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A stochastic differential equation (SDE) dX(t) = μ(X,t)dt + σ(X,t)dW(t) describes a process whose evolution has both a deterministic drift μ and a random diffusion σ driven by Brownian motion. Under Lipschitz and linear growth conditions on μ and σ, the SDE has a unique strong solution — the stochastic analogue of the Picard-Lindelöf theorem for ODEs. SDEs are the central modeling tool of stochastic analysis, describing everything from particle diffusion to financial asset prices.
A stochastic differential equation dX(t) = μ(X,t)dt + σ(X,t)dW(t) describes a system subject to both deterministic forces (the drift μ) and random perturbations (the diffusion σ times Brownian noise). The notation "dX = ..." is shorthand for the integral equation X(t) = X(0) + ∫₀ᵗ μ(X(s),s)ds + ∫₀ᵗ σ(X(s),s)dW(s), where the first integral is ordinary Lebesgue and the second is Itô. If you set σ = 0, you recover an ordinary differential equation — SDEs generalize ODEs by adding continuous random forcing.
The fundamental existence and uniqueness theorem mirrors the Picard-Lindelöf theorem from ODE theory. If μ and σ are globally Lipschitz in x (|μ(x,t) - μ(y,t)| + |σ(x,t) - σ(y,t)| ≤ K|x-y|) and satisfy a linear growth bound (|μ(x,t)| + |σ(x,t)| ≤ K(1+|x|)), then for any square-integrable initial condition X(0), the SDE has a unique strong solution. The proof constructs the solution via Picard iteration X_{n+1}(t) = X₀ + ∫μ(X_n)ds + ∫σ(X_n)dW and shows convergence in L² using the Itô isometry and Gronwall's inequality. The Lipschitz condition prevents branching (uniqueness); the linear growth condition prevents explosion (global existence).
The distinction between strong and weak solutions is subtler than anything in ODE theory. A strong solution is adapted to the filtration generated by the given Brownian motion W — it is a deterministic functional of W. A weak solution only requires that some probability space with some Brownian motion and some process exists satisfying the SDE. Tanaka's equation dX = sgn(X)dW demonstrates the gap: the process |W(t)| (reflected Brownian motion) is a weak solution, but no strong solution exists. In practice, most applications work with strong solutions, but weak solutions are the right framework for problems involving change of measure (Girsanov's theorem).
SDEs are solved explicitly only in special cases — linear SDEs, geometric Brownian motion, the Ornstein-Uhlenbeck process. For nonlinear SDEs, one typically studies qualitative properties: does the solution stay positive? Does it have a stationary distribution? What are its moment bounds? Itô's formula is the primary tool: to analyze f(X(t)), apply the formula to get the SDE for f(X) and read off its drift and diffusion. Numerical methods (Euler-Maruyama, Milstein) discretize the SDE for simulation, with convergence rates governed by the regularity of the coefficients.