Chapter 07

Hard

Stochastic Differential Equations

00 · Symbol Glossary

$dX_t = \mu(X_t,t)\,dt + \sigma(X_t,t)\,dW_t$a stochastic differential equation

Shorthand for an integral equation (Section 01). $\mu$ and $\sigma$ may depend on $X_t$ itself, unlike Chapter 06's examples where they were fixed functions of time alone.

$X_0$initial condition

A fixed (or independent random) starting value. As in ordinary differential equations, an SDE plus an initial condition is what pins down a single solution process.

$\mu(x,t),\,\sigma(x,t)$drift and diffusion coefficients

Functions of state and time, not fixed numbers. Lipschitz continuity in $x$ (Section 02) is the condition that keeps the equation from blowing up or having multiple solutions.

$X_t^{\text{strong}}$ vs. law of $X_t$strong vs. weak solution

A strong solution is a specific process adapted to the given $W_t$ 's filtration; a weak solution only needs some Brownian motion (possibly a different one) making the equation hold. Section 04 makes this precise.

01 · An SDE Is an Integral Equation in Disguise

The differential notation $dX_t=\mu(X_t,t)\,dt+\sigma(X_t,t)\,dW_t$ is convenient but not literally meaningful — $dW_t$ is not a well-defined increment the way $dx$ is in ordinary calculus (Chapter 03). What it actually means is the integral equation

X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,dW_s

with the second integral understood in the Itô sense of Chapter 05. A solution is a process $X_t$ , adapted to the same filtration as $W_t$ , satisfying this equation for every $t$ (almost surely). Solving an SDE means finding $X_t$ explicitly as a function of $t$ and the path of $W$ — not just verifying that some candidate satisfies the equation, though verification is often the easier direction, as the worked examples below show.

Every ordinary differential equation is the special case $\sigma\equiv 0$ : the integral equation collapses to $X_t=X_0+\int_0^t\mu(X_s,s)\,ds$ , an ordinary Riemann integral with no probabilistic content. Everything new in this chapter comes from the second term, and specifically from the fact that $\int_0^t\sigma(X_s,s)\,dW_s$ cannot be evaluated by antiderivatives the way the first term can.

Bridging from Calc DE1

An ordinary first-order differential equation $dy/dx=f(x,y)$ (see the Calculus chapter Differential Equations — First Order) is the $\sigma\equiv 0$ special case: no noise term, and the integrating-factor technique from that chapter solves the linear case $dy/dx+p(x)y=q(x)$ . Section 03 below reuses exactly that technique, adapted to handle the extra $\sigma\,dW_t$ term.

02 · Existence and Uniqueness (Informally)

Not every choice of $\mu,\sigma$ has a well-behaved solution. The standard sufficient condition mirrors the ODE case.

Lipschitz and growth conditions

If $\mu(x,t)$ and $\sigma(x,t)$ are Lipschitz in $x$ (there exists $K$ with $|\mu(x,t)-\mu(y,t)|\leq K|x-y|$ , same for $\sigma$ , uniformly in $t$ ) and satisfy a linear growth bound ( $|\mu(x,t)|\leq K(1+|x|)$ , same for $\sigma$ ), then the SDE has a unique strong solution, and $\mathbb{E}\!\int_0^T X_t^2\,dt<\infty$ . This is the stochastic analogue of the Picard–Lindelöf condition for ODEs: Lipschitz coefficients prevent solutions from separating too fast or exploding in finite time. Every SDE in this subject — geometric Brownian motion (Chapter 08), the Ornstein–Uhlenbeck process (Chapter 09) — satisfies these conditions comfortably; the condition mainly matters for ruling out pathological coefficients like $\sigma(x)=x^2$ growing too fast, which can produce solutions that blow up in finite time.

Uniqueness matters as much as existence for practical work: without it, a numerical scheme could converge to a solution that is not the one the model intended, with no warning sign in the output.

