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# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### How to express a process using Itos formula

Let $F(t,x)$ be the solution to the PDE $$F_t(t,x)=aF_x(t,x)+\frac{1}{2}F_{xx}(t,x),t>0$$ $$F(0,x)=g(x)$$ for some function $g$. Let $X_t$ be a process defined by $$dx_t=aX(t)dt+dW(t)$$ Now ...
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### Integral of Wiener process over time

This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$: $$d(\int W_t dt ) = W_t dt$$ I ...
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### Derive a mathematical equation for Eurodollar future rate

If we suppose that r(t) follows a Vasicek model, which is: $$dr(t) = (\mu - \kappa r(t))dt + \sqrt\sigma dW(t)$$ How to derive an expression for Eurodollar future rate?
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### Levy process and random measure

I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed.
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### Characteristic function and distribution of a random variable

This is exercise 4.3 in Bjork, Arbitrage Theory in Continous Time. $$X_t = \int^t_0 \sigma(s)dW_s$$ $\sigma$ is a deterministic function and $W_t$ is brownian motion. I am asked to find the ...
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### For the Brownian motion integrate

I want to calculate $$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$ I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to ...
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### Statistical estimation vs Stochastic calibration of models

I have never been able to deduce the precise differences between model building from the statistical perspective and the stochastic processes/calibration perspective. I can only infer that these are ...
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### Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...