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B0635
Title: Existence in the inverse Shiryaev problem Authors:  Yoann Potiron - Keio University (Japan) [presenting]
Abstract: The inverse first-passage Shiryaev problem is considered, i.e. for $(W_t)_{t \geq 0}$ a standard Brownian motion and any upper boundary continuous function $g: \mathbb{R}^+ \rightarrow \mathbb{R}$ satisfying $g(0) \geq 0$, we define $\tau_g^W := \inf \{t \in \mathbb{R}^+ \text{ s.t. } W_t \geq g(t)\},$ and $f_g^W (t)$ its related density. For any target density function of the form $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ satisfying some smooth assumptions and any arbitrarily big horizon time $T > 0$, we show the existence of a related boundary $g_{f,T} : \mathbb{R}^+ \rightarrow [0,T]$, with $g_{f,T}(0) \geq 0$, which satisfies $f_{g_{f,T}}^W (t) = f(t)$ for $0 \leq t \leq T.$ As an example, the exponential distribution $f(t) = \lambda \exp (- \lambda t)$ for $\lambda > 0$ satisfies the assumptions. As $g_{f,T}$ is exhibited as a limit boundary of a subsequence of a piecewise linear boundary, we do not obtain any explicit formula for $g_{f,T}$ as a function of $f$, nor the unicity of the solution. The results are also proved in the symmetrical two-dimensional boundary case.