03 · Solving Strategies

There is no general closed-form solution method — most SDEs used in practice are solved by matching them to one of a small number of recognizable patterns, the same way a first course in ODEs teaches separable, linear, and exact equations as named categories rather than one universal algorithm. A useful first classification: additive noise ( $\sigma$ independent of $X_t$ , as in the Ornstein–Uhlenbeck equation) usually falls to the integrating factor directly; multiplicative noise ( $\sigma$ proportional to $X_t$ , as in geometric Brownian motion) usually falls to a log or power transform first, after which what remains is additive.

Step-by-step — Common solving patterns

Guess a transform, apply Itô's Lemma: if

\mu,\sigma

are proportional to

X_t

(as in GBM), try

f(X_t)=\log X_t

and use Chapter 06 to turn the equation into one with constant coefficients.

Integrating factor: for linear SDEs

dX_t=(a(t)X_t+b(t))\,dt+\sigma(t)\,dW_t

, multiply by

e^{-\int a(t)\,dt}

— the direct stochastic generalization of the ODE integrating-factor trick — to collapse the drift term into a total differential (Chapter 09 works this out fully for Ornstein–Uhlenbeck).

Match to a known SDE: GBM and OU cover a large share of quantitative-finance use cases; recognizing the form often beats deriving from scratch.

Fall back to numerics: when no closed form exists (most SDEs with nontrivial

\mu,\sigma

), simulate instead — see the pointer below.

Example — Verifying a candidate solution

Claim: $X_t=X_0e^{-\lambda t}+\sigma\int_0^t e^{-\lambda(t-s)}\,dW_s$ solves $dX_t=-\lambda X_t\,dt+\sigma\,dW_t$ . Differentiate the claimed $X_t$ using the product rule extended to stochastic integrals (justified rigorously via Itô's Lemma applied to $e^{\lambda t}X_t$ ): $dX_t = -\lambda X_0 e^{-\lambda t}\,dt - \lambda\sigma\!\int_0^t e^{-\lambda(t-s)}\,dW_s\,dt + \sigma\,dW_t = -\lambda X_t\,dt+\sigma\,dW_t$ . It checks out — and this is precisely the Ornstein–Uhlenbeck solution that Chapter 09 derives from scratch using the integrating factor.

Example — Building the solution, not just checking it

For $dX_t = -\lambda X_t\,dt+\sigma\,dW_t$ , multiply by the integrating factor $e^{\lambda t}$ : $e^{\lambda t}\,dX_t+\lambda e^{\lambda t}X_t\,dt = \sigma e^{\lambda t}\,dW_t$ , and the left side is exactly $d(e^{\lambda t}X_t)$ by the product rule for Itô processes (no second-order correction appears here, since $e^{\lambda t}$ is deterministic — its own quadratic variation is zero). Integrating both sides from $0$ to $t$ and solving for $X_t$ reproduces the candidate solution verified above, this time derived rather than guessed. This is the pattern behind Step 2 in general.

04 · Strong vs. Weak Solutions

Definition — Strong and Weak Solutions

A strong solution is a process $X_t$ , adapted to the filtration generated by a given Brownian motion $W_t$ , satisfying the integral equation of Section 01 for that specific $W_t$ . A weak solution is a pair $(X_t, \tilde W_t)$ — possibly on a different probability space — where $\tilde W_t$ is some Brownian motion making the equation hold; only the law of $X_t$ is pinned down, not a pathwise map from a fixed $W_t$ .

Plain language: strong existence means "plug in this specific noise path and get this specific answer" — a genuine function of $\omega$ . Weak existence only guarantees a process with the right distributional behavior exists, built possibly from different randomness than the one you started with. The Lipschitz conditions of Section 02 give strong solutions; some SDEs with discontinuous or badly-behaved coefficients have a weak solution but no strong one. For every SDE in this subject, the distinction does not bite — Lipschitz coefficients throughout — but it is the reason the general existence theory has two separate branches, and it resurfaces later when Girsanov's Theorem (Chapter 10) changes the measure under which $W_t$ is a Brownian motion: the new process is a weak solution of the same-looking SDE under the new measure.

