# Questions tagged [wienerprocess]

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### Integrated Brownian motion

I occasionally see a post here: Integral of brownian motion wrt. time over [t;T]. This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$. However, here is my derivation which is ...
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### Feymann Kac pde with correlated process

I have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
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### Calculate value of Integral of Wiener process $\int_{0}^t e^{\lambda u } dZ_u$

I am not quite sure how to solve this integral to be able to do numerical calculations with it. $\lambda$ is a constant, $u$ is time, and $Z_u$ is a wiener process. Can anyone provide some direction ...
1 vote
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### Is this the right way to accelerate my Monte-Carlo Simulation

I am trying to develop a pricer for Call VS Call and I'm using MonteCarlo method to do so because my stocks are correlated between each others. Basically my inputs are ...
1 vote
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### Integral of brownian motion wrt. time over [t;T]

From the post Integral of Brownian motion w.r.t. time we have an argument for $$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$ However, how does this generalise for the interval $[t;T]$? I.e. ...
1 vote
129 views

A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process. The problem is to find the probability of the ...
1 vote
156 views

### Why the Esscher transform is the right transform for pricing formula?

A Wiener process has infinitely many states of the world at any time step. Does that not mean that there are infinitely many EMM's for any model that uses the Wiener process? But then if there is only ...
1 vote
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### Regression of stochastic integral on Wiener process

This question is a follow-up from the following: conditional expectation of stochastic integral so I won't repeat myself regarding assumptions and notation. Using Brownian bridge approach, we know ...
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### 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 ...
297 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 ...
187 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?
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### 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 ...
1 vote
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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 $\... 4 votes 1 answer 308 views ### On the reflection of a stochastic integral Let${(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},\...
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 ...