Chapter 4 — SDE Definition and Existence

Guiding question

Given a drift function $b(t,x)$ and a diffusion function $\sigma(t,x)$, does the stochastic differential equation $dX_t = b(t,X_t)\,dt + \sigma(t,X_t)\,dW_t$ have a solution? Is it unique? What conditions on $b$ and $\sigma$ are needed?

An SDE is a prescription for how a process evolves in time: a deterministic drift plus a random diffusion. The existence question asks whether such a prescription is self-consistent — whether a process with the prescribed dynamics can actually be constructed. Uniqueness asks whether the dynamics determine the process completely.

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4.1 The SDE as an integral equation

The shorthand $dX_t = b(t,X_t)\,dt + \sigma(t,X_t)\,dW_t$ is notation for the integral equation:

$$X_t = X_0 + \int_0^t b(s,X_s)\,ds + \int_0^t \sigma(s,X_s)\,dW_s, \qquad 0\le t\le T.$$

The first integral is a Lebesgue integral (pathwise, since $b$ is a drift), and the second is an Itô integral. Finding $X$ means finding a continuous adapted process that satisfies this identity for all $t$.

The SDE is implicit

The unknown process $X$ appears inside the integrals on the right side. This makes the SDE an implicit equation. The existence theorem constructs $X$ as the fixed point of a contraction map in the $L^2$ sense.

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4.2 Strong and weak solutions

There are two notions of solution.

A strong solution is a process $X$ defined on the given probability space $(\Omega,\mathcal F,\mathbb P)$, adapted to the filtration generated by $W$, that satisfies the integral equation pathwise.

A weak solution is a triple $(\Omega,\mathcal F,\mathbb P, W, X)$ — possibly a new probability space with a new Brownian motion — on which the integral equation holds.

Every strong solution is a weak solution. The converse is false in general.

Analogy with ODEs

For an ODE $\dot x = f(x)$, a strong solution constructs the trajectory from the given vector field $f$ and starting point. A weak solution allows changing the underlying space (redefining what "random" means) as long as the marginal dynamics are preserved.

In finance, the distinction matters for hedging: a strong solution gives a trading strategy that replicates under the original measure, while a weak solution only gives pricing without a hedging prescription.

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4.3 Lipschitz conditions

The standard hypothesis for existence and uniqueness of strong solutions is:

Standard conditions (Lipschitz and linear growth)

Suppose $b,\sigma:[0,T]\times\mathbb R\to\mathbb R$ satisfy:

1. Lipschitz in space: there exists $K>0$ such that for all $t\in[0,T]$ and $x,y\in\mathbb R$,

$$|b(t,x)-b(t,y)| + |\sigma(t,x)-\sigma(t,y)| \le K|x-y|.$$

2. Linear growth: there exists $C>0$ such that

$$|b(t,x)|^2 + |\sigma(t,x)|^2 \le C(1+x^2).$$

Under these conditions and any $\mathcal F_0$-measurable $X_0$ with $\mathbb E[X_0^2]<\infty$, the SDE has a unique strong solution satisfying $\mathbb E[\sup_{t\le T}X_t^2]<\infty$.

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4.4 Picard iteration: constructing the solution

The proof uses the same idea as the classical Picard–Lindelöf theorem for ODEs.

Define the sequence $X^{(0)}_t = X_0$ (constant), and

$$X^{(n+1)}_t = X_0 + \int_0^t b(s,X^{(n)}_s)\,ds + \int_0^t \sigma(s,X^{(n)}_s)\,dW_s.$$

Each $X^{(n+1)}$ is well-defined because $X^{(n)}$ is adapted and square-integrable. We show the sequence converges.

Convergence of Picard iterates

Let $Z^{(n)}_t = X^{(n+1)}_t - X^{(n)}_t$ and $\varepsilon_n(t) = \mathbb E[\sup_{s\le t}|Z^{(n)}_s|^2]$.

