# Estimates on perturbation of drift of SDEs

Let $$\mu_1,\mu_2:\mathbb{R}^n\rightarrow \mathbb{R}^n$$ and $$\sigma:\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$$ be Lipschitz functions, of at-most linear growth; i.e. $$\|\sigma(x)\|\lesssim \|x\|,\|\mu_i(x)\|\lesssim \|x\|$$. Let $$W_{\cdot}$$ be an $$n$$-dimensional Brownian motion.

Fix $$x_0\in \mathbb{R}^n$$ and let $$X^i_{\cdot}$$ be a strong solution to the SDE $$X_t^i = x_0 + \int_0^t\, \mu_i(X_s^i)ds + \int_0^t\,\sigma(X_s^i)dW_s.$$

Fix $$T>0$$. Are there estimates bounding $$\mathbb{E}[\sup_{1\le t\le T}\|X_t^1-X_t^2\|]$$ above a function of $$\|\mu_1-\mu_2\|_{\infty}$$ where $$\| .\|_{\infty}$$ is the sup-norm.

The difference is

$$X_t^1-X_t^2=\int_0^t\, \mu_1(X_s^1)-\mu_2(X_s^2)ds + \int_0^t\,\sigma(X_s^1)-\sigma(X_s^2)dW_s.$$

For the second term, you can use that it is a martingale and apply Burkholder-Davis-Gundy (see Revuz-Yor 2.4 theorem in SDE section) to bound

$$E[\sup_{t\leq T}|\int_0^t\,\sigma(X_s^1)-\sigma(X_s^2)dW_s|]$$

$$\leq E[(\int^{T} |\sigma(X_s^1)-\sigma(X_s^2)|^{2}ds)^{1/2}]$$

$$\leq L_{\sigma}E[(\int^{T} |X_s^1-X_s^2|^{2}ds)^{1/2}]$$

$$\leq L_{\sigma} T^{1/2} E[\sup_{s\leq T}|X_s^1-X_s^2|]$$

for $$L_{\sigma}$$ the Lipschitz constant for $$\sigma$$. So if $$L_{\sigma} T^{1/2}<1$$ we can bound

$$E[\sup_{s\leq T}|X_s^1-X_s^2|]\leq \frac{1}{1-L_{\sigma} T^{1/2}}E[\sup_{s\leq T}|\int_0^t\, \mu_1(X_s^1)-\mu_2(X_s^2)ds|]$$

$$\leq \frac{1}{1-L_{\sigma} T^{1/2}}E[\int_0^T\, |\mu_1(X_s^1)-\mu_2(X_s^2)|ds].$$

Now we need to add and subtract $$\mu_{2}(X_s^1)$$ and use that its $$L_{\mu_{2}}$$-Lipschitz to bound

$$\leq \frac{1}{1-L_{\sigma} T^{1/2}}E[\int_0^T\, |\mu_1(X_s^1)-\mu_2(X_s^1)|ds]+\frac{TL_{\mu_{2}}}{1-L_{\sigma} T^{1/2}}E[\sup_{s\leq T}|X_s^1-X_s^2|].$$

So again if $$\frac{TL_{\mu_{2}}}{1-L_{\sigma} T^{1/2}}<1$$ we bound

$$E[\sup_{s\leq T}|X_s^1-X_s^2|]\leq c_{1}c_{2}T\|\mu_{1}-\mu_{2}\|_{\infty},$$

for $$c_{1}:=\frac{1}{1-L_{\sigma} T^{1/2}},c_{2}:=(1-\frac{TL_{\mu_{2}}}{1-L_{\sigma} T^{1/2}})^{-1}.$$

Just as a side note, if you have some comparison $$\mu_{1}\leq \mu_{2}$$, then you can get comparison for the solutions eg. see "On pathwise uniqueness and comparison of solutions of one-dimensional stochastic differential equations".

• I don't quite see how did you se the fact that $L_{\sigma}\,T^{1/2}$; what inequality are you appealing to? Commented Nov 5, 2023 at 18:12
• I am assuming this in order to go through with the bounds. For large T this bound might be false. Commented Nov 5, 2023 at 18:17