We analyse optimal portfolio selection problem of maximizing the utility of an agent who invests in a stock and money market account in the presence of proportional transaction cost $\lambda>0$ and foreign exchange rate. The stock price follows a (generalized) Geometric It\^{o}-L\'{e}vy process. The utility function is $U(c)={c^{p}}/{p}$ for all $c\geq0$, $p<1$, $p\neq0$.

We consider a problem of maximizing the utility of an agent who invests in a stock, money market account and an index bond incorporating life insurance, deterministic income, and consumption. The stock is assumed to be a generalized geometric Itô-Lévy process. Assuming a power utility function, we determine the optimal investment-consumption-insurance strategy under inflation risk for the investor in a jump-diffusion setting using martingale approach.

We find the possible risk minimizing portfolio strategies in a two dimensional market consisting of a risk asset and risk-less asset. The investor in the market is subjected to consumption, purchasing of life insurance and stochastic income with inflation risk. The problem is formulated as zero sum game problem between the market and the investor. The strategies are determined for the different generations of the life of an investor, that is before the investor dies and after the investor dies. We used the concept of convex risk measures and monetary utility maximizing problem-concept studied before finding the risk minimizing portfolios which was solved using the game theoretic approach to obtain the strategies explicitly given in the propositions in the study.

We analyze the optimal portfolio selection problem of maximizing the utility of an agent who invests in a stock and a money market account in the presence of transaction costs. The stock price follows a geometric process. The preference of the investor is assumed to follow the constant relative risk aversion (CRRA). We further investigate the risk minimizing portfolio through a zero-sum stochastic differential game (SDG). To solve this two-player SDG we use the Hamilton–Jacobi–Bellman–Isaacs (HJBI) for general zero-sum SDG in a jump setting.

We consider a problem of maximizing the utility of an agent who invests in a stock and a money market account incorporating proportional transaction costs λ > 0 and foreign exchange rate fluctuations. Assuming a HARA utility function U c = c p / p for all c ≥ 0 , p < 1 , p ≠ 0 , we suggest an approach of determining the value function. Contrary to fears associated with exchange rate fluctuations, our results show that these fluctuations can bring about tangible benefits in one’s wealth. We quantify the level of these benefits. We also present an example which illustrates an investment strategy of our agent.