By Vincenzo Capasso, David Bakstein
"This ebook is an advent to the speculation of continuous-time stochastic strategies. A stability of thought and functions, the paintings positive aspects concrete examples of modeling real-world difficulties from biology, drugs, finance, and assurance utilizing stochastic tools. An advent to Continuous-Time Stochastic strategies could be of curiosity to a wide 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.
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Extra info for An Introduction to Continuous Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine
This is obvious from the linearity of the integral. 3. 28). 4. This property follows from the fact that for all B ∈ F : B Y dP = Y dP , with Y F -measurable and P -integrable. B 5. Here we use the fact that every convex function φ is of type φ(x) = supn (an x + bn ). Therefore, deﬁning ln (x) = an x + bn , for all n, we have that ln (E[Y |F ]) = E[ln (Y )|F ] ≤ E[φ(Y )|F ] and thus φ(E[Y |F ]) = sup ln (E[Y |F ]) ≤ E[φ(Y )|F ]. 129. If Y ∈ Lp (Ω, F, P ), then E[Y |F ] is an element of Lp (Ω, F , P ) and E[Y |F ] p ≤ Y p (1 ≤ p < ∞).
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. 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.
15) 3. Minkowski’s inequality: Np (X + Y ) ≤ Np (X) + Np (Y ) for X, Y ∈ Lp (P ), (p ≥ 1). 148. The mapping Np : Lp (P ) → R+ (p ≥ 1) has the following properties: 1. Np (αX) = |α|Np (X) for X ∈ Lp (P ), α ∈ R; 2. X = 0 ⇒ Np (X) = 0. 147, we can assert that Np is a seminorm on Lp (P ), but not a norm. 149. Let (Xn )n∈N be a sequence of elements of Lp (P ) and let X be another element of Lp (P ). Then the sequence (Xn )n∈N converges to X in mean of order p, if limn→∞ Xn − X p = 0. Convergence in Distribution Now we will deﬁne a diﬀerent type of convergence of random variables, which is associated with its partition function (see Lo`eve (1963) for further references).