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Increments (probability theory) of a stochastic processes Poisson–Dirichlet distribution F-related vector fields Section (mathematics) Gaussian drift Continuity in probability

Continuity in probability =

In probability theory, a stochastic process is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge. ^[1]^[2]

Definition

Let $X=(X_{t})_{t\in T}$ be a stochastic process in $\mathbb {R} ^{n}$ . The process $X$ is continuous in probability when $X_{r}$ converges in probability to $X_{s}$ whenever $r$ converges to $s$ .^[2]

Examples and Applications

Feller processes are continuous in probability at $t=0$ . Continuity in probability is a sometimes used as one of the defining property for Lévy process^[1]. Any process that is continuous in probability and has independent increments has a version that is càdlàg^[2]. As a result, some authors immediately define Lévy process as being càdlàg and having independent increments^[3].

References

^ ^a ^b Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
^ ^a ^b ^c Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 286.
^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.

Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer.

F-related Vector fields

F-relatedness is a relation between two Vector Fields in Differential Geometry, a subdiscipline f Mathematics

Definition

Let $M$ and $N$ be Manifolds, and let $F$ be a smooth map from $M$ to $N$ . Let $X$ be a vector field on $N$ . Then a vector field $Y$ on $N$ is called F-related to $X$ if the differential of $F$ maps $X$ to $Y$

Properties

Existence

Give a map F and a vecotr field X, there might not be a single vectorfield in M that is F-related to X. This happens for example when F is not surjectoive, and its differnetial maps ????. If F is not injective, then two differnet vecotrs are mapped to the same point on N, shwing

Naturality with respect to the Lie Bracket

Applications

Characterization of Tangen Submanifolds

Relatednes of Vecotor fields can be used to define what it means for a Vector field to be tangent to a submanifold. Statement. Let S be an an immersed submanifold of M and let i be the inclusion i colon M to S. If Y is a tangent to S, there is a uniquw vector field on S that is i-related to X then the inclusion is noted Y|S

Definitino of the Pushforward

If F is a diffeomorphism, then the Pushforward of X under F is the unique Vecotri field in N that is F-related to X. It can be defined explicitly by defining p=F^-1(q) Y_q= dF_p(X_p) Note that

[applebaum-1] Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.

[KallebergFoundations286-2] Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 286.

[KallebergFoundations290-3] Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.

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