Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Inverse First Passage Time Problem for killed Brownian motion   The classical inverse first passage time problem asks whether, for a Brownian
motion $(B_t)_{t\geq 0}$ and a positive random variable $\xi$, there exists a
barrier $b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{P}\{B_s>b(s), 0\leq s
\leq t\}=\mathbb{P}\{\xi>t\}$, for all $t\geq 0$. We study a variant of the
inverse first passage time problem for killed Brownian motion. We show that if
$\lambda>0$ is a killing rate parameter and $\mathbb{1}_{(-\infty,0]}$ is the
indicator of the set $(-\infty,0]$ then, under certain compatibility
assumptions, there exists a unique continuous function
$b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{E}\left[-\lambda \int_0^t
\mathbb{1}_{(-\infty,0]}(B_s-b(s))\,ds\right] = \mathbb{P}\{\zeta>t\}$ holds
for all $t\geq 0$. This is a significant improvement of a result of the first
two authors (Annals of Applied Probability 24(1):1--33, 2014).
  The main difficulty arises because $\mathbb{1}_{(-\infty,0]}$ is
discontinuous. We associate a semi-linear parabolic partial differential
equation (PDE) coupled with an integral constraint to this version of the
inverse first passage time problem. We prove the existence and uniqueness of
weak solutions to this constrained PDE system. In addition, we use the recent
Feynman-Kac representation results of Glau (Finance and Stochastics
20(4):1021--1059, 2016) to prove that the weak solutions give the correct
probabilistic interpretation.

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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: cs
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat