An Introduction to Continuous-Time Stochastic Processes: by Vincenzo Capasso, David Bakstein

By Vincenzo Capasso, David Bakstein

Expanding at the first variation of An creation to Continuous-Time Stochastic Processes, 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, medication, commercial functions, finance, and coverage utilizing stochastic tools. No past wisdom of stochastic methods is required.

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Extra info for An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine (Modeling and Simulation in Science, Engineering and Technology)

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N, πi : Rn → R be the ith projection, and thus Xi = πi ◦ X, i = 1, . . , n, be the ith component of X. Then the following statements are equivalent: 1. X is a random vector of dimension n. 2. For all i ∈ {1, . . , n}, Xi is a random variable. Proof . 18. 50. Under the assumptions of the preceding proposition, the probability measure Bi ∈ BR → PXi (Bi ) = P (Xi−1 (Bi )) ∈ [0, 1], 1 ≤ i ≤ n, is called the marginal law of the random variable Xi . The probability PX associated with the random vector X is called the joint probability of the family of random variables (Xi )1≤i≤n .

113. Let X be a multivariate normal vector valued in Rk for k ∈ N∗ such that X ∼ N (μX , ΣX ). Given a matrix D ∈ Rp×k , with p ∈ N∗ , and a vector b ∈ Rp , the random vector Y = D X + b is itself a Gaussian random vector: Y ∼ N (D μX + b, D ΣX DT ). Proof . The proof is not difficult and is left as an exercise. We may at any rate notice that, for well-known properties of expected values and covariances, E(Y) = D μX + b, whereas ΣY = D ΣX D T . We may now notice that, if Σ is a positive-definite square matrix, from the theory of matrices it is well known that there exists a nonsingular square matrix P ∈ Rk×k such that Σ = P PT.

Let X be a real-valued random variable on a probability space (Ω, F , P ), and let h : R → R+ ; then P (h(X) ≥ λ) ≤ E[h(X)] λ ∀λ ∈ R∗+ . Proof . , Jacod and Protter (2000, p. 22). 79. 1. If X is a P -integrable continuous random variable with density f , where the latter is symmetric around the axis x = a, a ∈ R, then E[X] = a. 2. If X is a Gaussian variable, then E[X] = m and V ar[X] = σ 2 . 3. If X is a discrete, Poisson-distributed random variable, then E[X] = λ, V ar[X] = λ. 4. If X is binomially distributed, then E[X] = np, V ar[X] = np(1 − p).

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