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Unformatted text preview: stic Optimal Control: ICDNS Using diﬀusions to solve a large class of control problems Without proof (though it is not complicated: see [3]) it turns out
2
that ∂t ψ = V − b T ∂x − 1 Tr (ν (t , x , u )∂x ) ψ can be solved by
λ
2
solving the diﬀusion process:
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∂t ρ = − V − ∂x (b ρ) + 1 ij νij ∂ x∂∂ xj ρ . We discover we can use
λ
2
i
solutions ρ of the preceding to ﬁnd J (x , t ) using the form
J (x , t ) = −λlog dy ρ(y , T x , t )exp (−φ(y )/λ).
This is great: we can thus evolve ρ forward in time to solve for
J (x , t ). In particular we can construct J (x , t ) by running Monte
Carlo simulations consistent with the dynamics of ρ (initialized at
x , t ) and then taking their weighted sum at time T . Nick Jones Elements of Stochastic Optimal Control: ICDNS Using diﬀusions to solve a large class of control problems II ∂t ρ = − V − ∂x (b ρ) +
λ 1
2 2 ij νij ∂ x∂∂ xj ρ .
i J (x , t ) = −λlog dy ρ(y , T x , t )exp (−φ(y )/λ).
This is great: we can thus evolve ρ forward in time to solve for
J (x , t ). In particular we can construct J (x , t ) by running Monte
Carlo simulations consistent with the dynamics of ρ (initialized at
x , t ) and then taking their weighted sum at time T . We thus
initialize a particle i at (x , t ), record its location yi at T and
weight it by exp(−φ(yi )/λ). If a particle is abs...
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 Spring '14
 NickJones
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