Suppose X is a solution to dX t = σ(X t ) dW t , where W is a d-dimensional Brownian motion, σ(x) is a d × d matrix-valued function that is bounded, and σ T σ is positive definite, uniformly in x....

Suppose X is a solution to dX_t
= σ(X_t) dW_t, where W is a d-dimensional Brownian motion, σ(x) is a d × d matrix-valued function that is bounded, and σ T σ is positive definite, uniformly in x. Prove the following estimate for the time to leave a ball: there exist constants c₁
and c₂
not depending on x₀
such that

Where

Chapter 40