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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 $$l$$.

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