[...]... consider as the domain of I only ys being less than the jump point of I to ∞ in the approximating method below Thus, the above condition (iii) does not narrow models which we can treat in this section Let {In }n≥1 be an increasing sequence of continuous functions which converges to I pointwise We take each In for n ≥ 2 to satisfy the following: I (x) = In (x) on R+ \Jn , I (x) ≥ In (x) on Jn and l k... Since each In is continuous, we can find a λn > 0 satisfying E[Φn (In (λn ϕ))] = δ How much can investors discount? 9 and 1 {δ + E[Ψn (λϕ)]} = EQ [In (λn ϕ)] λ>0 λ We shall prove a key lemma as follows: inf Lemma 1 There exists a random variable A such that Φn (In (λn ϕ)) → Φ(A) in L1 , taking a subsequence if necessary, that is, E[Φ(A)] = δ Proof We prove firstly the uniformly integrability of {Φn (In. .. continuously differentiable The mathematical model that appears in this case is a system of equations involving continuously differentiable functions in their left-hand sides and an equilibrium price is just a solution of the system If the numbers of equations and unknowns coincide (which does happen in systems arising in Debreu’s analysis) determinacy and regularity are equivalent properties by the inverse... measures In: Delbaen, F., Rasonyi, M., Stricker, C (eds.) Optimality and risk: modern trends in mathematical finance The Kabanov Festschrift Springer, Berlin (2009) 4 F¨ llmer, H., Leukert, P.: Quantile hedging Finance Stochast 3, 251–273 o (1999) 5 F¨ llmer, H., Schied, A.: Convex measures of risk and trading constraints o Finance Stochast 6, 429–447 (2002) 6 Karatzas, I., Kou, S.: Hedging American contingent... belongs to the image of F In case of a single-valued mapping we do not mention y when speak about regular or critical points Note that the definitions reduce to their classical counterparts if F is single-valued smooth The only difference worth noting is that regular and critical points in the last definitions are points of the graph of the mapping rather than elements in the domain space It easily follows... over U ∈ U under the constraint “x ≤ a constant” In particular, they solved it by using the Neyman–Pearson lemma In addition, they treated the problem minimizing the cost for a given probability of success, that is, minimizing x ∈ R such that there exists a U ∈ U satisfying P (x + U ≥ X) ≥ c, where c is a given constant in (0, 1) Moreover, the Black–Scholes model was discussed in [4] as a common example... a convex risk measure in the framework of bounded claims and discrete time trading Arai in [1] and [2] extended their result to the framework of Orlicz spaces and continuous time trading In this section, we focus on introducing robust representation results of ρl on Orlicz spaces Now, we need to prepare terminologies and concepts on Orlicz spaces A left-continuous non-decreasing convex non-trivial... of (i)–(iii) is actually valid even for set-valued mappings between metric spaces Certain forms of it were mentioned in some publications in the beginning of the 80s [9, 17], first proves were given in [4, 26], for a short prove see [18] For a C 1 mapping the implication (iv) ⇒ (i) is the famous Lusternik–Graves theorem actually contained in the main result of [15], the implication (iv) ⇒ (ii) likewise... prepare the following lemma: Lemma 2 Taking a subsequence if necessary, In (λn ϕ)) → A in L1 (Q), that is, EQ [A] = al (Q) How much can investors discount? 11 Proof Since Φ −1 is a continuous function, Lemma 1 implies that Φ −1 (Φn (In (λn ϕ))) → A a.s., by taking a subsequence if necessary For any ε1 > 0, there exists a sufficient large number n0 such that P ({ |In (λn ϕ) − Φ −1 (Φn (In (λn ϕ)))| < ε1... determinacy problems from the viewpoint of variational analysis We consider the same type of a problem as in the original paper by Debreu [6] in which the given data are individual demands (rather than utilities or preference relations) and we are not interested in how they have been obtained In particular we do not consider the question of existence of equilibrium prices which mathematically is not actually . free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The present volume of Advances in Mathematical Economics is a collection of. will also invite articles which might be considered too long for publication in journals. S. Kusuoka, T. Maruyama (Eds.) Advances in Mathematical Economics The Workshop on Mathematical Economics. new challenging stimuli from eco- nomic theories and those economists who are seeking effective mathematical tools for their research. The scope of Advances in Mathematical Economics includes,