# Questions tagged [brownian-motion]

In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

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### Construction of Ito Integral - simple integrands [closed]

I am trying to understand the application of the below concept: Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] = gain? In this calculation the purchase price is not taken into ...
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### Construction of Itos integral

I am trying to understand the below: Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss? In this calculation the purchase price is not taken ...
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### Python - Problem of random numbers in MC simulation

I am interested in estimating the price of a European Call Option using the Montecarlo simulation, to get a good approximation of the analytical Black Scholes formula, so a very simple task. ...
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### Infill lower frequency data: Brownian Bridge

Given monthly returns data, I would like to infill those to get daily returns. Roughly estimates imply that annual volatility is about 1.5x of SPY. One option that came up in my initial research was ...
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### Second variation of a Brownian motion under jump-diffusion process

I am trying to solve exercise 15.3 from the book The concepts and practice of mathematical finance where it is asked Suppose the $\log S_t$ follows a Brownian motion over the period $[0, 1]$ except ...
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### Ito's lemma $f(t,W_t^2)$

Let $f$ be a function of $t$ and $W_t^2$. a)Find a function $f$ such that $f(t,W_t^2)$ is a $F_{t^-}$ martingale, with $F$ the Brownian filtration. b)Use Ito's lemma to show that $f(t,W_t^2)$ is a ...
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### Sampling change in the driving brownian motion of a CIR process

I have volatility driven by a CIR process: $$\mathrm{d}v_t = \kappa (\bar{v}-v_t)\mathrm{d}t + \omega \sqrt{v_t}\mathrm{d}W_v\text{.}\tag{1}$$ I am working with several (complicated) approximations of ...
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Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying: $dB(t)=r(t)B(t)dt$ and $dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$ where $r,b,\... • 129 2 votes 1 answer 159 views ### Solving an SDE using Ito's Lemma Suppose that$Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$with$\theta()=\sigma^{-1}()[b()-r()]$,$\sigma()>0$and invertable and$W()$a Wiener process There is also ... • 129 3 votes 1 answer 207 views ### Brownian Bridge general case The SDE for the Brownian bridge is the following:$dY_t=\frac{b-Y(t)}{1-t}dt+dW(t)$with solution:$Y(t)=Y(0)(1-t)+bt+(1-t)\int_0^t \dfrac{dW(s)}{1-s}$Can someone help me on proving that $$\lim_{t\... • 129 8 votes 0 answers 259 views ### On a time integral of Brownian motion up to the hitting time Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space (\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\... • 81 3 votes 0 answers 112 views ### MGF of Generalised Itô Integral The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on t. Consider W_t ... • 31 1 vote 0 answers 64 views ### Distribution and Analytical solution of a GBM with stochastic interest rate? We model the exchange rate S_t with a geometric Brownian motion and the USD and EUR interest rates r_u and r_e each according to the Vasicek model. Under the domestic equivalent martingale ... • 409 1 vote 2 answers 298 views ### Drift Term in Black-Scholes Model Martingale How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof). For instance if Im looking at:$$dS_{t} = \mu S_{t} + \... • 21 0 votes 1 answer 102 views ### Sum of discretely sampled BM If an underlying follows lognormal GM with no drift$dS_t = \sigma S_t dW_t $and$A_N = \Sigma_{i=1}^{N} S_{t_i}$. How to compute variance of$A_N$? • 89 1 vote 1 answer 126 views ### Deriving Law of Motion by Ito's Lemma I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below: I've managed to take the partial ... 4 votes 1 answer 349 views ### Conditional expectation of integral of brownian motion I am trying to calculate $$\mathbb{E}\biggl[\biggl(\int_s^t W_u du\biggl)^2 \biggl|W_s=x, W_t=y\biggl]$$ where$W$is a Standard Brownian Motion and$s\leq u \leq t$. Any help or tips would be ... • 65 -2 votes 1 answer 90 views ### Monte carlo simulations giving biased output [closed] I wrote code to simulate the stock price using geometric brownian motion. My code is as follows: ... • 55 3 votes 1 answer 129 views ### EMM for Bachelier model The stock price is assumed to evolve as$S_{t}=S_{0}+\mu t+\sigma B_{t}$, where$S_{0}>0, \mu>0$and the process$B_{t}$is Brownian motion. The saving account is assumed to be$\beta_{t}=e^{r t}...
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During revision, I came across the following question in a past paper: Suppose $(B_t, t\geq0)$ is a standard Brownian motion. Compute for $0<s<t$ the covariance $$cov(tB_{3t}-B_{2t}+5, B_s-1).$$ ...