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We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If <italic>X</italic>(<italic>t</italic>) is a one-dimensional diffusion with jumps, starting from a random position <italic>η</italic> ∈ [<italic>a, b</italic>], let be <italic>τ</italic>a,b the time at which <italic>X</italic>(<italic>t</italic>) first exits the interval (<italic>a, b</italic>), and <italic>π</italic>a = <italic>P</italic> (<italic>X</italic>(<italic>τ</italic>a,b) ≤ <italic>a</italic>) the probability of exit from the left of (<italic>a, b</italic>). Given a probability <italic>q</italic> ∈ (0, 1), the problem consists in finding the density <italic>g</italic> of <italic>η</italic> (if it exists) such that <italic>π</italic>a = <italic>q</italic>; it can be seen as a problem of optimization.
</abstract>

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If X(t) is a one-dimensional diffusion with jumps, starting from a random position η ∈ [a, b], let be τa,b the time at which X(t) first exits the interval (a, b), and πa = P (X(τa,b) ≤ a) the probability of exit from the left of (a, b). Given a probability q ∈ (0, 1), the problem consists in finding the density g of η (if it exists) such that πa = q; it can be seen as a problem of optimization.

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

Control and Cybernetics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a...

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

Published Online: Aug 12, 2022

Page range: 31 - 42

Received: Nov 01, 2020

Accepted: Feb 01, 2022

DOI: https://doi.org/10.2478/candc-2022-0003

Keywords
jump-diffusion process, first-passage place, inverse first-passage place problem

© 2022 Mario Abundo, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

Published Online: Aug 12, 2022

Page range: 31 - 42

Received: Nov 01, 2020

Accepted: Feb 01, 2022

DOI: https://doi.org/10.2478/candc-2022-0003

Keywordsjump-diffusion process, first-passage place, inverse first-passage place problem

© 2022 Mario Abundo, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
jump-diffusion process, first-passage place, inverse first-passage place problem