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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63 views

Fractional Brownian Motion's Covariance Proof

Let's have the non independent Brownian motion such : $B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$ ...
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1answer
61 views

Integral of Brownian motion w.r.t. time and integral not starting at zero

I'm new to stochastic calculus and try to calculate (1) mean and (2) variance of $$\int_s^t W_u du$$ where $W_u$ is a Brownian motion. I already found this helpful answer, where it was shown that $\...
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1answer
86 views

Hermite polynomials as martingales [closed]

Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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1answer
85 views

Mutual variation of Brownian motions

Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
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1answer
149 views

What does it mean to “compute” an Itô integral?

I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
3
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1answer
154 views

Help on solving a stochastic differential equation

I am trying to solve the following SDE $$dX(t)=rdt+aX(t)dW(t),\ t>0$$ $$X(0)=x$$ where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor $$F(t)=...
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2answers
148 views

Proving that a stochastic process is a martingale using Ito's Lemma

Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F $$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$ Can ...
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1answer
126 views

Likelihood ratio and pathwise sensitivity method for coupled SDEs

I have two coupled SDEs \begin{align*} dS_t=rS_tdt+V_tdW_t^{(1)},\\ dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\ \end{align*} where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data ...
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1answer
56 views

Reason why a European binary call should be worth half of its American counterpart when driftless and out-of-the-money

Exercise 11 of chapter 8 of Mark Joshi's "The concepts and practice of mathematical finance", asks to compare prices of an American and a European digital (binary) calls when out-of-the-...
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1answer
54 views

Simplifying the expectation of the product of two stochastic integrals

Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
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1answer
87 views

Default intensity in Black-Cox model

Consider the model by Black and Cox (Journal of Finance, 1976). The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
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1answer
98 views

What is the expectation of a change in Brownian motion? [closed]

I know $E[W_T-W_t]=0$ but I have a solution which implies this is wrong. Question Answer
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51 views

How to show the following conditional expectation relation holds for a Brownian motion?

Suppose that $B_k$ stands for a standard Brownian motion process. \begin{equation} \mathbb{E}\Big(e^{-w\int_{t}^{S}B_k dk\, -uB_T}\Big| B_t = x\Big) \end{equation} where $w$ and $u$ are constants, and ...
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1answer
57 views

How to prove that the following is still a Brownian motion [closed]

Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
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52 views

Future price in continous time

I am in the following continuous time market: $S_t^0 = rS_t^0dt$ $S_t^1 = (\mu - \delta) S_t^1dt + \sigma S_t^1 dB_t$ where $r, \mu, \delta$ and $\sigma$ are constant values in $\mathbb{R}$. $\delta$...
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1answer
150 views

Conditional probability of Brownian motion (with drift and scaling) hitting barrier

I am trying to understand the pricing of barrier options, and am considering the Brownian motion $\mathrm{d}X_t=a\mathrm{d}t+b\mathrm{d}W_t$, $a$ and $b$ constant. I am trying to: derive the ...
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2answers
239 views

Calculate Ito integral $\int_0^t W_s^2\text dW_s$ from first principles

I am stuck on the 1st equation of the solution where the Wiener process $W_{t_i}^2$ is expanded so that the Itô integral (in terms of infinite sums) looks like the RHS of the first equation of the ...
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38 views

Do we model stock prices using non-Markovian processes in continuous setting?

In a continuous setting, is it common to model stock prices using non-Markovian processes ? If so, do you have some examples of models ? Or is Markovianity something "embedded" in the ...
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1answer
126 views

Quasi Monte Carlo and Brownian bridge (how to combine them)

I am trying to understand how quasi Monte Carlo (QMC) and the Brownian bridge (BB) can be combined to price an asset, but I am having a hard time understanding how. I am just considering a European ...
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2answers
55 views

How to calculate expectation and variance of smooth function applied to brownian motion [closed]

I applied a smoothing function to a Brownian equation and obtained a stochastic differential equation by using Ito's lemma. The smoothing function is exp(Bt). How do I get the expected value and ...
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83 views

Non attainable claim - Incomplete market

I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market. As an example, let us have the following market ($B = (B^1, B^2, B^3)$...
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1answer
93 views

How to show that a Brownian motion is normally distributed and that the covariance is zero? [closed]

I need help under standing this question. So i have the following given the logarithm of the price of a share of stock is given by \begin{align*} p(t)=p(0)+\mu t+\sigma W(t), \quad t \in[0, T] \end{...
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158 views

Summary of Stochastic Derivatives, Integrals, Expectations, and Variances

I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
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1answer
118 views

