Some approximation issues in the mathematics of finance

This thesis addresses two approximation problems. The first one is the problem of approximating random variables, in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale X. The criterion is minimization of L2-distance, or "least-squares". This problem has a straightforward and well-known solution when X is a Brownian motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure P. We address the general, semimartingale case by means of a duality approach; the adjoint variables in this duality are signed measures, absolutely continuous with respect to P, under which X behaves like a martingale. It is shown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it "has no gap" (i.e., the values of the primal and dual problems coincide); that the signed measure which is optimal for the dual problem can be easily identified whenever it exists; and that the duality is also "strong", in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented here both simplifies and extends the extant work on the subject. It has also natural connections and interpretations in terms of the theory of "variance-optimal" and "mean-variance efficient" portfolios in Mathematical Finance, pioneered by H. Markowitz and then greatly extended by H. Follmer, D. Sondermann and most notably M. Schweizer.;The second problem is to approximate the early exercise boundary for American Put options. There is no known explicit computation for this free-boundary problem, so one has to resort to representations and to approximations. We derive a new integral representation for the boundary, as well as for calculating the prices of American Put options. Our new representation is a one-dimensional integral equation, which constitutes a reduction in dimension from the commonly used integral representation. Our new representation is shown to have some different numerical properties and is possibly more tractable analytically than the integral representation currently found in the literature.