Stochastic stability of fractional \((B,S)\)-securities markets (Q2784988)

The authors investigate stochastic stability of fractional \((B,S)\)-markets, i.e. financial markets with stochastic behavior that is caused by a random process with long-range dependence (fractional Brownian motion). Three financial models are considered. The model of fractional \((B,S)\)-market proposed by \textit{M. Zähle} [Probab. Theory Relat. Fields 111, No. 3, 333-374 (1998; Zbl 0918.60037)] is defined by the system of SDE NEWLINE\[NEWLINEdB_t=rB_t dt, \quad dS_t=S_t(\mu dt+\sigma \delta B_t^H), \;H\in(0,1),NEWLINE\]NEWLINE where \(B_t\) and \(S_t\) correspond to bonds and stocks, resp., \(r\) is the interest rate, \(\mu\in R\) is the appreciation rate, \(\sigma\) is the volativity coefficient. NEWLINENEWLINENEWLINEIn the model of fractional \((B,S)\)-securities market proposed by \textit{Y. Hu} and \textit{B. Øksendal} [``Fractional white noise analysis and applications to finance'' (Preprint, Univ. Oslo, 1990)] the bond price \(B(t)\) is described by the equation \(dB(t)=rB(t) dt\), \(B(0)=1\), \(0\leq t\leq T\), and the stock price \(S(t)\) satisfies the equation \(dS(t)=\mu S(t) dt+\sigma S(t) dB_t^H\), \(S(0)=S_0\), \(H\in(1/2,1).\) In the fractional \((B,S)\)-securities market in the scheme by \textit{R. Elliott} and \textit{I. Van der Hoek} [``A general fractional white noise theory and applications to finance'' (Preprint, Univ. Adelaide, 2000)] the bond price \(B(t)\) is described by the equation \(dB(t)=rB(t) dt\), \(B(0)=1\), \(0\leq t\leq T\), and the stock price \(S(t)\) satisfies the equation \(dS(t)=\mu S(t) dt+\sigma S(t) dB_M(t)\), \(S(0)=S_0,\) where \(B_M(t)\) is a linear combination of fractional Brownian motions. NEWLINENEWLINENEWLINEThese models of financial market appeared as a result of different approaches to the notion of stochastic integral with respect to fractional Brownian motion. Stochastic stability of fractional markets with jumps is also considered.