# Questions tagged [wienerprocess]

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### CIR model. Is there a closed-form solution or even a good proxy of analytical solution?

Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE \begin{equation} dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1} \end{equation} ? Notice that $\{r_t\}$ is our ...
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### Expectation on a function of Wiener Process

If $W_t$ is a standard Wiener Process, then how should I prove that $E \left[ \int\limits_{0}^{t} \frac{1}{1+W_s^2} dW_s \right] = 0$?
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### How to expand lognormal approximation of Brownian motion

How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion. If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
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### Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)

For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
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### Calculation of a process's drift

Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift. The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint): I ...
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### What is the difference between standard deviation, volatility and quadratic variation?

What is the difference between standard deviation, volatility and quadratic variation? As I know, volatility is the standard deviation of the log returns, so they are basically the same. (One of ...
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### Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u$

Let $X_t$ be a stochastic process such that $$X_{t} =\frac{1}{t}\int_0^t u dW_u$$ I know that for $$Y_{t} =\int_0^t u dW_u$$ $Y_t-Y_s$ is independent of $Y_s$ where $t>s$. But is this also true ...
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### Probability Density Function of a Wiener Process Minimum

Let $W_t$ be a standard Wiener process. Find the probability density function of $m_T = min_{t\in [0,T ]}W_t$. I know that it is based of the concept of the reflection principle, but I wasn't too ...
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### Can anyone explain to how Hull get's from the stock returns to continuously compounded stock returns?

I'm reading Chapter 13 of Hull's book and am stuck on how he got from stock returns to continuously compounded stock returns. As a recap, he built the generalized Wiener Process, which describes a ...
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### Stochastic process theory question

*S follows a process $dS= mSdt + oSdz$ where m and o are constant. What is the probability followed by $Y=(Se)^{(r-t)}$. If S follows a process $dS= k (b-S) dt + oSdz$ where k, b, o are ...
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### Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
### Determine $E[W_p W_q W_r]$
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let 0 < p < q < r. Determine $E[W_p W_q W_r]$. ...
Have a problem regarding the expected value of the Wiener process inside a function, namely: Compute $E[cos(W_t)]$. To extend my question, what is the general method of computing these E´s when it ...