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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.

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