Abstract
We commence along the lines of the founding work of Kolmogorov by regarding stochastic processes as a family of random variables defined on a probability space and thereby define a probability law on the set of trajectories of the process. More specifically, stochastic processes generalize the notion of (finite-dimensional) vectors of random variables to the case of any family of random variables indexed in a general set T. Typically, the latter represents “time” and is an interval of \(\mathbb{R}\) (in the continuous case) or \(\mathbb{N}\) (in the discrete case). For a nice and elementary introduction to this topic, the reader may refer to Parzen (1962).
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Notes
- 1.
By the Feller property.
- 2.
For simplicity, but without loss of generality, we will assume that E[X t ] = 0, for all t. In the case where E[X t ] ≠ 0, we can always define a variable \(Y _{t} = X_{t} - E[X_{t}]\), so that E[Y t ] = 0. In that case \((Y _{t})_{t\in \mathbb{R}_{+}}\) will again be a process with independent increments, so that the analysis is analogous.
References
Anderson, W.J.: Continuous-Time Markov Chains: An Application-Oriented Approach. Springer, New York (1991)
Applebaum, D.: Levy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)
Ash, R.B., Gardner M.F.: Topics in Stochastic Processes. Academic, London (1975)
Baldi, P.: Equazioni differenziali stocastiche. UMI, Bologna (1984)
Bauer, H.: Probability Theory and Elements of Measure Theory. Academic, London (1981)
Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)
Billingsley, P.: Probability and Measure. Wiley, New York (1986)
Bosq, D.: Linear Processes in Function Spaces. Theory and Applications. Springer, New York (2000)
Breiman, L.: Probability. Addison-Wesley, Reading, MA (1968)
Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Heidelberg (1981)
Chung, K.L.: A Course in Probability Theory, 2nd edn. Academic, New York (1974)
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Wiley-Interscience, New York (1966)
Çynlar, E.: Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs, NJ (1975)
Daley, D., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer, Berlin (1988)
Daley, D., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Volume II: General Theory and Structure. Springer, Heidelberg (2008)
Devijver, P.A., Kittler, J.: Pattern Recognition. A Statistical Approach. Prentice-Hall, Englewood Cliffs, NJ (1982)
Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Levy Processes with Applications to Finance. Springer, Berlin/Heidelberg (2009)
Doob, J.L.: Stochastic Processes. Wiley, New York (1953)
Dynkin, E.B.: Markov Processes, vols. 1–2. Springer, Berlin (1965)
Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)
Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1971)
Ghanen, R.G., Spanos, P.D.: Stochastic Finite Elements. A Spectral Approach. Revised Edition. Dover Publication, Mineola, NY (2003)
Gihman, I.I., Skorohod, A.V.: The Theory of Random Processes. Springer, Berlin (1974)
Grigoriu, M.: Stochastic Calculus: Applications to Science and Engineering. Birkhäuser, Boston (2002)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Kodansha (1989)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer Lecture Notes in Mathematics. Springer, Berlin (1987)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (2003)
Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (1997)
Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes. Academic, New York (1975)
Karlin, S., Taylor H.M.: A Second Course in Stochastic Processes. Academic, New York (1981)
Karr, A.F.: Point Processes and Their Statistical Inference. Marcel Dekker, New York (1986)
Klenke, A.: Probability Theory. Springer, Heidelberg (2008)
Klenke, A.: Probability Theory. A Comprehensive Course, 2nd edn. Springer, Heidelberg (2014)
Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg (2014)
Lamperti, J.: Stochastic Processes: A Survey of the Mathematical Theory. Springer, New York (1977)
Last, G., Brandt, A.: Marked Point Processes on the Real Line: The Dynamic Approach. Springer, Heidelberg (1995)
Mercer, J.: Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. Roy. Soc. Lond. 209, 415–446 (1909)
Métivier, M.: Notions fondamentales de la théorie des probabilités. Dunod, Paris (1968)
Meyer, P.A.: Probabilités et Potentiel. Ilermann, Paris (1966)
Mikosch, T.: Non-Life Insurance Mathematics, 2nd edn. Springer, Berlin/Heidelberg (2009)
Parzen, E.: Statistical inference on time series by Hilbert space methods,Technical Report no. 23, Statistics Department, Stanford University, Stanford, CA, Jan 1959
Parzen, E.: Stochastic Processes. Holden-Day, San Francisco (1962)
Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin (1990). Second Edition 2004
Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, Berlin (1990). Second Edition 2004
Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, 2nd edn. Springer, New York (2005)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Heidelberg (1991)
Robert, P.: Stochastic Networks and Queues. Springer, Heidelberg (2003)
Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, vol. 1. Wiley, New York (1994)
Sato, K.I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Schuss, Z.: Theory and Applications of Stochastic Processes: An Analytical Approach. Springer, New York (2010)
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Capasso, V., Bakstein, D. (2015). Stochastic Processes. In: An Introduction to Continuous-Time Stochastic Processes. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-2757-9_2
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