ITM Web Conf.
Volume 34, 2020International Conference on Applied Mathematics and Numerical Methods – third edition (ICAMNM 2020)
|Number of page(s)||8|
|Published online||03 December 2020|
Analytical solution to an LQG homing problem in two dimensions
Department of Mathematics and Industrial Engineering, Polytechnique Montréal, C.P. 6079, Succ. Centre-ville, Montréal, Québec H3C3A7, Canada
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An analytical solution is found to the problem of maximising the time spent in the ﬁrst quadrant by the two-dimensional diﬀusion process (X(t), Y(t)), where Y(t) is a controlled Brownian motion and X(t) is proportional to its integral. Moreover, we force the process to exit the ﬁrst quadrant through the y-axis. This type of problem is known as LQG homing and is very diﬃcult to solve explicitly, especially in two or more dimensions. Here the partial differential equation satisﬁed by a transformation of the value function is solved by making use of the method of separation of variables. The exact solution is expressed as an inﬁnite sum of Airy functions.
Key words: ﬁrst-exit time / Brownian motion / partial diﬀerential equations / method of separation of variables
© The Authors, published by EDP Sciences, 2020
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