Journal of Mathematics Research; Vol. 12, No. 4; August 2020 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education The Semimartingale Equilibrium Risk Premium for a Risk Seeking Investor George M. Mukupa 1 , Elias R. Offen 2 1 School of Science, Engineering and Technology, Mulungushi University, Kabwe, Zambia 2 Falculty of Science, Department of Mathematics, University of Botswana, Gaborone, Botswana Correspondence: George M. Mukupa, School of Science, Engineering and Technology, Mulungushi University, Kabwe, Zambia Received: March 29, 2020 Accepted: June 5, 2020 Online Published: July 6, 2020 doi:10.5539/jmr.v12n4p13 URL: https://doi.org/10.5539/jmr.v12n4p13 Abstract In this paper, we consider jump amplitudes which are arbitrary and normal to study the risk seeking investor’s equilibrium risk premium in the semimartingale market. We realize that, there is no optimal consumption for this investor in the market. The investor’s premium differ significantly with risk aversion in both martingale and semimartingale markets in that the risk seeking investor has no optimal consumption and the wealth process only affects the rare-event premia with no effect on the diffusive premia. The compensation for this investor is highly attractive compared to risk aversion in this market. Keywords: semimartingale, risk seeking, risk premium, jump diffusion 1. Introduction The risk premium is the compensation an investor recieves for risk taking in the stock market. This premium is found by subtracting the estimated bond return from the estimated stock return. Infact, the size of the premium varies as the risk in a particular stock or in the stock market at large. It is generally true that high risk investments are compensated with a higher equity premium and vice-versa. The reason behind this premium is to entice investors to engage themselves in riskier investments because of the higher rate of return attached to them (Simon & John, 2005). However, an investment in stocks is less guaranteed as companies usually suffer downturns or go out of business. Infact, Aswath (2012) observes that this premium reflects fundamental judgments investors make about how much risk one sees in the market and what price one attaches to that risk. In this regard, investors can base their payment for a share of stock on the risk perceived and anticipated stock return. This will enable them calibrate their investments in a manner that properly compensates them for the excess risk they are taking. If the consumer has reasonable preferences then it is possible to use utility function to describe these preferences (Bellamy & Jeanblanc, 2000). As (Eberlein & Jacod, 1997) states, utility (happiness) depends on consumption. The more the consumption, the more the utility. However, the increase in utility which results from the increase in consumption is smaller the more consumption you already have. This implies that utility is a concave function (Mukupa & Offen, 2015). This follows directly from the fact that people do not like risks and so to induce people to substitute to a risky alternative, it is vital to compensate them by making the riskless alternative unfair and that is by giving less than its expected value. This makes investors indifferent between undertaking the risky or non risky investment.The certainty equivalent (C) for some investment whose outcome is a random variable Z is U(C) = E[U(Z)]. In this paper, the investor with utility function U has current wealth less than C so that the investment is attractive. It is important to consider that much of the work in finance has been based on martingale markets whose future is deemed fair and unpredictable by normalizing prices. This gives investors a fair chance to either gain or lose out on their invest- ments. In this paper, we consider a market X t to allow a decomposition X t = X 0 + M + A, such that M = ( M t ) 0≤t≤T is a square-integrable martingale with M 0 = 0 and A = (A t ) 0≤t≤T is a predictable process of finite variation | A| with A 0 = 0. 13