By the Lipschitz condition and the Burkholder–Davis–Gundy inequality (Section 2.9):

$$\varepsilon_{n+1}(t) \le C\int_0^t \varepsilon_n(s)\,ds.$$

Starting from $\varepsilon_0(t) = \mathbb E[\sup_{s\le t}|X^{(1)}_s - X_0|^2] \le C_0 t$ (from linear growth), iterating:

$$\varepsilon_n(t) \le \frac{C^n t^n}{n!}C_0.$$

Therefore $\sum_n\varepsilon_n(T)^{1/2} < \infty$, and the Borel–Cantelli argument gives almost sure and $L^2$ convergence.

Picard iteration converging to OU solution dX = −X dt + dW

Iterations: 3Path seed: 0

Picard iteration builds X⁽ⁿ⁺¹⁾ by plugging X⁽ⁿ⁾ into the drift integral. Starting from the constant x₀, each iterate adds one more term of the exponential expansion of e^−t. Under Lipschitz conditions the iterates converge in L² — this is the constructive proof of existence.

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4.5 Uniqueness via Gronwall's inequality

Pathwise uniqueness

Under the Lipschitz condition, if $X$ and $Y$ both solve the SDE with the same initial condition and the same Brownian motion, then $X_t = Y_t$ almost surely for all $t$.

Proof

Let $Z_t = X_t - Y_t$. Then

$$Z_t = \int_0^t [b(s,X_s)-b(s,Y_s)]\,ds + \int_0^t [\sigma(s,X_s)-\sigma(s,Y_s)]\,dW_s.$$

By Lipschitz and the Itô isometry/BDG inequality:

$$\mathbb E\!\left[\sup_{s\le t}Z_s^2\right] \le C\int_0^t \mathbb E[Z_s^2]\,ds \le C\int_0^t \mathbb E\!\left[\sup_{r\le s}Z_r^2\right]ds.$$

Let $u(t) = \mathbb E[\sup_{s\le t}Z_s^2]$. Then $u(t)\le C\int_0^t u(s)\,ds$ with $u(0)=0$.

Gronwall's inequality: If $u(t)\le A + C\int_0^t u(s)\,ds$ with $A=0$, then $u(t)\le 0$. Therefore $u\equiv 0$, so $Z_t = 0$ a.s.

Gronwall as an exponential estimate

Gronwall's inequality says: if a non-negative function grows no faster than a linear fraction of its own past accumulation, then it must be zero (when starting from zero). It is the integral analogue of the comparison principle for ODEs: if $\dot u \le Cu$, then $u(t)\le u(0)e^{Ct}$. When $u(0)=0$, $u\equiv 0$.

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4.6 The Markov property of SDE solutions

Under the standard conditions, the strong solution of

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

inherits the Markov property from Brownian motion: the future evolution depends on the past only through the current value $X_t$.

More precisely, for $s\le t$ and measurable $f$:

$$\mathbb E[f(X_t)\mid\mathcal F_s] = \mathbb E[f(X_t)\mid X_s].$$

This follows from pathwise uniqueness and the fact that the increments $W_t - W_s$ are independent of $\mathcal F_s$.

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4.7 Beyond Lipschitz: non-standard cases

The Lipschitz condition is sufficient but not necessary. Several important models fail it but still have unique solutions:

Model	Issue	Resolution
CIR: $dX = \kappa(\theta-X)dt + \sigma\sqrt{X}dW$	$\sigma(x) = \sigma\sqrt{x}$ is not Lipschitz at $x=0$	Yamada–Watanabe uniqueness theorem
GBM: $dS = \mu S\,dt + \sigma S\,dW$	Lipschitz fails globally but holds locally	Local Lipschitz + linear growth suffice
Absolute value: $dX =	X	^\alpha dW$

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4.8 A worked example: verification

Verify that $X_t = x_0 e^{(\mu-\sigma^2/2)t + \sigma W_t}$ solves $dX = \mu X\,dt + \sigma X\,dW$.

