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I am currently stuck on the following problem. Consider the SDE $$\text{d}X_t=\sigma(X_t)\text{d}W_t+\mu(X_t)\text{d}t,$$ with $|\mu(x)|^2+|\sigma(x)|^2\leq A(1+|x|^2)$, where $A$ is a finite constant, $X_0=0$, and the drift and diffusion terms are Lipschitz functions.

I need to show that

  • $\mathbb{E}\sup_{t\leq1}|X_t|^2\leq\text{e}^{8A}$,

  • the solution $\{X_t\}_{t\leq1}$ exists even if the drift and diffusion terms are only locally Lipschitz.

I'm so confused, I don't know how to begin! I've stared at https://math.stackexchange.com/questions/3251957/calculating-the-expecation-of-the-supremum-of-absolute-value-of-a-brownian-motio in hopes of getting some better insight, but I'm lost -- could someone guide me in the right direction please? Thank you!

user107224
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1 Answers1

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The solution satisfies $$x_t = \int_0^t \mu(X_u)du + \int_0^t \sigma(X_u)dW_u.$$ Using the inequality $|a+b+c|^2 \leq 3|a|^2+3|b|^2+3|c|^2,$ we have $$|X_t|^2 \leq 3 \left|\int_0^t \mu(X_u)du \right|^2 + 3 \left| \int_0^t \sigma(X_u)dW_u \right|^2.$$ By the Cauchy–Schwarz inequality the first integral above satifies $$\left|\int_0^t \mu(X_u)du \right|^2 \leq t \int_0^t \mu^2(X_u)du.$$ So, \begin{equation} \tag{1} E \sup_{t \le 1} |x_t|^2 \le 3 \left|\int_0^1 \mu^2(X_u)du \right|^2+3E\sup_{t \leq 1} \left| \int_0^t \sigma(X_u)dW_u \right|^2. \end{equation} By Doob's martingale inequality first and the Ito isometry later, the second integral of Equation (1) satisfies \begin{equation} \tag{2} E\sup_{t \leq 1} \left| \int_0^t \sigma(X_u)dW_u \right|^2 \leq 4E\left| \int_0^1 \sigma(X_u)dW_u \right|^2=4E \int_0^1 \sigma^2(X_u)du. \end{equation} Plugging (2) in (1) and using the linear growth condition (given in the problem), we have $$E \sup_{t \le 1} |x_t|^2 \leq 3E\int_0^1\mu^2(X_u)du +12E\int_0^1\sigma^2(X_u)du \leq 12AE\int_0^1 (1+ |X_u|^2)du. $$ Thus $$1+ E \sup_{t \le 1} |x_t|^2 \leq 1+ 12AE\int_0^1 (1+ |X_u|^2)du$$ and by the Gronwall inequality $$E \sup_{t \le 1} |x_t|^2 \leq e^{12A}.$$

UBM
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  • wow... I would’ve never thought of using any of these! do you have any tips on patterns/key ideas to look out for when proving inequalities like this? – user107224 Oct 21 '20 at 08:49
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    All the ideas I've used in this answer are from the proof of existence and uniqueness of the solution to a SDE. I would study that proof carefully. For example in Oksendal or Mao books. – UBM Oct 21 '20 at 10:44