Black Scholes price of exotic claim

Given a time horizon N, I want to know the time-$t$ Black-Scholes fair price of $$\int_0^T S_u du$$ where $S_u$ denotes the time-$u$ stock price. I have used the formula I have been given as follows: $...
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1answer
129 views

Using geometric brownian motion for stock price forecasting [closed]

I am doing a dissertation in finance on a maths degree. I wanted to forecast stock prices using artifcial neural networks but none of my tutors are able to supervise so I'm having to do something else....
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3answers
686 views

Expectation of exponential of 3 correlated Brownian Motion

Consider, are correlated Brownian motions with a given I want to calculate the, , I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
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1answer
222 views

Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]

I am trying to calculate the expectation of $$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$ where $(W_t)$ is a Wiener process. I was told that the value of this expectation is zero. Can someone ...
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1answer
41 views

Multiple underlying brownian motions

I'm trying to find a way to price a triple product forward with payoff XYZ at time T using risk-neutral pricing. But I don't really have a math background and I have trouble finding a way to account ...
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54 views

Probability of Hitting time of Brownian motion

Let $B =\{ B(t); t \ge 0\}$ be Brownian motion. What is the probability that $B$ hits state one and then state minus one before time one? My take: Let $T_x = \inf \{ t\ge 0 : B(t) = x\}$, the first ...
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2answers
216 views

Integral of the square of Brownian motion using definition of variance

Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
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76 views

Continuous option pricing: Brownian Bridge

I have a question on the proof of the formula of Sup(S) between 2 simulation points. Do you know how the prove the following formula? Thanks
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38 views

Rate of return of a bond price

Assume $B(t, T)$ is a zero coupon bond price and assume that it has dynamics $dB(t, T) = B(t, T)[\mu(t, T)dt + \sigma(t, T)dW_t]$, where $W_t$ is a Brownian motion under $(\Omega ;F; P)$ and $P$ is an ...
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1answer
71 views

Let $W_t$ denote a standard Brownian motion. Evaluate this integral [closed]

$$ \int_{0}^{t}d(W_{u}^2) $$ How can I deal with this kind of problem? If there is no function given to apply Itô's formula.
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1answer
130 views

Long-Term Energy Price Modelling: Log Returns, Distributions, Time-Weighting

I wish to forecast energy prices in the long-term (ca. 20 years) for energy-efficiency investments. While I understand that the energy carriers are particularly sensitive to external (geo-political) ...
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2answers
194 views

Proof of Feller condition for CIR square root process. Any reference?

Could you please give me some reference for the proof of the so-called Feller condition as to a stochastic differential equation of the form: $$dr_t=a(b-r_t)dt+\sigma\sqrt{r_t}dB_t\tag{1}$$ with $\...
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0answers
40 views

Relationship between risk and return for GBM and riskless bond

Suppose we have $S$, a stock following geometric Brownian motion ($dS_t = S_t (\mu dt + \sigma dZ_t)$ for $Z =$ Brownian motion) and $B$, a zero coupon bond with rate $r$, i.e. $dB_t = rB_t dt$. In ...
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0answers
96 views

Estimating constant and local volatility based on passage times

Consider a Brownian motion B_t with constant instantaneous volatility σ and zero drift where ...
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1answer
385 views

If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?

If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance? Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?
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2answers
156 views

Simulating artificial asset prices: Random walk vs Brownian motion?

How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each: Random walk with drift Random walk ...
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1answer
70 views

General Dynamics of a Tradable Asset under the Risk Neutral Measure

Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then $$\...
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0answers
91 views

Expectation of number of hits by a brownian motion

If we denote $\tau_i$ the sequence of stopping times defined by: $\tau_i = \inf(t>\tau_{i-1} : |B(t)-B(\tau_{i-1})| > a)$, $\tau_0=0$. If we denote N the number of stopping times below T. What ...
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0answers
69 views

Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown

I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2]. I do not ...
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0answers
181 views

Geometric Brownian Motion and Energy-Efficiency Investments

Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion: $$ dX/X = \mu dt + \sigma dz\ , $$ for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
2
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0answers
133 views

law of absolute of max of brownian motion

What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion? Any references for properties of this process?
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0answers
70 views

Brownian function and Clark's formula

I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage : Let $X_t$ be a Brownian motion, $F_t$ its filtration and $g$ a real-valued function. ...
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72 views

Alternatives to Lognormality for negative Prices

If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ...
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1answer
69 views

Modelling Power Prices

Since electricity prices involve strong seasonality, jump components as well as negative prices and can not be modelled by the GBM, what models/distributions exist, which would allow for modelling ...
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1answer
83 views

Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process

We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
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4answers
804 views

Ito Integral of functions of Brownian motion

How does one show that: $$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$ For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
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0answers
43 views

Simulating correlated stock paths to calculate VaR

So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...

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