Some estimates on exponentials of solutions to stochastic differential equations (Q1773285)

The author proves two ``negative'' results connected with Novikov condition of exponential integrability. One of them concerns the exponential estimate for the integral functional of the short interest rate satisfying the general Hull-White model, and the other deals with similar exponential estimate for risk premium. These results tell us that in the context involving equivalent matingale measures and/or arbitrage freeness, one has to be careful to use Hull-White model as the short-interest rate model. Then, the general problem is formulated. Connections with mathematical finance serve as its motivations. The author considers the exponential estimates for the solution of multidimensional stochastic differential equation of the form \(dX(t)=b(t,X(t))dt+\sigma(t,X(t))dW(t)\), \(t\geq 0\). The conditions on the coefficients and on the form of the functional are established to guarantee the finiteness of the mathematical expectations like \(E[\sup_{t\in[0,T]} e^{\varphi(X(t))}]\), where \(\varphi\) is some functional.