Stochastic differential equations and solution concepts

Definition Let $W$ be a standard Brownian motion with respect to a filtration $(\mathcal{F}_t)$ under $\mathbb{P}$, let $b$ and $\sigma$ be measurable functions on $[0, T] \times \mathbb{R}$, and let $\xi$ be an $\mathcal{F}_0$-measurable initial condition. The stochastic differential equation

with $X_0 = \xi$ is shorthand for the integral equation

required to hold almost surely for all $t \in [0, T]$ at once. A solution is a process $X$ with continuous paths, adapted to $(\mathcal{F}_t)$, such that $\int_0^T \lvert b(s, X_s) \rvert \, ds < \infty$ and $\int_0^T \sigma(s, X_s)^2 \, ds < \infty$ almost surely, so that both integrals are defined, and such that the integral equation holds. Brownian paths are nowhere differentiable, so $dW_t$ is not the derivative of $W$ times $dt$, and the equation is not an ordinary differential equation along each path. Equations in several dimensions are defined in the same way, with a vector drift and a matrix diffusion coefficient.

Definition Fix in advance the filtered probability space, the Brownian motion $W$ and the initial condition $\xi$, which is $\mathcal{F}_0$-measurable and hence independent of $W$. A strong solution is a solution $X$ on this space, driven by this $W$, that is adapted to the filtration generated by $\xi$ and $W$ and completed with the null sets. Its value at time $t$ is then determined, up to a null set, by $\xi$ and the Brownian path up to time $t$, so the given inputs determine the output. The word strong describes how the solution depends on its inputs. It says nothing about the accuracy of a numerical method.

Definition Suppose that $b$ and $\sigma$ are measurable on $[0, T] \times \mathbb{R}$ and that there are constants $L$ and $K$ such that for all $t \in [0, T]$ and all real $x$ and $y$

which is the global Lipschitz condition, and

which is the linear growth condition. Let $W$ be a Brownian motion for $(\mathcal{F}_t)$ and let $\xi$ be $\mathcal{F}_0$-measurable with $\mathbb{E}[\xi^2] < \infty$. Then the equation has a strong solution $X$ on $[0, T]$ with $\mathbb{E}[\sup_{t \le T} X_t^2] < \infty$, and pathwise uniqueness holds. Pathwise uniqueness means that any two solutions $X$ and $Y$ on the same space, with the same Brownian motion and the same initial condition, satisfy $X_t = Y_t$ for all $t \in [0, T]$ almost surely. The theorem gives sufficient conditions, not a definition of a solution and not the only route to existence. For coefficients that do not depend on $t$, the Lipschitz condition implies the growth condition. The proof is sketched here. Picard iteration starts from $X^{(0)}_t = \xi$ and sets

The Cauchy-Schwarz inequality bounds the drift integrals, the Itô isometry with Doob’s maximal inequality bounds the stochastic integrals, and the Lipschitz condition turns differences of coefficients into differences of states. The resulting bound on $\mathbb{E}[\sup_{s \le t} \lvert X^{(k+1)}_s - X^{(k)}_s \rvert^2]$ is of order $(Ct)^k / k!$ for a constant $C$, which is summable in $k$, so the iterates converge to a solution. For two solutions, the same estimates applied to $u(t) = \mathbb{E}[\sup_{s \le t} \lvert X_s - Y_s \rvert^2]$, after stopping both processes so that $u$ is finite, give $u(t) \le C \int_0^t u(s) \, ds$, and Gronwall’s inequality forces $u = 0$.

Definition A weak solution with initial law $\nu$ consists of a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$, a Brownian motion $W$ for $(\mathcal{F}_t)$ and a continuous adapted process $X$, all constructed together, such that $X_0$ has law $\nu$ and the integral equation holds almost surely on that space. Every strong solution is a weak solution. A weak solution still solves the integral equation with its own driver, so it is more than a process with the right distribution. The equation has uniqueness in law when any two weak solutions with the same initial law give processes $X$ with the same law on the space of paths. Pathwise uniqueness implies uniqueness in law, and together with the existence of a weak solution it implies the existence of a strong solution, the Yamada-Watanabe theorem, stated here without proof.

Definition For constants $\kappa > 0$, $\theta$ and $\sigma > 0$ and a deterministic initial value $x_0$, the Ornstein-Uhlenbeck equation is

with $X_0 = x_0$. Its coefficients are globally Lipschitz, so it has a unique strong solution. To find it, apply the product rule to $e^{\kappa t} X_t$. The factor $e^{\kappa t}$ has no Brownian part, so there is no covariation term, and

because $\kappa X_t \, dt$ cancels the $-\kappa X_t \, dt$ in the drift. Integrating from $0$ to $t$ and dividing by $e^{\kappa t}$ gives

The stochastic integral has a deterministic integrand, so it is normal with mean $0$, and the Itô isometry gives its variance $\sigma^2 \int_0^t e^{-2\kappa(t - s)} \, ds$. Hence $X_t$ is normal with

As $t \to \infty$ the law tends to the normal law with mean $\theta$ and variance $\sigma^2/(2\kappa)$. The same formulas with $X_s$ in place of $x_0$ and $t - s$ in place of $t$ give the conditional law of $X_t$ given $\mathcal{F}_s$, which is the exact transition. The solution formula also holds for $\kappa \le 0$, where the variance is $\sigma^2 t$ when $\kappa = 0$, but for $\kappa < 0$ deviations from $\theta$ grow instead of decaying.

Definition For the equation $dX_t = b(t, X_t) \, dt + \sigma(t, X_t) \, dW_t$ and a step $h > 0$ with grid times $t_n = nh$, the Euler step, also called the Euler-Maruyama step, freezes both coefficients at the start of the step,

with independent standard normal variables $Z_n$, so that $\sqrt{h}\, Z_n$ has the law of a Brownian increment over one step. It approximates the transition of the solution and in general is not that transition. For coefficients that do not depend on $t$ and satisfy the global Lipschitz condition, with $\mathbb{E}[X_0^2] < \infty$ and with $\sqrt{h}\, Z_n$ taken to be the increment $W_{t_{n+1}} - W_{t_n}$ of the driving Brownian motion, the root-mean-square error at the grid times is of order $h^{1/2}$, a result stated here and studied with numerical methods later. Pathwise uniqueness, uniqueness in law and convergence of a scheme answer three different questions, and none of them replaces another.