Vincenzo Capasso, David Bakstein's An Introduction to Continuous-Time Stochastic Processes: PDF

By Vincenzo Capasso, David Bakstein

ISBN-10: 0817632344

ISBN-13: 9780817632342

ISBN-10: 0817644288

ISBN-13: 9780817644284

This concisely written publication is a rigorous and self-contained advent to the speculation of continuous-time stochastic techniques. A stability of thought and functions, the paintings beneficial properties concrete examples of modeling real-world difficulties from biology, drugs, business functions, finance, and coverage utilizing stochastic tools. No past wisdom of stochastic methods is required.

Key themes coated include:

* Interacting debris and agent-based types: from polymers to ants

* inhabitants dynamics: from beginning and loss of life procedures to epidemics

* monetary industry versions: the non-arbitrage precept

* Contingent declare valuation versions: the risk-neutral valuation conception

* threat research in coverage

An creation to Continuous-Time Stochastic Processes may be of curiosity to a extensive viewers of scholars, natural and utilized mathematicians, and researchers or practitioners in mathematical finance, biomathematics, biotechnology, and engineering. compatible as a textbook for graduate or complex undergraduate classes, the paintings can also be used for self-study or as a reference. must haves contain wisdom of calculus and a few research; publicity to likelihood will be important yet no longer required because the worthy basics of degree and integration are provided.

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

Example text

A stochastic process (Xt )t∈R+ is right-(left-)continuous if its trajectories are right-(left-)continuous almost surely. 23. A stochastic process (Xt )t∈R+ is said to be right-continuous with left limits (RCLL) or continu ` a droite avec limite ` a gauche (c` adl` ag) if, almost surely, it has trajectories that are RCLL. The latter is denoted Xt− = lims↑t Xs . 24. Let (Xt )t∈R+ and (Yt )t∈R+ be two RCLL processes. Xt and Yt are modifications of each other if and only if they are indistinguishable.

8. Let X be a random variable with characteristic function φ. ), if for any n ∈ N∗ , there exists a characteristic function φn such that φ(s) = (φn (s))n for any s ∈ R. 1. d. 2. d. 3. d. 4. d. 5. d. 6. d. characteristic function never vanishes. 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) < +∞.

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)). 12. We will say that two distribution functions F and G on R are of the same type if there exist two constants a ∈ R∗+ and b ∈ R such that F (ax + b) = G(x) for any x ∈ R.

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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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