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Let $b: \mathbb R_+\to\mathbb R_+$ and $\sigma: \mathbb R_+\times \mathbb R\to\mathbb R_+$ be Lipschitz and bounded. Assume further $\sigma$ is elliptic, i.e. $\inf_{(t,x)}\sigma(t,x)>0$. For each $s>0$ and $y\in\mathbb R$, let $g_{\alpha}(\cdot,\cdot,s,y):[0,s]\times \mathbb R\to\mathbb R$ be the fundamental solution to

$$\partial_t g_{\alpha}(t,x,s,y) + \frac{\sigma(t,x)^2}{2\big(1+\alpha(t)\big)^2}\partial^2_{xx}g_{\alpha}(t,x,s,y) + b(t)\partial_{x}g_{\alpha}(t,x,s,y)=0,\quad \forall t\in [0,s),~ x\in\mathbb R$$

together with $g_\alpha(s,x,s,y)=\delta_y(x)$, where $\alpha:\mathbb R_+\to\mathbb [0,1]$ is Holder continuous. Does there exist a continuous function $\theta:\mathbb R_+\to \mathbb R_+$ with $\theta(0)=0$ s.t.

$$\left|\int_0^{\infty}g_\alpha(0,x,s,y)dy-\int_0^{\infty}g_\beta(0,x,s,y)dy\right|\le \theta(T)\|\alpha-\beta\|_s,\quad \forall s\le T,~ \forall x\in\mathbb R$$

and

$$\int_0^s\left|\frac{d}{dt}\left(\int_0^{\infty}g_\alpha(t,0,s,y)dy-\int_0^{\infty}g_\beta(t,0,s,y)dy\right)\right|dt\le \theta(T)\|\alpha-\beta\|_s,\quad \forall s\le T,$$

where $\alpha,\beta: \mathbb R_+\to [0,1]$ are both Holder continous and $\|\alpha-\beta\|_s:=\max_{0\le u\le s}|\alpha(u)-\beta(u)|$.

Any answer, comments and references are highly appreciated.

PS : A probabilistic interprectation of $g_{\alpha}$ is as follows : Consider the parametric stochastic differential equation (SDE)

$$dX_t=b(t)dt + \frac{\sigma(t,X_t)}{1+\alpha(t)}dW_t,\quad \forall t\ge 0.$$

Then $g_{\alpha}$ is the conditional density of $X_s$ knowing that $X_t=x$, i.e.

$$\mathbb P[X_s\in dy|X_t=x]\sim g_{\alpha}(t,x,s,y)dy.$$

However, I don't know any probability tools to handle the density regularity of the solution to a SDE...

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    $\begingroup$ It looks to me like you could differentiate the equation in the variable alpha and get an equation for $\partial_\alpha g_\alpha$, with some terms that you can throw on the right side. It that looks to me like it should be relatively ok. Presumably you could get a bound in L^1 for this function, especially if you only want an estimate only up to a fixed finite time $T$. Did you try that? $\endgroup$ Commented Feb 4, 2022 at 2:40
  • $\begingroup$ @ScottArmstrong Thanks a lot for the hint which seems very appealing. I didn't think of that, but the differentiation w.r.t. $\alpha$ may yield the terms $\sigma\partial_{\alpha}\sigma\partial_{xx}g_{\alpha}$ and $\partial_{\alpha}b\partial_{x}g_{\alpha}$ which contain $\partial_{x}g_{\alpha}$ and \partial_{xx}g_{\alpha}. How could we handle these terms by putting them on the right side? I suppose I don't know a priori any estimate on them... Could you please detail your idea by writting down an answer? I do appreciate! $\endgroup$
    – GJC20
    Commented Feb 4, 2022 at 7:20
  • $\begingroup$ Have you tried the Malliavin calculus? $\endgroup$
    – user420828
    Commented Feb 6, 2022 at 12:19
  • $\begingroup$ @Philo18 Could you please specify a bit more? I know very few on Malliavin calculus $\endgroup$
    – GJC20
    Commented Feb 6, 2022 at 13:23
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    $\begingroup$ @GJC20 I think you might refer to the book "The Malliavin Calculus and Related Topics", while the non constant term in front of $W$ makes your problem more complicated... $\endgroup$
    – user420828
    Commented Feb 7, 2022 at 18:52

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