A stochastic differential equation specifies local drift and covariance and asks for a global random path realizing them. Well-posedness has several levels: existence in law, pathwise construction from a given Brownian motion, uniqueness in law, and pathwise uniqueness. Lipschitz coefficients give all of them; weaker coefficients require these notions to be kept distinct.
Strong and weak solutions
Definition. Let \(B\) be \(m\)-dimensional Brownian motion, \(b:[0,T]\times\mathbb R^d\to\mathbb R^d\), and \(\sigma:[0,T]\times\mathbb R^d\to\mathbb R^{d\times m}\). The notation \[ \mathrm{d}X_t=b(t,X_t)\,\mathrm{d} t +\sigma(t,X_t)\,\mathrm{d}B_t \] means the integral equation \[ X_t=X_0+\int_0^tb(s,X_s)\,\mathrm{d} s +\int_0^t\sigma(s,X_s)\,\mathrm{d}B_s. \]
Definition.
- A strong solution is adapted to the completed filtration generated by the prescribed pair \((X_0,B)\).
- A weak solution allows the probability space and Brownian motion to be part of the unknown.
- Pathwise uniqueness means that two solutions driven by the same Brownian motion with the same initial value are indistinguishable.
- Uniqueness in law means that all weak solutions have the same path distribution.
Lipschitz well-posedness
Theorem. Suppose there is \(L<\infty\) such that, for all \(t,x,y\), \[ \|b(t,x)-b(t,y)\| +\|\sigma(t,x)-\sigma(t,y)\|_{\mathrm{HS}} \le L\|x-y\|, \] and \[ \|b(t,x)\|^2+\|\sigma(t,x)\|_{\mathrm{HS}}^2 \le L^2(1+\|x\|^2). \] If \(X_0\in L^2\), then the SDE has a unique strong solution and \[ \mathbb E\sup_{t\le T}\|X_t\|^2 \le C_T(1+\mathbb E\|X_0\|^2). \]
Proof. Define Picard iterates \(X_t^{(0)}=X_0\) and \[ X_t^{(n+1)} = X_0+\int_0^tb(s,X_s^{(n)})\,\mathrm{d} s +\int_0^t\sigma(s,X_s^{(n)})\,\mathrm{d}B_s. \] Cauchy-Schwarz and the BDG inequality give \[ \mathbb E\sup_{u\le t} \|X_u^{(n+1)}-X_u^{(n)}\|^2 \le C L^2\int_0^t \mathbb E\sup_{v\le s} \|X_v^{(n)}-X_v^{(n-1)}\|^2\,\mathrm{d} s. \] Iteration yields a factorial bound \(C_T^{n}/n!\), so the differences are summable in \(L^2(\Omega;C[0,T])\). The limit is adapted and satisfies the equation by continuity of the deterministic and stochastic integrals. For two solutions \(X,Y\), the same estimate gives \[ \mathbb E\sup_{u\le t}\|X_u-Y_u\|^2 \le C L^2\int_0^t \mathbb E\sup_{v\le s}\|X_v-Y_v\|^2\,\mathrm{d} s. \] Gronwall implies equality. Applying the estimate to \(X\) and zero, using linear growth, gives the moment bound.
Proposition (stability in initial data). If \(X^x,X^y\) solve the same Lipschitz SDE from \(x,y\), then \[ \mathbb E\sup_{t\le T}\|X_t^x-X_t^y\|^2 \le C_T\|x-y\|^2. \]
Proof. Repeat the uniqueness estimate but retain the initial difference, then apply Gronwall.
Local coefficients and explosion
Theorem. If \(b,\sigma\) are locally Lipschitz, there is a unique maximal strong solution up to an explosion time \[ \zeta=\lim_{R\to\infty}\inf\{t:\|X_t\|\ge R\}. \] A linear-growth bound implies \(\zeta=\infty\) almost surely.
Proof outline. Truncate the coefficients outside balls to obtain globally Lipschitz equations. Pathwise uniqueness makes their solutions agree until they leave the smaller ball, so they patch into a maximal solution. The stopped moment estimate gives \[ P(\tau_R\le T) \le R^{-2}\mathbb E\sup_{t\le T}\|X_{t\wedge\tau_R}\|^2 \le C_T/R^2, \] and \(R\to\infty\) rules out explosion under linear growth.
Yamada-Watanabe principle
Theorem. Weak existence plus pathwise uniqueness implies existence of a unique strong solution. In particular, pathwise uniqueness upgrades uniqueness in law.
Proof outline. Disintegrate the joint law of a weak solution \((X,B)\) conditionally on the Brownian path and initial value. Couple two conditionally independent draws from this regular conditional law over the same \(B\). Pathwise uniqueness forces the two draws to agree almost surely, so the conditional law is a point mass. A measurable-selection argument therefore writes \(X\) as a measurable functional of \((X_0,B)\), giving a strong solution and uniqueness in law.
Generator and martingale problem
Theorem. For time-homogeneous coefficients and \(f\in C_c^2(\mathbb R^d)\), define \[ Lf(x) = b(x)^\top\nabla f(x) +\frac12\operatorname{tr} \{a(x)\nabla^2f(x)\}, \qquad a(x)=\sigma(x)\sigma(x)^\top. \] Then \[ f(X_t)-f(X_0)-\int_0^tLf(X_s)\,\mathrm{d} s \] is a local martingale.
Proof. Ito's formula gives exactly the displayed finite-variation term plus \[ \int_0^t\nabla f(X_s)^\top\sigma(X_s)\,\mathrm{d}B_s, \] a local martingale.
Proposition (Fokker-Planck equation). If \(X_t\) has a smooth density \(p(t,x)\), then formally \[ \partial_tp=L^*p = -\sum_i\partial_i(b_ip) +\frac12\sum_{i,j}\partial_{ij}(a_{ij}p). \]
Proof. For a test function \(f\), \[ \frac{\mathrm{d}}{\mathrm{d}t}\int f(x)p(t,x)\,\mathrm{d} x =\mathbb E Lf(X_t) =\int Lf(x)p(t,x)\,\mathrm{d} x. \] Integrate by parts to transfer derivatives from \(f\) to the coefficients times \(p\), identifying the adjoint \(L^*\).
Langevin diffusion
Example. For a smooth potential \(U\), \[ \mathrm{d}X_t=-\nabla U(X_t)\,\mathrm{d} t+\sqrt2\,\mathrm{d}B_t \] has formal invariant density \[ \pi(x)\propto e^{-U(x)}. \] Indeed, with \(b=-\nabla U\) and \(a=2I\), \[ L^*\pi=\nabla\mathbin{\cdot}(\pi\nabla U)+\Delta\pi=0. \] This diffusion is the continuous-time basis of Langevin Monte Carlo. Its Euler discretization inherits the consistency-versus-stability questions developed for numerical initial-value problems, but strong and weak stochastic errors require additional martingale estimates.