Stochastic control in discrete and continuous time by Atle Seierstad

By Atle Seierstad

This e-book offers a accomplished advent to stochastic regulate difficulties in discrete and non-stop time. the fabric is gifted logically, starting with the discrete-time case prior to continuing to the stochastic continuous-time types. significant subject matters are dynamic programming in discrete time and HJB-equations in non-stop time. issues lined comprise stochastic greatest ideas for discrete time and non-stop time, even for issues of terminal stipulations. quite a few illustrative examples and workouts, with recommendations on the finish of the publication, are incorporated to augment the knowledge of the reader. through interlinking many fields in stochastic keep watch over, the fabric supplies the coed the chance to determine the connections among varied fields and the underlying rules that unify them.

This textual content will profit scholars in utilized arithmetic, economics, engineering, and similar fields. necessities comprise a direction in calculus and straightforward chance concept. No wisdom of degree idea is assumed.

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Example text

Let {us } be any given sequence ut , . . ” Let zt1 and zt2 , t < t be two sequences 1 ,z ,Z of values of zt ,t < t. For uˆs (zt , Zt+1 , . . , Zs ) := us (z10 , . . , zt−1 t t+1 , . . , Zs ), 2 2 1 1 we have that J{uˆs } (t, z0 , . . , zt−1 , zt ) = J{us } (t, z0 , . . ) and us give the same Xs ’s, s > t, given zt and the zt1 ’s. Then, surely, we 2 ,z ) ≥ J 1 1 2 2 have J(t, z20 , . . , zt−1 t {us } (t, z0 , . . , zt−1 , zt ), hence even J(t, z0 , . . , zt−1 , zt ) ≥ 1 1 J(t, z0 , . .

We saw above that a Markov control that is optimal among all Markov controls is also optimal among all history-dependent controls. Let us compute the effect on the criterion by changing from the sequence { u∗s : s = 0, 1, . . , T } to the sequence { us : s = 0, 1, . . , T }. 1). This will be the same as the path obtained when wt (Zt∗ ) + ut∗ (Zt∗ ) is inserted, where Zt∗ = (Xt∗ (V→t ),Vt ). We then obtain that T T ∆ := E ∑ f0 (t, Xt , ut∗ (Zt∗ ) + wt (Zt∗ )) − E ∑ f0 (t, Xt∗ , ut∗ (Zt∗ )) t=0 t=0 T = E ∑ f0 (t, Xt , ut∗ (Zt∗ ) + wt (Zt∗ )) − f0 (t, Xt∗ , ut∗ (Zt∗ )) .

Now, given a sequence of optimal policies {ut (yt , vt )}t , in the current case the term E[J(t,Yt ,Vt )|yt−1 , vt−1 ] in the optimality equation equals +1 F(s,Ys , us (Ys ,Vs ),Vs ))|yt−1 , vt−1 ]. , disregarding the term ut (yt , vt ) = ut (ut−1 , vt ), we get that 0 = F3 (t − 1, yt−1 , ut−1 , vt−1 ) + E[F2 (t, ut−1 , ut (yt ,Vt ),Vt )|yt−1 , vt−1 ]. 31). 17 (History dependence versus Markov dependence*). Why don’t we need history-dependent controls? Some new arguments will now be presented. ) = 0, xt is a constant, which we then ignore.

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