition of Brownian motion,
the increment \(w_s - w_u\) for \(u \leq s\) is centered Gaussian with variance
\(s - u\), hence has the law of \(w_{s - u}\), and the exponential moment identity of
the previous section applies to it. In particular, writing
\(Z_s F_u = e^{\alpha ( w_s - w_u ) - \frac{1}{2} \alpha^2 ( s - u )}\),
\[
\begin{align*}
\mathbb{E}\bigl[ Z_s^2 \bigr]
&= \mathbb{E}\bigl[ e^{2 \alpha w_s} \bigr]\, e^{- \alpha^2 s}
= e^{\alpha^2 s} , \\\\
\mathbb{E}\bigl[ ( Z_s F_u )^2 \bigr]
&= \mathbb{E}\bigl[ e^{2 \alpha ( w_s - w_u )} \bigr]\, e^{- \alpha^2 ( s - u )}
= e^{\alpha^2 ( s - u )}
\leq e^{\alpha^2 s} .
\end{align*}
\]
For the integral term, expanding the square of a nonnegative integral as a double
integral and applying \(2ab \leq a^2 + b^2\) gives the elementary bound
\(\bigl( \int_0^s g\, du \bigr)^2 \leq s \int_0^s g^2\, du\), and
Tonelli's theorem
moves the expectation inside the time integrals. Combining with
\((a + b)^2 \leq 2 a^2 + 2 b^2\),
\[
\begin{align*}
\mathbb{E}\bigl[ Y_s^2 \bigr]
&\leq 2 Y_0^2\, \mathbb{E}\bigl[ Z_s^2 \bigr]
+ 2 r^2\, \mathbb{E} \Bigl[ \Bigl( \int_0^s Z_s F_u\, du \Bigr)^2 \Bigr] \\\\
&\leq 2 Y_0^2\, e^{\alpha^2 s}
+ 2 r^2 s \int_0^s \mathbb{E}\bigl[ ( Z_s F_u )^2 \bigr]\, du \\\\
&\leq \bigl( 2 Y_0^2 + 2 r^2 s^2 \bigr)\, e^{\alpha^2 s} ,
\end{align*}
\]
which is bounded on \([0, T]\), so
\(\mathbb{E} \int_0^T \alpha^2 Y_s^2\, ds \lt \infty\) and
\(\alpha Y \in \mathcal{V}(0, T)\) for every \(T \gt 0\).
The product rule.
All hypotheses of the product rule are now verified: \(Z\) and \(A\) are Itô
processes, and the combined coefficient \(\alpha Y\) lies in \(\mathcal{V}(0, T)\)
for every \(T\). Since \(Z_s F_s = 1\), the rule gives, for every \(t \geq 0\),
almost surely,
\[
\begin{align*}
Y_t
&= Z_0 A_0
+ \int_0^t \bigl( Z_s\, u^A + A_s\, u^Z + v^Z v^A \bigr)\, ds
+ \int_0^t \bigl( Z_s\, v^A + A_s\, v^Z \bigr)\, dw_s \\\\
&= Y_0 + \int_0^t r\, Z_s F_s\, ds + \int_0^t \alpha\, Z_s A_s\, dw_s \\\\
&= Y_0 + r t + \int_0^t \alpha Y_s\, dw_s .
\end{align*}
\]
The drift coefficient of the equation is the constant \(r\), trivially pathwise
integrable, and the diffusion coefficient \(\alpha Y_s\) has just been placed in
\(\mathcal{V}(0, T)\), so \(Y\) is an Itô process of exactly the equation's form,
which is claim (i).
The mean.
The second moment bound makes \(Y_t\) integrable, and the summands of the expanded
form are handled separately. The first has mean
\(Y_0\, \mathbb{E}\bigl[ Z_t \bigr] = Y_0\), by the exponential martingale
identity invoked above. For the second, the integrand
\(Z_t F_s\) is positive, so Tonelli exchanges expectation and integral, and the
increment identity with \(c = \alpha\) gives
\(\mathbb{E}\bigl[ Z_t F_s \bigr]
= e^{\frac{1}{2} \alpha^2 ( t - s )} e^{- \frac{1}{2} \alpha^2 ( t - s )} = 1\).
Hence
\[
\mathbb{E}[Y_t]
= Y_0 + r \int_0^t \mathbb{E}\bigl[ Z_t F_s \bigr]\, ds
= Y_0 + r t ,
\]
which is claim (ii).
The expanded form of the solution reads like a ledger. The initial capital \(Y_0\) is
carried from time \(0\) to time \(t\) by the homogeneous growth factor
\(Z_t\), and each infinitesimal deposit \(r\, ds\) made at time \(s\) is carried from
\(s\) to \(t\) by the ratio \(Z_t / Z_s = e^{\alpha ( w_t - w_s ) - \frac{1}{2} \alpha^2 ( t - s )}\),
which depends only on the noise accumulated after the deposit. The solution is a
superposition of contributions, each propagated forward by the noise it actually
experienced. The mean in (ii) repeats the lesson of the previous section. The noiseless
skeleton \(dy/dt = r\) has solution \(Y_0 + rt\), and under the Itô reading the
proportional noise leaves this mean exactly in place. The same computation with a
time-dependent noise coefficient \(c(t)\) in place of \(\alpha\) works verbatim, with the
integrating factor \(\exp \bigl( - \int_0^t c\, dw_s + \tfrac{1}{2} \int_0^t c^2\, ds \bigr)\),
and we record the method without pursuing the general statement.
Between this section and the last, two placements of the unknown have been solved.
Noise proportional to the state with drift proportional to the state gave geometric
growth, and noise proportional to the state with constant drift gave the ledger above. The
remaining basic configuration reverses the roles. When the drift is proportional
to the state while the noise enters additively, with constant strength, the integrating
factor loses its randomness and the classical \(e^{-\mu t}\) returns intact. That
configuration is the Ornstein-Uhlenbeck family, and it has earned
a page of its own, where the closed form opens into
Gaussian process structure and the stochastic engine of diffusion models.