# Questions tagged [sde]

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### Simulation of Geometric Brownian Motion in R

Using R, I would like to simulate a sample path of a geometric Brownian motion using \begin{equation*} S(t) = S(0) \exp\left(\left(\mu - \frac{\sigma^{2}}{2}\right)t + \sigma B_{t}\right), \end{...
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### Baxter and Rennie: A question on Notation

On page 56 of Baxter and Rennie (Financial Calculus), we have The definition of a continuous stochastic process, in terms of the drift $\mu_s$ and volatality $\sigma_s$. Its important to keep in ...
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### Valuation of Cash-Or-Nothing option

Studying options pricing, I'm stuck with the following problem: The price of a stock is described by the dynamic: $$dS_t = \mu\, dt + \sigma\,dW_t$$ Compute the fair price of a Cash or Nothing ...
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### Expectation of Stochastic Differential

First of all, I am a mathematician, so I apologize for my ignorance regarding stochastic calculus. What exactly does an expression like: $$\mathbb{E}[dX_tdY_t]$$ here $X_t,Y_t$ are stochastic ...
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### Vasicek model and spot interest rate parametrised by reversion rate

By solving an SDE I want to derive the analytical results for mean and variance of the process of extended Vasicek model. $$dr(t) = \left(\eta - \gamma r(t) \right)dt + c dX(t)$$ where $\gamma$ ...
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### How to solve these SDE Problems

Quuestion1. I make a solution $r(t)$ used by Ito's lemma $r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$ Is this right? and I try to make ...
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### How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?

I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112: I am unable to understand: How does instability arise when we use the Euler scheme on X(t)? What change does ...
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### How to determine the order of convergence of the Euler-Maruyama method?

To make this simple let us consider the Geometric Brownian Motions. My questions: 1. How can I show that the Euler-Maruyama Method is convergent using GBM? 2. How can I determine the order of ...
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### Expected value of stochastic optimization

I have a optimization problem where the SDE is: $$dX(t) = [X(t)(u(t)-\beta(t))+\theta(t)]dt+X(t)u(t)\sigma dW(t), t \in [0,T], X(0) = X_0$$ where $\beta(t)$ and $\theta(t)$ are deterministic ...
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### Expectation in a stochastic differential equation

I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = \exp(W_t)$, with $W_t$ a Wiener process. I used Ito's Lemma is arrive at the SDE: \begin{align} d(X_t) = \frac{1}{2}X_t dt ...
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### Proving Flow Property of Stochastic Differential Equation

I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d ...
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### Correlated stock prices and geometric Brownian motion

I have two uncorrelated stocks which follow geometric Brownian motion, as follows \begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned} ...
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### Bounded solution for a SDE

I have this SDE $$dX(t) = [X(t)(u(t)(\delta-r)+r-\beta(t))+\theta(t)(1-\alpha(t))]dt+X(t)u(t)\sigma dW(t), t \in [0,T] \\ X(0) = X_0(1-\alpha(0))$$ I've checked some books and I find the solution ...
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### Time integral of geometric brownian motion

Suppose $S_t$ is a geometric brownian motion. Then how to understand its time integral, i.e., $Y_t=\int_0^{t}S_udu$? Is $Y_t$ still a stochastic process? How to compute the expectation of $Y_t$? ...
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