SDE with $a(t,x) = |x|^{\alpha}$ and $a(t,x) =a(x)$

Question

• Let $a,b : [0,T] \times \mathbb{R} \to \mathbb{R}$ be Borel functions.
• Let $X$ be the solution of

$$ X_t= \int_{0}^t a(s,X_s) dW_s + \int_{0}^t b(s,X_s) ds , t \in[0,T]$$ We call it $SDE(a,b)$

1. If $a(t,x) = |x|^{\alpha}$ with $\alpha > \dfrac12$ and $b$ Lipschitz, does $SDE(a,b)$ admit pathwise uniqueness? What about uniqueness in law?

2. Suppose $a(t,x)=a(x)$ and $b(t,x)=0$. Does $SDE(a,0, \delta_{x_0})$ admit existence and uniqueness in law?

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Some conditions are given here and in Singular Stochastic Differential Equations by Alexander S. Cherny and Hans-Jürgen Engelbert.

1. Yamada-Watanabe criterion?

2. We use the Engelbert-Schmidt criterion. For a Borel function, we define

• $Z(\sigma)= \{x \in\mathbb{R} | \sigma(x)=0 \}$
• $I(\sigma)$ the set of real numbers $x$ such that $\int_{x - \epsilon}^{x + \epsilon} \dfrac{dy}{ \sigma^2(y) }= \infty, \forall \epsilon >0$
• $I(a) \subset Z(a)$ is a necessary and sufficient condition for existence in law.
• $I(a) = Z(a)$ is a necessary and sufficient condition for existence and uniqueness in law

Answer 1

The Watanabe criterion states that if

• We suppose that $b$ is Lipschitz and

• there exists $h : \mathbb{R}^+ \to \mathbb{R}^+$ strictly increasing continuous such that

• $h(0)=0$

• $\int_{x - \epsilon}^{x + \epsilon} \dfrac{dy}{ h^2(y) }= \infty, \forall \epsilon >0$

• $ |a(t,x) - a(t,y)| \leq h(x-y)$

Then we have pathwise existence.

With $\alpha \geq \frac12$ and $a(t,x)= x^{\alpha}$ the conditions are verified according to this inequality