In this paper, we propose a new risk sensitive (RS) control law of finite time for linear systems with stochastic system parameters. Stochastic parameters entail a nonlinear RS controller that is not compatible with linear systems, and can be thus unreliable. Moreover, it is difficult to obtain a globally defined exact solution due to the nonlinearity. To solve these problems, we derive a novel problem setting based on standard RS control. The standard RS type cost function is converted to a weighted sum of quadratic functions such that resulting feedback is linear in the state variable. How to convert the cost function without losing the characteristic of RS control is the main topic of this paper. We derive a novel weight independently of the state and input to obtain a linear RS control law. This allows one to derive the exact solution that is globally defined similarly to the well-known linear quadratic control problem. Such a linear controller offers high reliability and safe implementation without restricting the control region of the state. A numerical simulation demonstrates the effectiveness of the proposed method.