Generalized Bounds for Convex Multistage Stochastic Programs
This work was completed during my tenure as a scientific assistant and d- toral student at the Institute for Operations Research at the University of St. Gallen. During that time, I was involved in several industry projects in the field of power managemen
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Daniel Kuhn
Generalized Bounds for Convex Multistage Stochastic Programs
(Q,B(Q)), where f2 denotes the state space comprising the so-called observations. In the context of stochastic programming, Q is usually taken to be a compact subset of R M , and B{Q) denotes the Borel field1 of the state space. Furthermore, a stochastic process is understood to be a family of random variables {w t } t€T , (f2t,B(f2t))- Again, fit is supposed to be a compact subset of MM*. By convention, a stochastic process is called Tl-adapted or T1'-previsible if u>t is measurable with respect to T1 or Jrt~1, respectively. In many applications, u>o is measurable with respect to the trivial cr-algebra {0, Q}. This implies roughly that there is no information at time 0. By definition, a sequence of {0, f?}-measurable random variables represents a deterministic process. A filtration {.F'}teT is said to be induced by a process {d)t}4€T if JF1 coincides with the (T-field generated by the sets \J-S^Q{G)~X{A)\A G B(QS)}. The induced cr-algebra J-1 describes the information which is available at time t by only observing the underlying process. Figure 2.1 shows an exemplary discrete stochastic process {6Jt}t€r with four time steps. Every different path of the corresponding scenario tree, i.e. every possible sequence of observations, is assigned to one element of the sample space, which is chosen to be Q = {o>i,... ,&&}• The atoms of the induced corresponding to the stochastic
process
{ujt}teT. • • •
x
x
• • •
(2.1)
x
Every element of the state space J? can be identified with an equivalence class of outcomes in the sample space. Thus, {£2,7^) naturally inherits the probability measure defined on (&,F) through P(A) := P(u)~1(A)), A G J- '•= B{Q). By construction, P is a regular probability measure on (fi,T). Moreover, the state space is equipped with a filtration {J-"*}tgr, which is given through {A x Qt+i x • • • x QT \ A e
x
•x
Qt)}.
Instead of the abstract probability space (/), P, P), we may equivalently consider the induced probability space [Q,T, P), which we will use henceforth. With this convention, Gs reduces to the identity map, while u>t becomes a specific coordinate projection for every t G r. Moreover, the terms 'outcome' and 'observation' will from now on be used synonymously. From a conceptual point of view, it is important to distinguish random variables u>t and their realizations, which will be denoted by oJt below. By convention, E(-) denotes expectation over the probability measure P. Conditional expectations Et(-) '•= E^P) on (i?,^7, P) are defined up to an equivalence relation; i.e. there can be many versions of Et(-), which differ on P-null sets. In this work, Et(-) is taken to be a regular conditional expectation being representable as an indefinite integral with respect to a regular conditional probability. Such regular conditional probabilities exist since T is the Borel field on Q and P is a regular Borel measure [66, Sect. 27].
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2 Basic Theory of Stochastic Optimization
2.2 Policies In the sequel w
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