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Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark. (English) Zbl 1047.91025
The author considers the model which consits of following \(k+2\) underlying processes: 1) \(k\) risky assets \(S_{t}^{(i)}, i=\overline{1,k}\), which satisfies the stochastic differential equation \[ dS_{t}^{(i)}=\mu_{i}S_{t}^{(i)}dt +\sum_{j=1}^{k}\sigma_{ij}S_{t}^{(i)}dW_{t}^{(j)}, i=\overline{1,k}, \] where \(\mu_{i}, i=\overline{1,k}, \sigma_{ij}, i,j=\overline{1,k}\) are constants, and \(W_{t}^{(j)},j=\overline{1,k}\) are standard independent Brownian motions; 2) The price \(B_{t}\) of the risk-free asset is assumed to evolve according to \(dB_{t}=rB_{t}dt\), where \(r\geq 0, \mu_{i}>r, i=\overline{1,k}\); 3) The stochastic benchmark \(Y_{t}\), which satisfies the stochastic differential equation \[ dY_{t}=\alpha Y_{t} dt+ \sum_{i=1}^{k}b_{i}Y_{t}dW_{t}^{(i)}+\beta Y_{t} dW_{t}^{(k+1)}, \] where \(W_{t}^{(k+1)}\) is the Brownian motion independent on \(W_{t}^{(i)}, i=\overline{1,k}\). The investor may invest only in the risky stocks and the bond. Let \(X_{t}^{f}\) denote the wealth of the investor at time \(t\), if it follows policy \(f=(f_{t}^{(i)},t\geq0, i=\overline{1,k})\), where \(f_{t}^{(i)}\) is the proportion of the investor’s wealth invested in the risky stock \(i\) at time \(t\). For numbers \(l,u\) with \(lY_{0}<X_{0}<uY_{0}\) we say that performance goal \(u\) is reached if \(X_{t}^{f}=uY_{t}\), for some \(t>0\) and that performance shortfall level \(l\) occurs if \(X_{t}^{f}=lY_{t}\), for some \(t>0\). Author considers following active portfolio management problems: 1) Maximizing the probability performance goal \(u\) is reached befor shortfall \(l\) occurs; 2) Minimizing the expected time until the performance goal \(u\) is reached; 3) Maximizing the expected time until shortfall \(l\) is reached; 4) Maximizing the expected discounted reward obtained upon achieving goal \(u\); 5) Minimizing the expected discounted penalty paid upon falling to shortfall level \(l\).

91G10 Portfolio theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E20 Optimal stochastic control
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