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Bounds on process of contingent claims in an intertemporal economy with proportional transaction costs and general preferences. (English) Zbl 0935.91014
The authors consider an economy with two securities, a bond with price \(B(t)\) and stock with price \(S(t)\) at time \(t\geq 0\). The bond pays no coupons, is default free and has price dynamics \(B_{t}=e^{rt}B_0\), \(t\geq 0\), where \(r\) is the constant rate of interest. The stock price is the diffusion process \[ S_{t}=S_0\exp\left\{\left( \mu-{\sigma^2\over 2}\right)t+ \sigma W_{t}\right\}, \] where the mean rate of return \(\mu\) and the volatility \(\sigma\) are constants such that \(\mu>r\), \(\sigma\neq 0\), \(W_{t}\) is a one-dimensional standard Brownian motion. The investor holds \(x_{t}\) dollars of the bond and \(y_{t}\) dollars of the stock at date \(t\), and consumes at the rate \(c_{t}\) dollars out of the bond account. The stock account process, starting with \(y_0=y\), is \[ y_{t}=y+\int_0^{t}\mu y_{\tau} d\tau+ \int_0^{t}\sigma y_{\tau} dW_{\tau} +L_{t}-M_{t}, \] where \(L_{t}\) represents the cumulative dollar amount transferred into the stock account and \(M_{t}\) the cumulative dollar amount transferred out of the stock account, \(L_0=M_0=0\). The bond account process, starting with \(x_0=x\), is \[ x_{t}=x+\int_0^{t}\{r x_{\tau} -c_{\tau}\} d\tau-\beta L_{t}+\alpha M_{t},\quad 0<\alpha<1<\beta. \] Analytic bounds on the reservation write price of European-type contingent claims are derived. The option prices are obtained via a utility maximization approach by comparing the maximized utilities with and without the contingent claim.

91B28 Finance etc. (MSC2000)
93E20 Optimal stochastic control
60G40 Stopping times; optimal stopping problems; gambling theory
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