By Vincenzo Capasso, David Bakstein

ISBN-10: 0817632344

ISBN-13: 9780817632342

"This publication is an advent to the idea of continuous-time stochastic strategies. A stability of concept and purposes, the paintings positive factors concrete examples of modeling real-world difficulties from biology, drugs, finance, and assurance utilizing stochastic equipment. An creation to Continuous-Time Stochastic methods could be of curiosity to a large viewers of scholars, natural and utilized mathematicians, and researchers or practitioners in mathematical finance, biomathematics, biotechnology, physics, and engineering. appropriate as a textbook for graduate or complicated undergraduate classes, the paintings can also be used for self-study or as a reference.

**Read or Download An Introduction to Continuous Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine PDF**

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**Additional info for An Introduction to Continuous Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine**

**Example text**

Show that the maximal value of entropy within the set of real random variables with ﬁxed mean μ and variance σ 2 is attained by the Gaussian N (μ, σ 2 ). 7. We say that X is a compound Poisson random variable if it can be expressed as N Yk X= k=1 for N ∈ N∗ , and X = 0 for N = 0, where N is a Poisson random variable with some parameter λ ∈ R∗+ , and (Yk )k∈N∗ is a family of independent and identically distributed random variables, independent of N . Determine the characteristic function of X. 8.

7. d. 9. d. characteristic function with ﬁnite variance if and only if ln φ(s) = ias + R eisx − 1 − isx G(dx) for any s ∈ R, x2 where a ∈ R and G is an increasing function of bounded variation (the reader may refer to Gnedenko (1963)). 10. e. a measure deﬁned on R∗ such that R∗ min{x2 , 1}λL (dx) < +∞. The triplet (a, σ 2 , λL ) is called the generating triplet of the inﬁnitely divisible characteristic function φ. 11. A distribution is inﬁnitely divisible if and only if it is the weak limit of a sequence of distributions, each of which is compound Poisson (the reader may refer to Breiman (1968)).

2. The sum of two independent Poisson variables, distributed as P (λ1 ) and P (λ2 ), is distributed as P (λ1 + λ2 ) for any λ1 , λ2 ∈ R∗+ . 3. The sum of two independent binomial random variables distributed as B(r1 , p) and B(r2 , p) is distributed as B(r1 + r2 , p) for any r1 , r2 ∈ N∗ and any p ∈ [0, 1]. The Gaussian, Poisson, and binomial distributions are said to reproduce themselves. 90. Consider N independent and identically distributed random variables Xi , i = 1, . . , N, with common probability law P1 , belonging to a family G of probability laws.

### An Introduction to Continuous Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine by Vincenzo Capasso, David Bakstein

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