The purpose of this thesis has been to combine stochastic analysis, functional analysis, measure- and integration theory, real analysis and convex analysis to investigate, and prove, some results in mathematical nance. Background theory, such as convexity, convex analysis, the conjugate duality framework of Rockafellar [34], Lagrange duality and the general stochastic analysis frame-work for modeling a nancial market, has been covered. Then, these results and frameworks have been applied to various central problems in mathematical nance, for instance convex risk measures, utility maximization, pricing, and arbitrage problems. I believe the results are interesting, and that they indicate the potential of exploiting duality in mathematical nance, and more generally, in stochastic control problems.
A central technique of this thesis can be roughly summarized as follows:
Given a problem in mathematical nance, formulate the problem as an op-timization problem in a suitable space. Typically, it will be a constrained stochastic optimization problem. Call this the primal problem. Try to solve the primal problem by standard optimization techniques.
If one is unable to solve this primal problem as it is: Find a dual to the problem. The optimal value of the dual problem will provide bounds on the optimal value of the primal problem. Try to solve this dual problem.
If one is unable to solve this dual problem as well, it may help to transform either the primal or the dual problem (by basic algebra, exploiting that some inequalities must hold with equality in the optimum etc.).
Try to show that there is no duality gap, i.e. that the optimal primal and dual values coincide. There are hopes of this if one is working with a convex objective function and convex constraints.
However, as this thesis has shown, there are many types of duality and some are simpler to handle than others. For example, linear programming is
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less complicated than the conjugate duality framework. So how can one know which duality method to try?
If the time is nite and discrete, the scenario space is nite, and the ob-jective function as well as the constraints are linear: Linear programming (LP) can be applied. There will not be a duality gap from the linear pro-gramming duality theorem. Note that sometimes it may be easier to nd the Lagrange dual problem than nding the LP dual (LP is a special case of Lagrange duality).
If there are nitely many constraints (even though the scenario space is arbitrary and time is continuous) Lagrange duality can be applied. If the objective function and constraint functions are convex, the Slater condi-tion can be used to show that there is no duality gap. If the constraints are linear, this boils down to checking whether there exists a feasible solution to the primal problem.
If there are arbitrarily many constraints (for instance one constraint for eachω∈Ω), the conjugate duality framework may work. There are several ways to dene the perturbation space and the perturbation function, and hence there are several dierent dual problems. It may be necessary to try dierent perturbation spaces.
If the objective function and constraints are convex, there is hope that there is no duality gap. In order to show this, one must essentially show that the optimal value function ϕ is lower semi-continuous. However, there are several other theorems guaranteeing this, for instance the gen-eralized Slater condition of Example 4 in Rockafellar and Theorem 9 (Theorem 6.9) in Pennanen and Perkkiö [30].
Though closing the duality gap is a desirable result in many examples, it is not always possible. However, all is not lost. Even if one cannot close the duality gap, the dual problem gives bounds on the optimal primal value, and an iterative method where one computes primal and dual solutions every other time will give an interval where the optimal primal (and dual) value(s) must be.
Also, note the elegance and step-by-step approach that duality methods bring to many well-known problems in mathematical nance. Another advantage regarding most of the duality approaches of this thesis is that one does not have to make a lot of assumptions on the market structure.
However, even though duality methods provide a step-by-step approach, many diculties can occur. A lot of stochastic and functional analysis is nec-essary in order to study the functions that arise naturally from duality. When working with conjugate duality, it may be dicult to choose appropriate paired spaces, and an appropriate perturbation function, in order to attain useful prop-erties assuring, for instance, no duality gap. Sometimes, the primal and dual problems have to be transformed in order to apply duality methods, or per-haps in order to get results that can be interpreted. These transformations may require a lot of stochastic analysis, measure theory, and clever observations.
163 Many results in this thesis, in particular the results in the nal chapters, have been proved in discrete time. The reason for this is that this makes it simpler to apply the conjugate duality framework of Rockafellar [34]. However, it may be possible to generalize the results presented here to continuous time, possibly using a discrete time approximation. However, this is beyond the grasp of this thesis, and open for further research.
I hope this thesis has enlightened, and structured, the vast and (at least to me in the beginning) rather confusing topic of duality methods in mathematical nance. A topic which I feel has a lot of potential still to be exploited.