Questions tagged [sde]
The sde tag has no usage guidance.
93
questions
0
votes
1answer
122 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 ...
2
votes
0answers
21 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 $k$...
3
votes
1answer
165 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 ...
2
votes
1answer
89 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:
$$ R_t = \frac{\mathrm{d}S_t}{S_t} = \mu \...
2
votes
1answer
115 views
Finding Option Probability Density Using Local Volatility from Dupire Model
This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price?
I also asked this on Math ...
1
vote
0answers
41 views
Programming the Milstein method and computing the increments
In the wikipedia article on the Milstein method, the following python code to simulate a geometric Brownian motion is presented:
...
4
votes
1answer
108 views
Weak solution of a SDE
$\text { Consider the } \operatorname{SDE} d X_{t}=\operatorname{sign}\left(X_{t}\right) d t+d B_{t} \text { on } 0 \leq t \leq T, \text { where } \operatorname{sign}(x)=1\\
\text { for } x>0 \text ...
3
votes
0answers
63 views
Derivation of option pricing PIDE: Why does the drift need to be zero?
I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero.
...
0
votes
0answers
65 views
What is the difference between "stochastic" heat equation and just heat equation?
I am trying to understand the difference between the "stochastic" heat equation and the heat equation. Will i be wrong to say the stochastic heat equation is just the heat equationg with the ...
3
votes
1answer
202 views
Help on solving a stochastic differential equation
I am trying to solve the following SDE
$$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)=...
0
votes
2answers
122 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 $df = f_tdt+f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt$)
...
1
vote
0answers
63 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 ...
0
votes
0answers
87 views
Are these two expectations the same?
I'm studying Markov Processes and Ito diffusion, I'm just at the beginning but I can't understand the different formulation of the expectation formulated in two different books. I'm talking about ...
0
votes
1answer
99 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 ...
2
votes
0answers
133 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 ...
5
votes
2answers
255 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:
$$...
4
votes
0answers
156 views
Angular bracket notation (physics)
In a few papers I have seen the following notation:
$$
\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{\...
0
votes
0answers
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 ...
2
votes
0answers
45 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 ...
3
votes
2answers
330 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 ...
2
votes
0answers
28 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_{...
3
votes
1answer
211 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 $\...
1
vote
0answers
167 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 ...
2
votes
1answer
423 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 ...
2
votes
3answers
351 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 + \...
2
votes
1answer
3k 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{...
6
votes
1answer
239 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
$$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
3
votes
1answer
198 views
Computing Itô differential of conditional expectation process (Heston SDE)
Going through this article
on Heston's model, where the variance evolves following the SDE
\begin{equation}
\label{sd1}
d\sigma^2_t = \kappa \bigg( m - \color{red}{\sigma^2_t} \bigg)dt + \nu \sqrt {\...
2
votes
1answer
147 views
Problem of stochastic differential equation (SDE)
Please help to answer this stochastic differential equation (SDE). Thank you very much.
4
votes
1answer
202 views
Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
3
votes
0answers
317 views
Estimating Market Price of Risk
I need help with estimating market price of risk. Assume money market account and two risky assets which exposed to same two sources of risks follow process:
$dM(t)=rM(t)dt$
$dS_1(t)=S_1(t)(\mu_1dt+\...
1
vote
1answer
90 views
SDE Parameter Estimation
Have a question about "How to estimate parameters for SDE with multiple Brownian Motions ?"
Let's say $X_t$ follows the process:
$dX_t=\mu dt+\sigma_1 dW_t^1 + \sigma_2 dW_t^2 $
I think I've checked ...
6
votes
1answer
253 views
Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$
I want to solve the following SDE:
$$ 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} $$
$$ ...
0
votes
1answer
118 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 ...
2
votes
1answer
390 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 ...
4
votes
1answer
126 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 ...
0
votes
1answer
478 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
0answers
167 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 ...
2
votes
0answers
42 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 ...
3
votes
1answer
413 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 ...
3
votes
1answer
120 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 ...
3
votes
1answer
198 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 ...
2
votes
0answers
187 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 ...
0
votes
1answer
608 views
Correlated stock prices and geometric Brownian motion [closed]
I have two uncorrelated stocks which follow geometric Brownian motion, as follows
$$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$
...
3
votes
0answers
57 views
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 ...
1
vote
1answer
1k views
How to calculate mean and volatility parameters for Geometric Brownian motion?
Say I have a time series $S_K$ for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{...
8
votes
1answer
3k views
Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?
I found this great post deriving the solution to the Merton Jump-Diffusion SDE
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\prod_{j=0}^{N_t}V_j$$
The first part of ...
2
votes
1answer
473 views
Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model
I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation):
$$
dr_t = a(b-r_t)dt + \sigma dW_t
$$
Every solution is proceeding to multiply both sides of the equation by the ...
4
votes
2answers
528 views
SDE for option value
Given an SDE for an underlying:
$$dS(t) = \mu(S,t)dt+\sigma(S,t)dW(t)$$
the SDE for the value of the option $V=V(S,t)$ is given via Ito's lemma as:
$$dV = V_tdt+V_S\mu(S,t)dt+\frac{1}{2}V_{SS}\...
2
votes
1answer
218 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$
equals
$$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
