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

By Vincenzo Capasso, David Bakstein

Expanding at the first version of An creation to Continuous-Time Stochastic Processes, this concisely written ebook is a rigorous and self-contained creation to the idea of continuous-time stochastic tactics. A stability of idea and functions, the paintings good points concrete examples of modeling real-world difficulties from biology, medication, business functions, finance, and coverage utilizing stochastic equipment. No past wisdom of stochastic approaches 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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Xk ) , valued in Rk , is said to be multivariate normal or a Gaussian vector if and only if the scalar random variable, valued in R, defined by k Yc := c · X = ci X i , i=1 has a normal distribution for any choice of the vector c = (c1 , . . , ck )T ∈ Rk . 5 Gaussian Random Vectors 31 Given a random vector X = (X1 , . . , Xk ) , valued in Rk , and such that Xi ∈ L2 , i ∈ {1, . . , k}, it makes sense to define the vectors of the means μX = E(X) := (E(X1 ), . . , E(Xk )) and the variance–covariance matrix ΣX := cov(X) := E[(X − μX )(X − μX ) ].

Xn ) be a real-valued random vector with density f and probability PX that is absolutely continuous with respect to the measure μn . The following two statements are equivalent: • X1 , . . , Xn are independent. s. 61. From the previous definition it follows that if a random vector X has independent components, then their marginal distributions determine the joint distribution of X. 62. Let X be a bidimensional random vector with uniform density f (x) = c ∈ R for all x = (x1 , x2 ) ∈ R. If R is, say, a semicircle, then X1 and X2 are not independent.

Xn )dμn ∀B ∈ BR . 56. Under the assumptions of the preceding definition, defining Xi = πi ◦ X, 1 ≤ i ≤ n, then PXi is endowed with density with respect to Lebesgue measure μ on R and its density function fi : R → R+ is given by i fi (xi ) = i where we have denoted by the ith one. f (x1 , . . , xn )dμn−1 , the integration with respect to all variables but Proof . 4) we have that for all Bi ∈ BR PXi (Bi ) = PX (CBi ) = = R CBi dx1 · · · f (x1 , . . , xn )dμn dxi · · · Bi R f (x1 , . . , xn )dxn i = dxi f (x1 , .

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