In this note, we show how the dual approach in its particular form presented in Andersen and Broadie (Manag. Sci. 50:1222–1234, 2004) can be fitted into the framework of the recent work (Belomestny et al., Finance Stoch. 17:717–742, 2013).
We consider an optimal control problem for a linear stochastic integro-differential equation with conic constraints on the phase variable and with the control of singular–regular type. Our setting includes consumption-investment problems for models of financial markets in the presence of proportional transaction costs, where the prices of the assets are given by a geometric Lévy process, and the investor is allowed to take short positions. We prove that the Bellman function of the problem is a viscosity solution of an HJB equation. A uniqueness theorem for the solution of the latter is established. Special attention is paid to the dynamic programming principle.
We investigate models with negative risk sums when the company invests its reserve into a risky asset whose price follows a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to one identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome, namely the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.
A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (NA1NA1) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under NA1NA1, a local martingale numéraire may fail to exist. In this work, we establish that under NA1NA1, a supermartingale numéraire under the original probability PP becomes a local martingale numéraire for equivalent probabilities arbitrarily close to PP in the total variation distance.