Apply Itô's formula to $f(t,x) = x_0 e^{(\mu-\sigma^2/2)t+\sigma x}$:

$$\partial_t f = \left(\mu-\tfrac{\sigma^2}{2}\right)f,\quad \partial_x f = \sigma f,\quad \partial_{xx}f = \sigma^2 f.$$

Therefore

$$dX_t = \left[\left(\mu-\tfrac{\sigma^2}{2}\right)f + \tfrac12\sigma^2 f\right]dt + \sigma f\,dW_t = \mu X_t\,dt + \sigma X_t\,dW_t.$$

The SDE is satisfied. Combined with uniqueness, this is the unique strong solution.

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4.9 Problems

Problem 1: Lipschitz check

Check whether $b(x) = x^2$ satisfies the Lipschitz condition on $\mathbb R$. Does it satisfy linear growth? What does this mean for global existence of $dX = X^2\,dt + dW$?

Problem 2: Gronwall's lemma

Prove Gronwall's lemma: if $u:[0,T]\to[0,\infty)$ is continuous and $u(t)\le A + C\int_0^t u(s)\,ds$, then $u(t)\le A\,e^{Ct}$. (Hint: define $v(t) = e^{-Ct}(A+C\int_0^t u(s)\,ds)$ and differentiate.)

Problem 3: Additive noise SDE

Find the strong solution to $dX_t = -\alpha X_t\,dt + \sigma\,dW_t$ with $X_0 = x_0$. (Hint: use the integrating factor $e^{\alpha t}$ to reduce to an exact differential.)

Problem 4: Pathwise uniqueness for the OU process

Verify directly (without citing the general theorem) that any two solutions of $dX = -\alpha X\,dt + dW$ with the same $W$ and same $X_0$ are indistinguishable, by writing the difference equation and applying Gronwall.

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4.10 Selected solutions

Solution 1

$|b(x)-b(y)| = |x^2-y^2| = |x+y|\cdot|x-y|$. For large $|x|$ or $|y|$, the factor $|x+y|$ is unbounded, so Lipschitz fails globally. Also $|b(x)|^2 = x^4$ grows faster than $C(1+x^2)$, so linear growth fails. The SDE $dX = X^2\,dt + dW$ has solutions that may explode (reach $\pm\infty$) in finite time; global existence is not guaranteed.

Solution 2

Let $v(t) = e^{-Ct}[A + C\int_0^t u(s)\,ds]$. Then $v(0) = A$ and

$$v'(t) = -Ce^{-Ct}\left[A+C\int_0^t u\right] + Ce^{-Ct}u(t) = Ce^{-Ct}[u(t)-(A+C\int_0^t u)] \le 0,$$

since $u(t)\le A + C\int_0^t u$. So $v$ is decreasing, hence $v(t)\le v(0) = A$, giving $A + C\int_0^t u\le Ae^{Ct}$. Since $u(t)\le A + C\int_0^t u\le Ae^{Ct}$, Gronwall is proved.

Solution 3

Let $Y_t = e^{\alpha t}X_t$. By Itô's product rule:

$$dY_t = \alpha e^{\alpha t}X_t\,dt + e^{\alpha t}dX_t = \alpha e^{\alpha t}X_t\,dt + e^{\alpha t}(-\alpha X_t\,dt + \sigma\,dW_t) = \sigma e^{\alpha t}\,dW_t.$$

Integrating: $Y_t = Y_0 + \sigma\int_0^t e^{\alpha s}\,dW_s = x_0 + \sigma\int_0^t e^{\alpha s}\,dW_s$. Therefore

$$X_t = e^{-\alpha t}x_0 + \sigma\int_0^t e^{-\alpha(t-s)}\,dW_s.$$

This is the Ornstein–Uhlenbeck process. It is Gaussian, mean-reverting to zero with rate $\alpha$.

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4.11 What this chapter gives you

Concept	Content
SDE as integral equation	$X_t = X_0 + \int b\,ds + \int\sigma\,dW_s$
Strong solution	adapted to filtration of $W$, pathwise
Lipschitz + linear growth	sufficient conditions for existence and uniqueness
Picard iteration	constructive proof via $L^2$ contraction
Gronwall's inequality	uniqueness via integral comparison
Markov property	SDE solutions inherit Brownian Markov structure

The next chapter solves a collection of important SDEs explicitly using Itô's formula and integrating factors.