# Faculty

## Tsukasa Fujiwara

### Major research activity

My research area is in probability theory. The main theme of my research is the study of random processes from both a mathematical and a financial perspective.

I have been much interested in random processes, such as random walk, Brownian motion and Poisson process. In the theory of probability, they have been used as mathematical models to describe random motions that appear in the circumstances around us.

Among others, I have been studying Lévy processes, which are essentially processes with stationary and independent increments. The outstanding feature of them is the existence of jumps. Poisson process is a typical and fundamental example of Lévy processes. Lévy processes form a wide and rich class of random processes and therefore a robust mathematical theory on them exists at the present time. Furthermore, I have been studying stochastic differential equations with jumps, such as stochastic differential equations based on Lévy processes for the last thirty years. They can be considered as a model for describing the time evolution of a dynamical system with random shock noises.

On the other hand, random processes with jumps including Lévy processes have recently been finding increasing applications to mathematical finance. They form a rich class of processes that are well suited for mathematical modelling of financial phenomena where random discontinuities appear.

key words:

probability theory, stochastic analysis, mathematical finance

### Major relevant publications

- T. Fujiwara and Y. Miyahara, The minimal entropy martingale measures for geometric Lévy processes, Finance and Stochastics 7, 509–531 (2003).
- T. Fujiwara, On the exponential moments of additive processes with the structure of semimartingales, Journal of Math-for-Industry 2 (2010A-2) (2010), 13–20.
- T. Fujiwara, The minimal entropy martingale measures for exponential additive processes revisited, Journal of Math-for-Industry 2 (2010B-1) (2010), 115–125.