## Most likely path to the shortfall risk under the optimal hedging strategy

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In this dissertation, we search for the most likely path to the shortfall risk in hedging a long-term supply commitment with short-term futures contracts, which leads to a class of optimization problems. Motivated by a simple model initially discussed in Culp and Miller [3], Mello and Parsons [18] and Glasserman [10], the optimal hedging strategy provided by Larcher and Leobacher [12], and the simple discussion about the most likely path by Glasserman [10], we studied the following optimization problem: $$\min_{\phi \in \mathcal{A}*{x}} \frac{1}{2} \int_0^1 \left[ \dot{\phi}(t) \right]^2 dt$$ where $$\mathcal{A}*{x}=\left{ \phi: \sigma \int_0^t [G(s)+s-t] dW_s \leq -x, textrm{for some