❌ Simulating an SDE with a first-order Taylor step

The natural instinct is to discretize $dX_t\approx \mu(X_t,t)\Delta t + \sigma(X_t,t)\Delta W_t$ and step forward — the Euler–Maruyama scheme — and assume it converges the way Euler's method does for ODEs.

Why it breaks: it does converge, but only at strong order $0.5$ in $\Delta t$ (compare ODE Euler's order $1$ ), because the discretization drops a $\sigma\sigma'$ correction term related to the same $(dW_t)^2=dt$ effect that produced Itô's Lemma. Using an ODE-sized step size for an SDE systematically underestimates the error.

Consequence: naive step sizes carried over from deterministic simulation understate simulation error for SDEs; a finer discretization (or the higher-order Milstein scheme, which restores some of the dropped term) is often needed for the same accuracy. Full treatment of these schemes belongs to the Numerical Methods subject's ODE/SDE chapters — this chapter only flags that the analogy with ODE numerics is not exact.

05 · Exercises

EXERCISE 7.1

Write the SDE as its integral-equation form and identify $\mu,\sigma$ .

$X_t=X_0+\int_0^t r X_s\,ds+\int_0^t \sigma X_s\,dW_s$ , so $\mu(x,t)=rx$ , $\sigma(x,t)=\sigma x$ . Both are Lipschitz in $x$ with constant $\max(|r|,\sigma)$ and satisfy linear growth, so Section 02's conditions hold and a unique strong solution exists — the GBM equation of Chapter 08.

Write $dX_t=rX_t\,dt+\sigma X_t\,dW_t$ in integral form and verify the Lipschitz/growth conditions of Section 02 hold.

EXERCISE 7.2

Multiply both sides of $dX_t = (a-bX_t)\,dt+\sigma\,dW_t$ by $e^{bt}$ and check it collapses to $d(e^{bt}X_t)$ .

$d(e^{bt}X_t) = be^{bt}X_t\,dt+e^{bt}\,dX_t = be^{bt}X_t\,dt+e^{bt}(a-bX_t)\,dt+e^{bt}\sigma\,dW_t = ae^{bt}\,dt+\sigma e^{bt}\,dW_t$ . The $bX_t$ terms cancel — that cancellation is exactly what "integrating factor" means: the factor $e^{bt}$ is chosen so the drift's $X_t$ -dependence disappears.

For $dX_t=(a-bX_t)\,dt+\sigma\,dW_t$ , verify that multiplying by $e^{bt}$ turns the left side into a total differential $d(e^{bt}X_t)$ .

EXERCISE 7.3

Compare the definitions in Section 04: does a strong solution require a specific, given $W_t$ ?

No — every strong solution is automatically a weak solution (take $\tilde W_t = W_t$ , the given one), but not conversely: a weak solution only asserts existence of some Brownian motion making the equation hold, possibly not the one you started with. So strong existence is the stronger, more restrictive notion, despite what the names might suggest at first glance.

Is every strong solution automatically a weak solution? Is the converse true? Justify from the definitions in Section 04.

EXERCISE 7.4

Classify the noise as additive or multiplicative per Section 03, then pick the matching strategy.

$\sigma(x)=\sigma$ is constant, independent of $x$ : additive noise. The integrating-factor route of Exercise 7.2/Section 03 applies directly; no log or power transform is needed first, since there is no $X_t$ -dependence in $\sigma$ to remove.

For $dX_t = (a-bX_t)\,dt+\sigma\,dW_t$ , is the noise additive or multiplicative in the sense of Section 03? Which solving strategy does that suggest first?

06 · Chapter Summary

Concept	Meaning
SDE meaning	shorthand for an Itô integral equation
Lipschitz + growth	sufficient conditions for a unique strong solution
Guess-and-verify	apply Itô's Lemma to a candidate transform, check it matches
Integrating factor	cancels state-dependence in linear-drift SDEs
Strong solution	pathwise function of a given $W_t$
Weak solution	correct law, possibly different underlying noise
Euler–Maruyama	naive discretization; only strong order $0.5$ — see Numerical Methods

Next: Chapter 08 — Geometric Brownian Motion, the first SDE solved completely using the transform-and-Itô's-Lemma strategy of Section 03.