For standard sde such as $${\rm d}{X_t} = \mu {X_t}\,{\rm{d}}t + \sigma {X_t}{\mkern 1mu} {\rm{d}} B_t$$ the existence and uniqueness th. is : $$\begin{array}{l} \text{let } :T > 0\\ \mu :R^n \times [0,T] \to {\mathbb{R}^n};\\ \sigma :R^n \times [0,T] \to {\mathbb{R}^{n \times m}}; \end{array}$$ be measurable functions for which there exist constants C and D such that $$\begin{array}{l} |\mu (x,t)| + |\sigma (x,t)| \le C(1 + |x|);\\ |\mu (x,t) - \mu (y,t)| + |\sigma (x,t) - \sigma (y,t)| \le D|x - y|; \end{array}$$ Let $Z$ be a random variable that is independent of the σ-algebra generated by $B_s$, $s ≥ 0$, and with finite second moment:$$[|Z|^2] < + \infty $$Then the stochastic differential equation/initial value problem $$\begin{array}{l} {\rm{d}}{X_t} = \mu ({X_t},t){\mkern 1mu} {\rm{d}}t + \sigma ({X_t},t){\mkern 1mu} {\rm{d}}{B_t} \text{ for } t \in [0,T];\\ {X_0} = Z; \end{array}$$ has a Pr-almost surely unique t-continuous solution $(t, ω) \mapsto Xt(ω)$ such that $X$ is adapted to the filtration FtZ generated by $Z$ and $B_s$, $s ≤ t$, and$$\left[ \int_0^T |X_t|^2 {\mkern 1mu} {\rm{d}}t \right] < + \infty $$ .now my question is : when I have this mean reverting SDE $$dx_t=(\mu-x_t)\,dt+\sigma \,dB_t\\ \text{or} \\dx_t=\theta(\mu-x_t)\,dt+\sigma \, dB_t$$ how can I rewrite existence and uniqueness th. for that ? I know this a simple SDE ,but because of $\mu \,dt $ term in SDe ,I can't write it ... A hint to start is needed
I am thankful of your guide in advanced.
You are aware that in the geometric Browninan motion, $μ(x,t)=μ·x$ and $σ(x,t)=σ·x$? That is, you used the same letter twice, once for the constant factors and then for the full coefficient functions in $$ dX_t=μ(X_t,t)\,dt+σ(X_t,t)\,dB_t. $$
Thus in the second equation, you get $μ(x,t)=θ(μ−x)$ and $σ(x,t)=σ$. As both are linear functions, the assumptions of the existence theorem are trivially satisfied.