# Questions tagged [sde]

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### optimal stopping time problem

I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
621 views

### Dynamics of FX rate

I've see a couple of places where a FX rate, denoted $X$, such as EURUSD (quoted as "the number of USD needed to buy 1 EUR") is modeled with a diffusion process / Geometric Brownian Motion ...
1 vote
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### Simulating sum of squared brownian motions process

I'm trying to simulate the following stochastic process: \begin{equation} R_t = \sum_{i=1}^nB_{i,t}^2 \end{equation} which has the following dynamics: \begin{equation} \begin{aligned} dR_t = \sum_{...
1 vote
105 views

### Show that the solution to a SDE is strong

I have the following SDE \begin{equation} dX_t = - \frac{1}{1+t}X_t dt + \frac{1}{1+t}dB_t \end{equation} that has the solution: \begin{equation} \begin{aligned} X_t = \frac{X_0 + B_t}{1+t} = \frac{...
268 views

### Solving SDE using integration factor and Ito's lemma

I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential ...
24 views

### Existence of the solution for SDE with Gaussian Process

I'm interested in the existence of the solution for a non-Ito SDE. Sloppy notation but assume a SDE given by $\dot{x}=f(x),\quad f(x)\sim GP(0,k(x,x')),$ where $f$ is a Gaussian Process with kernel $k$...
277 views

### How to incorporate momentum in Ornstein Uhlenbeck to capture overshooting in financial markets?

In modelling asset prices, it is a good idea to model it using a fair value or target price concept. Recently Carr & Prado explored this idea to find optimal stop loss/take profit levels when the ...
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### SDE Exmaple (no drift) [closed]

Assume, $X_t := ∫^t_0e^{μs}dB_s$ ($B_s$ is Brownian motion) My Reference Said $dX_t = e^{μt}dB_t$. I tried to Ito's formula to solve this (that is $df = f_tdt+f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt$) ...
1 vote
68 views

### Geometric Brownian Motion SDE

I recently saw the clip : GBM which quantpie made. here is the link https://www.youtube.com/watch?v=98xF6b0PZpo In the time 1:33 of the clip, it naturally said $dX^2 = σ^2X_t^2dt$ To proof this it ...
130 views

### Differentiability of solutions of a stochastic differential equation

I would like to clarify a confusion I have. It is well known that a Wiener process (Brownian motion) is nowhere differentiable. I have no difficulty in understanding that. But I am wondering about the ...
139 views

### Is it possibile to use Ito Formula here?

I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$. Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
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### 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 ...
49 views

### Solution to Stock Price SDE with mean reversion [duplicate]

Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part): $$S_t := S_0 + \int_{h=t_0}^{h=t}\alpha(\mu -S_h)dh + \int_{h=t_0}^{h=t}\sigma S_h dW(h)$$. I actually don't know how ...
415 views

### SDE Jump-Diffusion

If you combine the compound Poisson process with the Brownian motion you obtain the simplest case of a Jump-diffusion. Let’s define $$X_t = \mu t + \sigma W_t + J_t$$ where $W_t$ is a Wiener process ...
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188 views

### Dynamic programming and Bellman equation to obtain the maximum

This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM" Suppose an endowment economy where the representative ...
572 views

### Idea of using logarithm for solving SDE in Black-Scholes model

In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$dS = \mu S dt + \sigma S\ dW$$ I was wondering, was it common at the time they work on this ...
459 views

### How can I prove that the solution to the Heston SDE is a Markov process?

Consider the Heston model expressed as \begin{align} dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\ dV_t &= \kappa(\theta - V_t)dt + \... 4k views

### 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{...
277 views

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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 ...
545 views

### 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 ...
144 views

### 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 ...
555 views

### 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 $\gamma$ ...
1 vote
187 views

### 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 ...
45 views

### 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 ...
444 views

### 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 ...
132 views

### 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 ...
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 ...
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 ...