By Ovidiu Calin

ISBN-10: 9814678937

ISBN-13: 9789814678933

The objective of this booklet is to give Stochastic Calculus at an introductory point and never at its greatest mathematical aspect. the writer goals to seize up to attainable the spirit of simple deterministic Calculus, at which scholars were already uncovered. This assumes a presentation that mimics comparable homes of deterministic Calculus, which allows figuring out of extra advanced subject matters of Stochastic Calculus.

Readership: Undergraduate and graduate scholars drawn to stochastic strategies.

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**Additional resources for An Informal Introduction to Stochastic Calculus with Applications**

**Example text**

Tower property (“the least information wins”): E[E[X|G]|H] = E[E[X|H]|G] = E[X|H], if H ⊂ G; 4. Positivity: E[X|G] ≥ 0, if X ≥ 0; 5. Expectation of a constant is a constant: E[c|G] = c. 6. An independent condition drops out: E[X|G] = E[X], if X is independent of G. 7 Prove the property 3 (tower property) given in the previous proposition. 8 Toss a fair coin 4 times. Each toss yields either H (heads) or T (tails) with equal probability. (a) How many elements does the sample space Ω have? (b) Consider the events A = {Two of the 4 tosses are H}, B = {The ﬁrst toss is H}, and C = {3 of the 4 tosses are H}.

7 Prove the following extension of Jensen’s inequality: If ϕ is a convex function, then for any σ-ﬁeld G ⊂ F we have ϕ(E[X|G]) ≤ E[ϕ(X)|G]. 8 Show the following: (a) |E[X]| ≤ E[|X|]; (b) |E[X|G]| ≤ E[|X| |G], for any σ-ﬁeld G ⊂ F; (c) |E[X]|r ≤ E[|X|r ], for r ≥ 1; (d) |E[X|G]|r ≤ E[|X|r |G], for any σ-ﬁeld G ⊂ F and r ≥ 1. 9 (Markov’s inequality) For any λ, p > 0, we have the following inequality: P (ω; |X(ω)| ≥ λ) ≤ 1 E[|X|p ]. λp Proof: Let A = {ω; |X(ω)| ≥ λ}. Then E[|X|p ] = Ω = λ |X(ω)|p dP (ω) ≥ |X(ω)|p dP (ω) ≥ A A dP (ω) = λ P (A) = λ P (|X| ≥ λ).

It is easy to show that V ar[Wt − Ws ] = |t − s|, V ar[Wt ] = t. 4 Show that a Brownian process Bt is a Wiener process. The only property Bt has and Wt seems not to have is that the increments are normally distributed. 2. From now on, the notations Bt and Wt will be used interchangeably. Inﬁnitesimal relations In stochastic calculus we often need to use inﬁnitesimal notation and its properties. If dWt denotes the inﬁnitesimal increment of a Wiener process in the time interval dt, the aforementioned properties become dWt ∼ N (0, dt), E[dWt ] = 0, and E[(dWt )2 ] = dt.

### An Informal Introduction to Stochastic Calculus with Applications by Ovidiu Calin

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