# Questions tagged [expected-value]

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### Infinitesimal Generators and Expectation of First Hitting Time as Solution of Differential Equation

I've been learning about Linear Diffusions and how their infinitesimal generators can be used to relate expectations and deterministic differential equations. Let $X$ be an one-dimensional diffusion ...
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### Limit of conditional expectations (when limit linked to the conditionning)

I am working with conditional expectations and am trying to derive a limit property. Consider $(Y_n)_{n \in \mathbb{N}}$ a sequence of square integrable random variables, that converge in $L^2$ to a ...
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### Properties of integrated GBM

(I asked this question on MSE but I think it might have more success here) Good day, I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
29 views

### Intuitive explanation for - Shreve Vol 1 Ch 3 Lemma 3.2.6

I need help in getting a more intuitive understanding of this lemma 3.2.6 in Vol 1 of Shreve's Stochastic Calculus for Finance: Here, in the context of multi-period binomial pricing model, Y is a ...
23 views

### Order of expectation versus expectation of order (error terms in Taylor expansion)

Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as $$E[F(X)] = F(a) + E[ O((X-a))]$$ Under what ...
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### Expectation of option value

Say we are in a BS world where the (conditional on t) price of a call is given by the usual $$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$ Now, what about the ...
$I(t)=\int_0^t \sqrt sdW_s$ What is $E(I(t)^4)$