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A stochastic differential equation has strong existence for specified coefficients and initial data if a strong stochastic solution can be constructed on every prescribed stochastic basis carrying the required initial variable and driving Brownian motion. This existence assertion is distinct from pathwise uniqueness and from a definition of the solution itself.

For bounded Borel $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, the stochastic differential equation $d{X}_t=b(X_t)dt+d{B}_t$ has strong existence and pathwise uniqueness. The identity diffusion matrix is nondegenerate. A Girsanov theorem argument alone establishes weak existence and uniqueness in law; the strong conclusion requires an additional theorem. This statement does not extend without further hypotheses to arbitrary path-dependent drift or arbitrary diffusion matrices. See the primary bounded-drift strong-solution theorem.

A Synchronous coupling drives two processes with the same fresh random inputs. For example, recursions $X_{n+1}=F(X_n,{\xi }_{n+1})$ and $Y_{n+1}=F(Y_n,{\xi }_{n+1})$ use the same input ${\xi }_{n+1}$ at each step. This can reveal synchronous contraction of a random affine recursion even when independently simulated trajectories would conceal the relationship.

A Synchronous coupling drives two solutions of a stochastic differential equation by the same Brownian motion. Their difference has noise coefficient $\sigma (X)-\sigma (Y)$. A dissipativity inequality for the drift and this coefficient gives the mean-square contraction of synchronously coupled diffusions.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.