Questions tagged [wienerprocess]

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6
votes
1answer
144 views

Integral of the OU (Ornstein Uhlenbeck) process conditioned on hitting a threshold value for the first time

Let say I have a zero-mean OU process as follows: $dX_t = -\alpha X_t + dW_t$ The process starts at $x_0 = 0%$ and I'm interested in the event in which the process hits the value $x_{\tau} = a$ for ...
5
votes
1answer
2k views

Integral of Wiener process w.r.t. time

I have a doubt with regards to the calculation of the below integral- $\int_0^t W_sds$ where $W_s$ is the Wiener Process. This has been solved very ably in the following page. It turns out to be a ...
4
votes
3answers
416 views

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 $B_t = W_t^3 - 3tW_t$. ...
4
votes
3answers
345 views

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 $E[W_p W_q W_r]$. ...
4
votes
1answer
264 views

How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$

How can I calculate? \begin{align} Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right) \end{align} Thank you for your attention.
4
votes
1answer
159 views

Stochastic Differential

Let $W_t$ be a Wiener process. It is clear to me that $dW_t$ is of size $\sqrt{dt}$. This can be seen because $$ \mathrm{Var}(W_{t+\Delta} - W_{t})=\Delta. $$ But am I allowed to actually write $(...
4
votes
2answers
279 views

Are two stochastic processes independent if the Wiener processes inside are uncorrelated

Assume there are two stochastic processes: $dx_t = \alpha_1(x_t,t)dt + \beta_1(x_t,t)dW^1_t$ and $dy_t = \alpha_2(y_t,t)dt + \beta_2(y_t,t)dW^2_t$. Does $dW^1_t\times{dW^2_t} = 0$ imply that $\...
4
votes
1answer
183 views

On the reflection of a stochastic integral

Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by $$ I_t=\int_{0}^{t}\theta_sdW_t, $$ where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq 0},\...
3
votes
1answer
228 views

Wiener process integral

Suppose that $W_{t}$ is a Wiener process. Assume $W_{0}=0.$ Is it true that $\int_{t=0}^{T}dW_{t}=W_{T}$? If so, why? Is one preferred to the other?
3
votes
1answer
151 views

Interpretation of IV and its use in stock movement prediction

I would like to validate my understanding of IV as a prediction tool. Black-Scholes model is based on the assumption that rate of return of a stock is a Wiener process: $$ \frac{dS_t}{S_t} =\mu \,...
3
votes
1answer
374 views

Solving a backwards heat equation using stochastic calculus

Given the PDE $$\frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2} = 0$$ with condition $F(T,x) = x^2$, one can use the Feynman-Kac formula to arrive at $$F(t,x) =...
2
votes
1answer
61 views

Can the differential operator be removed to get the mean/variance of an Ito process?

If $X_t$ is an Ito process, such that: $dX_t = \mu(t, X_t)dt + \sigma(t, Xt)dW_t$ where $W_t$ is a standard brownian motion. Then we can say that: $E(dX_t) = \mu(t, X_t)dt$ and that $Var(dX_t) = \...
2
votes
2answers
105 views

Bounded Stochastic discrete process

I just came across this stochastic process (link): $dY_t = (a-bY_t)dt + c \sqrt{Y_t(1-Y_t)}dW_t$, where $dW_t$ is a Wiener Process. According to the author under certain conditions this process is ...
2
votes
1answer
116 views

What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?

The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S: $dS=\mu S dt + \sigma S dX$ Clearly $S$ ...
2
votes
2answers
3k views

How to calculate the expected value of a function of a standard brownian motion (Wiener process)

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 ...
2
votes
0answers
127 views

Expected value of a wiener process on an infinite time horizon with a barrier

Say I have a wiener process with $X(0) = X_0>0$ and the dynamics \begin{equation} dX(t) = \begin{cases} -\mu dt + \sigma X(t) dW(t)^{\mathbb{Q}} & \mathrm{for\ } X(t)>0\\ 0 & \mathrm{...
1
vote
1answer
332 views

Is this process of Brownian motion?

Background Information: The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if i) $W_t$ is continuous, and $W_0 = 0$ ii) the value of $W_t$ is distributed, under $\mathbb{...
1
vote
2answers
118 views

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 ...
1
vote
1answer
68 views

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 ...
1
vote
0answers
60 views

Solving for roots of a stochastic pay-off function

I have a pay-off function for a derivative which is defined by the Heaviside difference between $G$ and $B$ shifted by $-F$. To find the value of $V_{t=0}$, I need to find $\tau$ when $\frac{dV}{dt} = ...
1
vote
1answer
525 views

Matlab implementation for modelling stock price process

I am trying to model the stock's price process. Let's assume volatility and risk-free rate is given. I've come up with the code below to try and model the price process with the geometrical Brownian ...
1
vote
1answer
379 views

Obtaining the drift of a Wiener process formed from a random walk

I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ...
0
votes
1answer
99 views

Two Wiener process under same martingale measure Q

Let $W_1,$ $W_2$ be to Wiener processes under the martingale measure $Q$. What can be said about $dW_1*dW_2$? I know that $$(dW_i)^2=dt$$ but what about the case with two different wiener processes?
-3
votes
2answers
282 views

Geometric Brownian Motion: Why is the Wiener process multiplied by volatility?

Below is the stochastic differential equation of the Geometric Brownian Motion: $$dS_t = S_t \mu dt + S_t\sigma dW_t$$ My understanding of the Wiener process is that the volatility component of an ...