Computing Near Optimal Strategies for Stochastic Investment Planning Problems

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4 where is the expected added gain for investing in the second unit of commodity for represents the difference from investing in the second unit of commodity at compared to the optimal one-unit strategy. It consists of two terms: the expected gain from holding the commodity for one step plus a correction term (a kind of expected loss or negative gain) for holding it one more step in the case we would like to sell it but sell constraint does not allow us to do that. Note that depends solely on the price of a commodity and also The other important feature is that by substituting (p, 0) from Equation 3, the value equals: which means that the integral part of the expression now becomes independent of the commodity held in the first step and disregards any units we were not able to sell. Intuitively, under expectations, the term allows us to pretend that we were able to sell both units at the end of the previous step without restriction, though in reality when the price climbs above we have to keep one unit and sell it later when the sell capacity is available. Using the fact that allows us to disregard any commodity left from previous steps we can express the value function for an arbitrary policy as For zero units of commodity the policy always equals its generating (see equation 5) and therefore (6) The theorem shows how to get the optimal policy from TI for the zero commodity start state. This policy can be applied also to the non-zero commodity state in a straightforward way. Up to this point we know how to find the best (optimal) policy from This policy is no worse than that trades only one unit (simply because However, this does not necessarily imply that the optimal two-unit strategy is also globally optimal. Theorem 3 is the optimal two-unit strategy. Proof To prove this we need to show that the optimal policy always resides in The opposite can happen only when: (1) the optimal strategy is not incremental (the optimal choice for the first unit does not imply the optimal choice for two units case) and (2) the optimal strategy depends on c, such that the relation is not captured by Equation 5 and relation). The incremental property follows from the fact that the maximum gain we can capture by any unit is which equals the expected gain for the first unit. Thus the policy must always trade the first unit optimally. Using the incremental result, the relation in Equation 5 can be violated only when the globally optimal policy at some point recommends to hold 2 units and does not, or vice versa. However, this cannot happen as it would mean that we choose to hold the second unit when is negative or not to hold it when it is positive. Thus must be the optimal policy. Threshold price for the second unit The optimal two-unit strategy says that we want to invest into the second unit only when The question now is if we can come up with a compact representation of this condition, similarly to the threshold price for the one-unit case. Indeed, we can show that if ever becomes positive, there is a unique threshold value such that when then is guaranteed. This follows directly from the monotonicity of which we prove next. Theorem 4 For it holds Proof In order to prove it is sufficient to show, using Equation 6, that for This means that if we want to find the optimal we can do it by simply finding the optimal from from Theorem 2 The optimal two-unit strategy maximizing is defined by a strategy maximizing Proof Using Equation 7, and the fact that the integral part becomes independent of the amount of commodity we actually held in the previous step, we can maximize the value of a policy by maximizing the sum of expected gains. That is: hold. This is trivial and follows from the monotonicity of and the fact that and define normal distributions with means and the same standard deviation. The main consequence of being monotonically decreasing is that there is a unique zero point such that for any 0. Therefore, the optimal generating strategy for trading two units of commodity (with the sell constraint 1) can be defined compactly using a set of threshold prices such that HAUSKRECHT, PANDURANGAN, AND UPFAL 1313

5 The optimal strategy is then compactly represented as: 5.4 Trading -units under sell limit In principle the same ideas as used for the two-unit case can be applied to find the optimal strategy for trading at most units of commodity. Such a policy is guaranteed to be no worse than the policy for trading smaller number of units. We summarize the main results and conclusions, the detailed analysis is deferred to the full paper. The expected added gain from holding unit of commodity is defined recursively as: A nice property of is that it is monotonically decreasing in p and The optimal generating policy equals: can be described compactly using a set of threshold values such that: where thresholds are unique zero points The optimal policy for the sell limit 1 can be then compactly represented as: 5.5 Solution for buy, sell and store limits The optimal policy for at most units of commodity and the sell constraint can be derived directly from the solution for the sell limit 1: Adding buy and store constraints to this result is easy and results in the following policy: 6 Finding optimal thresholds (8) The optimal policy for units can be represented compactly using a set of threshold prices The threshold price for unit is the zero of the the expected added gain for the unit = 0. As the value of depends only on threshold prices the set of threshold prices can be built incrementally. The main problem in this process is that there is no closed form solution for finding the zero point of Figure 1: The optimal generating policy with sell limit 1, 100, 0.6, 10, , 0.2. Stepwise changes reflect positions of thresholds. is the only exception). Thus, in order to find the threshold values for we resort to stochastic (Monte Carlo) approximation techniques (see [Kushner and Yin, 1997]). In particular we use the Robbins-Monro scheme, which finds the zero of the function iteratively using the update where, is the price value at the iteration, is the "noisy" estimate of the function at obtained by Monte Carlo sampling from Equation 8. The sequence e is used to "average out" the "noisy" estimates from the Monte-Carlo sampling. We use the standard sequence, which converges (under reasonable assumptions) to the zero value with probability one. Since is monotonous there is a unique root such that [Kushner and Yin, 1997] also give more details on the rate of convergence. The process for finding new thresholds can be applied incrementally to find the solution for an arbitrary number of units. The issue that remains open is that there can be an infinite number of thresholds to define the complete solution. However, thresholds for larger values of are less likely to be used, as they cover the range of prices with extremely small chance of occurrence. Thus if the initial price is in a reasonable range more complex policies tend to contribute less. To provide for more robustness, we propose an on-line algorithm that keeps building thresholds on the demand basis. That is, only when price encountered in not covered by a current set of thresholds, we start to work on thresholds for higher values of Note that this algorithm can be further refined into an anytime scheme [Dean, 1991], suitable for time critical settings. 7 Experimental results We have tested our approach on several sets of parameters. Figure 1 illustrates a typical policy. To find the zero points of expected added gain functions, we used the basic 1314 UNCERTAINTY AND PROBABILISTIC REASONING

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