Propagation of convexity by Markovian and martingalian semigroups
This paper is an extract from the author's Ph. D. thesis ``Propagation de la convexité par des semigroupes markoviens and martingaliens sur la demi-droite'' (Université d'Evry, 1995). The central question is to find criteria guaranteeing that a Markovian operator \(Q\) maps convex functions on the half-line \((0,\infty)\) into convex functions. This question originates from the study of volatility risks in mathematical finance. The present paper is of a purely theoretical nature; for connections with mathematical finance the reader is referred to the author's thesis. Let \(Q\) be a linear operator on a vector space \(G\) of functions over \((0,\infty)\) which contains all piecewise linear functions. A particular choice will be \[ G:= \Biggl\{g\in C((0,\infty)):\|g\|:= \sup{|g(x)\over 1+ x}< \infty\Biggr\}. \] The operator \(Q\) is called positive if \(f\geq 0\Rightarrow Qf\geq 0\), Markovian if \(Q1=1\) and martingalian if \(Qx=x\). \(Q\) is said to propagate convexity if the associated call function \(C_Q(x,y):= Q(\cdot-y)^+(x)\) is convex as a function of \(x\) (or, equivalently, if the put function \(P_Q(x, y):= Q(y-\cdot)^+(x)\) is convex in \(x\)); the convexity of \(C_Q(x,\cdot)\) is always fulfilled. If \(Q\) propagates convexity, the second (distributional) derivative \(\partial^2_z C_Q(dz, y)\) exists and induces a kernel \(\widehat Q(y, dz)\) which, in turn, allows to associate with \(Q\) the call-put dual operator via \[ \widehat Q f(y):= \text{boundary values at }0 \& \infty+ \int f(z) \widehat Q(y,dz). \] The relation between \(Q\) and \(\widehat Q\) is given by \[ \widehat Q(\cdot-x)^+(y)= Q(y-\cdot)^+(x).\tag{\(*\)} \] The main result of the first section is that a positive, Markovian and martingalian \(Q\) propagates convexity if and only if \(\widehat Q\) has the same properties. In the second section the author studies semigroups of positive, Markovian and martingalian operators \(\{Q_t\}_{t\geq 0}\) defined on the space \(G\). Notice that \((*)\) is an example of Liggett's \(h\)-duality of semigroups for \(h(x,y)= (x- y)^+\). Such semigroups are always strongly continuous on \(Q_0G\); the case where \(Q_0\neq \text{id}\) is studied in some detail. The semigroup \(\{Q_t\}_{t\geq 0}\) is said to propagate convexity if each operator \(Q_t\) does. This property is then expressed in terms of the infinitesimal generator \(A\) of the semigroup and its Banach-space adjoint \(A^*\). Moreover, if the semigroup propagates convexity, the generator satisfies a Lévy-Khinchin type decomposition and so does the call-put adjoint operator if the Radon measures of the form \(\varphi(x)dx\), \(\varphi\in C^\infty_c((0,\infty))\), are an operator core for \(A^*\). The paper closes with some concrete applications and examples, in particular, to one-dimensional stochastic differential equations of the form \(dS_t= \sigma(S_t)S_t dB_t\) (\(B_t\) is Brownian motion) with bounded, measurable \(\sigma(\cdot)\). In this case, the semigroup is symmetric and admits a call-put dual which coincides with the original semigroup. The call functions \(C_{Q_t}(\cdot, y)\) are continuous and, if \(\sigma\) is continuous, of class \(C^1\).
scientific article; zbMATH DE number 1286663