By Vincenzo Capasso, David Bakstein
This concisely written e-book is a rigorous and self-contained advent to the speculation of continuous-time stochastic methods. A stability of conception and purposes, the paintings positive factors concrete examples of modeling real-world difficulties from biology, drugs, business purposes, finance, and assurance utilizing stochastic tools. No prior wisdom of stochastic methods is required.
Key subject matters coated include:
* Interacting debris and agent-based types: from polymers to ants
* inhabitants dynamics: from delivery and loss of life strategies to epidemics
* monetary industry types: the non-arbitrage precept
* Contingent declare valuation versions: the risk-neutral valuation idea
* threat research in assurance
An creation to Continuous-Time Stochastic Processes might be of curiosity to a large viewers of scholars, natural and utilized mathematicians, and researchers or practitioners in mathematical finance, biomathematics, biotechnology, 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. necessities comprise wisdom of calculus and a few research; publicity to chance will be invaluable yet no longer required because the important basics of degree and integration are provided.
Read or Download An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine PDF
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Extra info for An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine
120. Let (Ω, F, P ) be a probability space, and let G be a subσ-algebra of F. We say that a real random variable Y on (Ω, F, P ) is independent of G with respect to the probability measure P if ∀B ∈ BR , ∀G ∈ G : P (G ∩ Y −1 (B)) = P (G)P (Y −1 (B)). 121. s. Proof: Let G ∈ G; then, by independence, Y dP = IG Y dP = E[IG Y ] = E[IG ]E[Y ] = P (G)E[Y ] = G E[Y ]dP, G from which the proposition follows. 122. Both the monotone and dominated convergence theorems extend in an analogous way to expectations conditional on σ-algebras.
A function f : R+ → R is right-continuous if for any t ∈ R+ , with s > t, lim f (s) = f (t). s↓t Instead, the function is left-continuous if for any t ∈ R+ , with s < t, lim f (s) = f (t). 22. 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.
14. Show that every stable law is inﬁnitely divisible. What about the converse? 15. Show that if φ is the characteristic function of a stable law which is symmetric about the origin, then there exist c ∈ R∗+ and α ∈]0, 2] such that φ(s) = e−c|s| for any x ∈ R. 16. If φ1 (t) = sin t, φ2 (t) = cos t are characteristic functions, then give an example of random variables associated with φ1 , φ2 , respectively. Let φ(t) be a characteristic function, and describe the random variable with characteristic function |φ(t)|2 .
An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine by Vincenzo Capasso, David Bakstein