By Ovidiu Calin
The objective of this e-book is to give Stochastic Calculus at an introductory point and never at its greatest mathematical aspect. the writer goals to catch up to attainable the spirit of basic deterministic Calculus, at which scholars were already uncovered. This assumes a presentation that mimics comparable houses of deterministic Calculus, which allows knowing of extra advanced themes of Stochastic Calculus.
Readership: Undergraduate and graduate scholars attracted to stochastic approaches.
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Extra info for An Informal Introduction to Stochastic Calculus with Applications
A random variable X : Ω → R is called square integrable if E[X 2 ] = Ω |X(ω)|2 dP (ω) = R x2 p(x) dx < ∞. 3 If X is a square integrable random variable, then it is integrable. Proof: Jensen’s inequality with ϕ(x) = x2 becomes E[X]2 ≤ E[X 2 ]. Since the right side is ﬁnite, it follows that E[X] < ∞, so X is integrable. 4 If mX (t) denotes the moment generating function of the random variable X with mean μ, then mX (t) ≥ etμ . Proof: yields Applying Jensen’s inequality with the convex function ϕ(x) = ex eE[X] ≤ E[eX ].
Let X be a random variable normally distributed with mean μ and variance 2 σ . It is known that its moment generating function is given by 1 2 2 σ m(t) = E[etX ] = eμt+ 2 t . Using the ﬁrst Chernoﬀ bound we obtain P (X ≥ λ) ≤ 1 2 2 m(t) = e(μ−λ)t+ 2 t σ , ∀t > 0, λt e page 35 May 15, 2015 36 14:45 BC: 9620 – An Informal Introduction to Stochastic Calculus Driver˙book An Informal Introduction to Stochastic Calculus with Applications which implies 1 min[(μ − λ)t + t2 σ 2 ] 2 . P (X ≥ λ) ≤ e t>0 It is easy to see that the quadratic function f (t) = (μ − λ)t + 12 t2 σ 2 has the λ−μ .
10). 7 The conclusion still holds true even in the case when there is a p > 0 such that E[|Xn |p ] → 0 as n → ∞. Limit in Distribution We say the sequence Xn converges in distribution to X if for any continuous bounded function ϕ(x) we have lim E[ϕ(Xn )] = E[ϕ(X)]. e. it is implied by it. An application of the limit in distribution is obtained if we consider ϕ(x) = itx e . In this case the expectation becomes the Fourier transform of the probability density E[ϕ(X)] = eitx p(x) dx = pˆ(t), and it is called the characteristic function of the random variable X.