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Let $\alpha:\mathbb R_+\to\mathbb R_+$ be a "nice" function with $\alpha(0)=1$. Define the process

$$Y_t=Y_0+t+\int_0^t\frac{dW_u}{1+\alpha(u)},\quad \forall t\ge 0,$$

where $Y_0>0$ has a density $\rho:\mathbb R_+\to\mathbb R_+$ and $(W_t)_{t\ge 0}$ is an independent Brownian motion. For $t\ge 0$ and $x\in\mathbb R$, let $(Y^{t,x}_s)_{s\ge t}$ be the conditional process given by $Y^{t,x}_s:=x+Y_s-Y_t$ for $s\ge t$. Does there exist $\alpha$ s.t.

$$\alpha(s) = \mathbb P[Y_s>0] + \int_0^s \dot{\alpha}(t)\mathbb P[Y^{t,0}_s>0] dt,\quad \forall s\ge 0?\quad\quad\quad\quad\quad\quad\quad\quad(\ast)$$

Any answer, comments or references are highly appreicated!


PS :

(1) Write $A_{t,s}:=\int_t^s du/(1+\alpha(u))^2$ for $0\le t\le s$, then $(\ast)$ can be rewritten as

$$\alpha(s) = \int_0^{\infty}\rho(x)\Phi\left(\frac{x+s}{\sqrt{A_{0,s}}}\right)dx + \int_0^s \dot{\alpha}(t)\Phi\left(\frac{s-t}{\sqrt{A_{t,s}}}\right)dt,\quad \forall s\ge 0,\quad\quad(\times),$$

where $\Phi:\mathbb R\to\mathbb R$ is defined by $\Phi(y):=\int_{-\infty}^y e^{-z^2/2}/\sqrt{2\pi}dz$;

(2) A formal integration by parts yields

$$\alpha(s) = \mathbb P[Y_s>0] - \mathbb P[Y^{0,0}_s>0] - \int_0^s \alpha(t)\partial_t\mathbb P[Y^{t,0}_s>0] dt=:\Gamma[\alpha](s),\quad \forall s\ge 0,\quad(\star).$$

My inital attemp is to prove, for small $T>0$, the existence of $\alpha$ s.t. $\alpha(s)=\Gamma[\alpha](s)$ for $s\in [0,T]$. The idea is to show that $\Gamma$ is a contraction map on some suitable subspace of $C([0,T])$, where $C([0,T])$ denotes the space of continuous functions starting at one endowed with the uniform norm. I believe it is tractable but I know very little about the inequalities about $\Phi$ and $\dot{\Phi}=:\phi$.

(3) If $(\ast)$ admits a solution at some finite horizon $[0,T]$. Can we extend to $[0,\infty)$?

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  • $\begingroup$ Does $\dot{\alpha}$ stand for the derivative of $\alpha$? $\endgroup$
    – user420828
    Jan 7, 2022 at 15:17
  • $\begingroup$ @Philo18 Exact. I corrected $\Phi'$ to $\dot{\Phi}$ to be consistent $\endgroup$
    – GJC20
    Jan 7, 2022 at 15:38

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