Some doubts on proof of pathwise uniqueness of a stochastic differential equation

Question

I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.

Starting from a stochastic differential equation of the form:

$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\tag{1}$$ with $\left\{W_t\right\}_{t\geq0}$ a standard Wiener process in the filtered probability space $\left(\Omega,\mathcal{F},\left\{\mathcal{F}_n\right\},\mathbb{P}\right)$. We assume $\alpha,\beta>0$ and $r_m<r_{\mu}<r_M$, which guarantee the existence of stationary distribution.

(1. Why do the assumption $\alpha,\beta>0$ and $r_m<r_{\mu}<r_M$ guarantee stationarity of distribution? What is exactly meant here?)

$\alpha$ represents the speed of reversion to the longrun mean $r_{\mu}$.
Then, for diffusion of $(1)$, set $\sigma(x)=\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}$. For any $x,y\in[r_m,r_M]$, it holds that: $$|\sigma^2(x)-\sigma^2(y)|=\beta|r_m+r_M-(x+y)||x-y|\leq\beta(r_M-r_m)|x-y|\tag{2}$$ This means that the diffusion coefficient function $\sigma(x)$ is Holder $\frac{1}{2}$ continous.

(2. Why does $(2)$ translate into the fact that $\sigma(x)$ is Holder $\frac{1}{2}$ continous? Doesn't $(2)$ denote instead that $\sigma^2(x)$ is Holder continous (with coefficient $1$)?)

Since $\mu(x)=\alpha(r_{\mu}-x)$ is Lipschitz continuous, the pathwise uniqueness of the stochastic differential equation is guaranteed by the general uniqueness theorem.

(3. Could you please give me some reference for such a theorem? In this specific case, does this theorem rely on Lipschitz condition on $\mu(x)$ combined with Holder continuity of $\sigma(x)$? If so, why do such two conditions guarantee the uniqueness of the path of SDE $(1)$?)

Answer 1

Concerning(3): If I am not mistaken, this type of condition is sometimes referred to as Yamada-Watanabe condition for pathwise uniqueness. Your particular case is just a special case of Thm 2.1 in these lecture notes: https://fam.tuwien.ac.at/~schmock/notes/Yamada-Watanabe.pdf

EDIT: If the drift coefficient $b$ is Lipschitz continuous and the diffusion coefficient $\sigma$ is $\frac{1}{2}$-Hölder, then they satisfy an inequality like $$ |b(x)-b(y)| + |\sigma(x)-\sigma(y)|^2 \leq K |x-y| $$ for some $K>0$. Now you can use this to show pathwise uniqueness for the SDE $$ dX_t = b(X_t)dt + \sigma(X_t)dB_t. $$ Suppose you have two strong solutions $X^1$ and $X^2$ for this SDE. Then you can apply Itô's formula to the process $\Delta_t:= X^1_t -X^2_t$ and the function $$ \phi_{\varepsilon}(x) = (\varepsilon + x^2)^{\frac{1}{2}}. $$ If you then take the limit $\varepsilon \to 0$ you can show via Gronwall's inequality that indeed $\mathbb{E}[|X^1_t - X^2_t|] = 0$ for all $t\geq 0$. The strategy for the proof in the more general case in the lecture notes I linked above is essentially the same, just a bit more technical.

Answer 2

Your equation (1) is a Jacobi diffusion equation. In particular, it matches equation (2.1) in this paper if you take there $X_t=r_t$, $\sigma=\beta$, $\beta=\alpha$, $m=(r_M+r_m)/2$, $z=(r_M-r_m)/2$, and $\gamma=\dfrac{r_\mu-m}z$. As explained in the linked paper, if the distribution of $X_0$ (that is, of your $r_0$) is the shifted and rescaled Beta-distribution on the interval $(m−z,m+z)=(r_m,r_M)$ given by formula (2.2) in the linked paper, then the distribution of the diffusion process is stationary. This answers your questions 1 and 3.

Concerning your question 2, for $C:=\beta(r_M-r_m)$ you have $$|\sigma(x)-\sigma(y)| =\frac{|\sigma^2(x)-\sigma^2(y)|}{\sigma(x)+\sigma(y)} \\ \le\min\Big(\frac{C|x-y|}{\sigma(x)+\sigma(y)},\sigma(x)+\sigma(y)\Big) \\ \le\sqrt{C|x-y|},\tag{*}$$ so that $\sigma$ is Hölder $1/2$-continuous. (The latter displayed inequality is a special case of the inequality $\min(\frac cu,u)\le\sqrt c$ for real $c,u>0$.)

Responses to the comments by the OP:

in Delbaen and Shirakawa (2002) it is not specified whether r0 follows a beta-distribution, but it is simply stated that "$\alpha,\beta>0$ and $r_m<r_\mu<r_M$ guarantee the existence of stationary distribution."

The logic here is quite simple: We can choose the distribution of $r_0$ to be the shifted and rescaled Beta-distribution on the interval $(m−z,m+z)=(r_m,r_M)$ given by formula (2.2) in the linked paper, and then the distribution of the diffusion process will be stationary. So, a stationary distribution exists.

As for the pathwise uniqueness, it was addressed in answer by jonask.

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Concerning the answer to question 2: Inequalities $|\sigma^2(x)-\sigma^2(y)|\le\sigma(x)+\sigma(y)$ and even $\sigma^2(x)-\sigma^2(y)\le\sigma(x)+\sigma(y)$ will be of course false in general, and I stated nothing of that sort. Rather, part of what display (*) says is that $$\frac{|\sigma^2(x)-\sigma^2(y)|}{\sigma(x)+\sigma(y)}\le\sigma(x)+\sigma(y),$$ and this is very simple algebra.