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Brownian motion played a central role throughout the twentieth century in probability theory. The same statement is even truer in finance, with the introduction in 1900 by the French mathematician Louis Bachelier of an arithmetic Brownian motion (or a version of it) to represent stock price dynamics. This process was pragmatically transformed by Samuelson in 1965 into a geometric Brownian motion ensuring the positivity of stock prices. More recently, the elegant martingale property under an equivalent probability measure derived from the no-arbitrage assumption combined with Monroe's theorem on the representation of semi martingales has led to write asset prices as time-changed Brownian... 

In this thesis, we are going to extend hoeffding's inequality to supermartingale with differences bounded from above and compare this inequality with the other ones which derived in the past. After defining basic thorems, we introduce upper bounds which are derived for the probability that the sum of some independent random variables exceeds its mean by a positive number and also, compare to each o ther. In addition , we use these bounds and derive upper bounds of probability such as above for dependent random variables. Then, we introduce a n inequality which cover Hoeffding's inequality for ... 

In this thesis, we study first passage times of a double exponential jump diffusion process to boundaries. This process consists of a continuous part which includes brownian motion and a jump part with jump sizes which have a double exponential distribution. We study explicit solutions obtained for the laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima. Additionally, several interesting probabilistic results are provided. Its results have finance applications, including pricing barrier and lookback options