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 ...
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5 votes
2 answers
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 ...
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1 vote
1 answer
36 views

SDE linear combination of stock price

Assume that $X_t$ is a process with dynamics $dX_t = \sigma X_t dW_t$ is where $W_t$ is a standard Brownian motion. Given two deterministic functions $p(t)$ and $q(t)$, compute $\mathbb{E}[p(t)X(t)+q(...
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2 votes
0 answers
91 views

How did Bachelier characterize the Brownian motion?

The model for a stock price $$ dS_t=\mu dt + \sigma dB_t $$ where $B_t$ is a Brownian motion on $(\Omega, \mathcal{F},P)$, is commonly attributed to the work that Bachelier has carried out in his PhD ...
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  • 221
1 vote
0 answers
49 views

Is there a closed form solution to the following system of SDEs?

Suppose we have the system \begin{align} dr_t=\alpha_r(x_t-r_t)dt+\sigma_rdW_t^r\\ dx_t=\alpha_x(\bar{x}-x_t)dt+\sigma_xdW_t^x\\ \end{align} As this system is affine, I believe there should be an easy ...
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  • 123
3 votes
1 answer
184 views

Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$

Assume $(X_t)_{t\geq 0}$ follows an SDE of the form: $$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$ where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
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  • 165
0 votes
0 answers
94 views

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_{...
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1 vote
1 answer
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{...
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0 votes
1 answer
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 ...
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2 votes
0 answers
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 $k$...
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  • 21
3 votes
1 answer
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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  • 31
2 votes
1 answer
96 views

Confused by derivation of variance swap payoff

I'm trying to follow https://en.wikipedia.org/wiki/Variance_swap#Pricing_and_valuation where it seems to me that they're just subtracting a simple return: $$ R_t = \frac{\mathrm{d}S_t}{S_t} = \mu \...
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2 votes
1 answer
174 views

Finding Option Probability Density Using Local Volatility from Dupire Model

This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price? I also asked this on Math ...
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1 vote
0 answers
63 views

Programming the Milstein method and computing the increments

In the wikipedia article on the Milstein method, the following python code to simulate a geometric Brownian motion is presented: ...
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  • 201
4 votes
1 answer
207 views

Weak solution of a SDE

$\text { Consider the } \operatorname{SDE} d X_{t}=\operatorname{sign}\left(X_{t}\right) d t+d B_{t} \text { on } 0 \leq t \leq T, \text { where } \operatorname{sign}(x)=1\\ \text { for } x>0 \text ...
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3 votes
0 answers
69 views

Derivation of option pricing PIDE: Why does the drift need to be zero?

I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero. ...
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0 votes
0 answers
71 views

What is the difference between "stochastic" heat equation and just heat equation?

I am trying to understand the difference between the "stochastic" heat equation and the heat equation. Will i be wrong to say the stochastic heat equation is just the heat equationg with the ...
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3 votes
1 answer
212 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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0 votes
2 answers
152 views

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 $df = f_tdt+f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt$) ...
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1 vote
0 answers
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 ...
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0 votes
1 answer
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 ...
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2 votes
0 answers
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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  • 33
5 votes
2 answers
277 views

Can a Process with a Stochastic Drift be a Martingale?

I have repeatedly come across the statement that "a process with a drift cannot be a martingale". Is this true also for stochastic drifts? Suppose I have a process with a stochastic drift: $$...
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  • 5,106
4 votes
0 answers
190 views

Angular bracket notation (physics)

In a few papers I have seen the following notation: $$ \langle X_t \rangle $$ Also, in Bergomi's book, at page 8, we have the following equality: $$ \biggr\langle \int_0^T e^{-rt}s^2 \frac{d^2P_{\hat{\...
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0 votes
0 answers
76 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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2 votes
0 answers
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 ...
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  • 5,106
3 votes
2 answers
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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  • 33
2 votes
0 answers
29 views

Expression for the expectation of Integrated variance in case of GARCH(1,1) process

I have the following SDE (GARCH(1,1)) for the instantaneous variance: $$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$ I would like to find an expression for $IV_t = E[\int_{...
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  • 489
3 votes
1 answer
236 views

Application of Ito's Lemma in expected utility theory

An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
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0 votes
0 answers
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 ...
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2 votes
1 answer
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 ...
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2 votes
3 answers
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 + \...
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2 votes
1 answer
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{...
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  • 101
6 votes
1 answer
277 views

Why is the numeraire in the LGM model tradeable?

I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to $$dX(t) = \alpha(t) dW^N(t)$$ wrt the numeraire $$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
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  • 183
3 votes
1 answer
232 views

Computing Itô differential of conditional expectation process (Heston SDE)

Going through this article on Heston's model, where the variance evolves following the SDE \begin{equation} \label{sd1} d\sigma^2_t = \kappa \bigg( m - \color{red}{\sigma^2_t} \bigg)dt + \nu \sqrt {\...
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  • 33
2 votes
1 answer
157 views

Problem of stochastic differential equation (SDE)

Please help to answer this stochastic differential equation (SDE). Thank you very much.
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4 votes
1 answer
207 views

Evaluating the SDE $dX_t = t\,dS_t$

The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
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3 votes
0 answers
329 views

Estimating Market Price of Risk

I need help with estimating market price of risk. Assume money market account and two risky assets which exposed to same two sources of risks follow process: $dM(t)=rM(t)dt$ $dS_1(t)=S_1(t)(\mu_1dt+\...
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1 vote
1 answer
101 views

SDE Parameter Estimation

Have a question about "How to estimate parameters for SDE with multiple Brownian Motions ?" Let's say $X_t$ follows the process: $dX_t=\mu dt+\sigma_1 dW_t^1 + \sigma_2 dW_t^2 $ I think I've checked ...
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6 votes
1 answer
284 views

Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$

I want to solve the following SDE: $$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$ Integrating, I get: $$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$ $$ ...
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0 votes
1 answer
126 views

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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2 votes
1 answer
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 ...
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  • 123
4 votes
1 answer
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 ...
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0 votes
1 answer
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 $\gamma$ ...
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1 vote
0 answers
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 ...
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  • 11
2 votes
0 answers
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 ...
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  • 31
3 votes
1 answer
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 ...
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  • 1,585
3 votes
1 answer
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 ...
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3 votes
1 answer
215 views

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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  • 489
2 votes
0 answers
219 views

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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  • 636