# Existence and Uniqueness of the solutions to SDE with locally Lipschitz coefficient and linear growth

The following theorem states the existence and uniqueness of a strong solution for a class of SDEs.

Theorem. For the SDE $$\mathrm{d} X=G(t, X(t)) \mathrm{d} t+H(t, X(t)) \mathrm{d} W(t), \quad X\left(t_{0}\right)=X_{0},$$ assume the following hold.
(1) Both $$G(t, x)$$ and $$H(t, x)$$ are continuous on $$(t, x) \in\left[t_{0}, T\right] \times \mathbb{R}$$.
(2) The coefficient functions $$G$$ and $$H$$ satisfy the Lipschitz condition $$|G(t, x)-G(t, y)|+|H(t, x)-H(t, y)| \leq K|x-y| .$$ (3) The coefficient functions $$G$$ and H satisfy a growth condition in the second variable, $$|G(t, x)|^{2}+|H(t, x)|^{2} \leq K\left(1+|x|^{2}\right),$$ for all $$t \in\left[t_{0}, T\right]$$ and $$x \in \mathbb{R}$$.
Then the SDE has a strong solution on $$\left[t_{0}, T\right]$$ that is continuous with probability 1 and $$\sup _{t \in\left[t_{0}, T\right]} \mathbb{E}\left[X^{2}(t)\right]<\infty,$$ and for each given Wiener process $$W(t)$$, the corresponding strong solutions are pathwise unique, which means that if $$X$$ and $$Y$$ are two strong solutions, then $$\mathbb{P}\left[\sup _{t \in\left[t_{0}, T\right]}|X(t)-Y(t)|=0\right]=1 .$$

(I took it from Dunbar, S.R. Mathematical Modeling in Economics and Finance: Probability, Stochastic Processes, and Differential Equations; Vol. 49, American Mathematical Soc., 2019.)

Anyway, in my case I have an SDE where $$H$$ is constant and $$G :\mathbb R_+\times\mathbb R\to\mathbb R$$ is such that $$G(t, X(t))=F(t)X(t)$$ where $$F(t)=\frac{1}{\cos t+ \sin t}$$. Let $$t_0=0$$, $$T>0$$ and let $$\mathcal A$$ be the domain of $$F(t)$$, i.e. $$\mathcal A=\left\{t|\cos t + \sin t\neq0\right\}$$. My question is: does the above SDE admits a unique strong solution?

My attempt

Point (1) is satisfied because $$G$$ is continuous $$(t, x) \in\left(\left[0, T\right]\cap\mathcal A\right) \times \mathbb{R}$$. Point (2) is not true globally but for a local Lipschitz condition if $$K=\sup_{t\in\left(\left[0, T\right]\cap\mathcal A\right)}F(t)$$ Similar consideration can be done for point (3).

Can I infer the existence of a strong solution on $$\left[0, T\right]\cap\mathcal A$$? I think that the answer is not so obvious because I replace a global Lipschitz condition, in point (2), with a local one. What do you think?