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#### Keywords

- Black-Scholes model (1)
- Capital-at-Risk (1)
- Optimal portfolios (1)
- Value-at-Risk (1)
- crash modelling (1)
- equilibrium strategies (1)
- generalized inverse Gaussian diffusion (1)
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- log-utility (1)
- optimal portfolios (1)

We consider some continuous-time Markowitz type portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the Capital-at-Risk. In a Black-Scholes setting we obtain closed form explicit solutions and compare their form and implications to those of the classical continuous-time mean-variance problem. We also consider more general price processes which allow for larger uctuations in the returns.

We consider the determination of optimal portfolios under the threat of a crash. Our main assumption is that upper bounds for both the crash size and the number of crashes occurring before the time horizon are given. We make no probabilistic assumption on the crash size or the crash time distribution. The optimal strategies in the presence of a crash possibility are characterized by a balance problem between insurance against the crash and good performance in the crash-free situation. Explicit solutions for the log-utility case are given. Our main finding is that constant portfolios are no longer optimal ones.

We consider investment problems where an investor can invest in a savings account, stocks and bonds and tries to maximize her utility from terminal wealth. In contrast to the classical Merton problem we assume a stochastic interest rate. To solve the corresponding control problems it is necessary to prove averi cation theorem without the usual Lipschitz assumptions.

In the Black-Scholes type financial market, the risky asset S 1 ( ) is supposed to satisfy dS 1 ( t ) = S 1 ( t )( b ( t ) dt + Sigma ( t ) dW ( t ) where W ( ) is a Brownian motion. The processes b ( ), Sigma ( ) are progressively measurable with respect to the filtration generated by W ( ). They are known as the mean rate of return and the volatility respectively. A portfolio is described by a progressively measurable processes Pi1 ( ), where Pi1 ( t ) gives the amount invested in the risky asset at the time t. Typically, the optimal portfolio Pi1 ( ) (that, which maximizes the expected utility), depends at the time t, among other quantities, on b ( t ) meaning that the mean rate of return shall be known in order to follow the optimal trading strategy. However, in a real-world market, no direct observation of this quantity is possible since the available information comes from the behavior of the stock prices which gives a noisy observation of b ( ). In the present work, we consider the optimal portfolio selection which uses only the observation of stock prices.