Abstract
Letp(t, x, y) be the fundamental solution of a linear, second order partial differential equation of parabolic type. The functionI = −logp satisfies a nonlinear parabolic equation, which is the dynamic programming equation associated with a control problem of stochastic calculus of variations type. This gives a stochastic variational formula forp. The proof depends on a result of Molchanov about the asymptotic behavior ofp(t, x, y) for smallt.
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Communicated by A.V. Balakrishnan
This research was partially supported by NSF under Grant No. MCS 8121940 and by AFOSR under Grant No. AF-AFOSR-81-0116-C.
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Fleming, W.H., Sheu, SJ. Stochastic variational formula for fundamental solutions of parabolic PDE. Appl Math Optim 13, 193–204 (1985). https://doi.org/10.1007/BF01442207
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DOI: https://doi.org/10.1007/BF01442207