Questions tagged [sde]
The sde tag has no usage guidance.
78
questions
4
votes
0answers
97 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:
$$
\biggr\langle \int_0^T e^{-rt}s^2 \frac{d^2P_{\hat{\...
0
votes
0answers
64 views
Alternatives to Lognormality for negative Prices
If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ...
2
votes
0answers
43 views
Solution to Stock Price SDE with mean reversion [duplicate]
Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part):
$$ 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) $$.
I actually don't know how ...
3
votes
2answers
144 views
SDE Jump-Diffusion
If you combine the compound Poisson process with the Brownian motion you obtain the simplest case
of a Jump-diffusion. Let’s define
$$X_t = \mu t + \sigma W_t + J_t$$
where $W_t$ is a Wiener process ...
2
votes
0answers
25 views
Expression for the expectation of Integrated variance in case of GARCH(1,1) process
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{...
3
votes
1answer
184 views
Application of Ito's Lemma in expected utility theory
An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests
A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
0
votes
0answers
22 views
Statistical test for comparing two different speed of mean reversion parameters for CIR model
I am trying to compare two different values of speed of mean reversion parameter for CIR model.
I would like to know if there exists a statistical test for comparing these two parameters.
the estimate ...
1
vote
0answers
115 views
Dynamic programming and Bellman equation to obtain the maximum
This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM"
Suppose an endowment economy where the representative ...
2
votes
1answer
110 views
Idea of using logarithm for solving SDE in Black-Scholes model
In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$
I was wondering, was it common at the time they work on this ...
2
votes
3answers
203 views
How can I prove that the solution to the Heston SDE is a Markov process?
Consider the Heston model expressed as
\begin{align}
dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\
dV_t &= \kappa(\theta - V_t)dt + \...
1
vote
1answer
1k views
Simulation of Geometric Brownian Motion in R
Using R, I would like to simulate a sample path of a geometric Brownian motion using
\begin{equation*}
S(t) = S(0) \exp\left(\left(\mu - \frac{\sigma^{2}}{2}\right)t + \sigma B_{t}\right),
\end{...
6
votes
1answer
150 views
Why is the numeraire in the LGM model tradeable?
I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to
$$dX(t) = \alpha(t) dW^N(t)$$
wrt the numeraire
$$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
3
votes
1answer
140 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
132 views
Problem of stochastic differential equation (SDE)
Please help to answer this stochastic differential equation (SDE). Thank you very much.
4
votes
1answer
192 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
217 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
80 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
211 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:
$$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$
$$ ...
0
votes
1answer
83 views
Baxter and Rennie: A question on Notation
On page 56 of Baxter and Rennie (Financial Calculus), we have
The definition of a continuous stochastic process, in terms of the drift $\mu_s$ and volatality $\sigma_s$. Its important to keep in ...
2
votes
1answer
186 views
Valuation of Cash-Or-Nothing option
Studying options pricing, I'm stuck with the following problem:
The price of a stock is described by the dynamic:
$$dS_t = \mu\, dt + \sigma\,dW_t$$
Compute the fair price of a Cash or Nothing ...
3
votes
1answer
98 views
Expectation of Stochastic Differential
First of all, I am a mathematician, so I apologize for my ignorance regarding stochastic calculus. What exactly does an expression like:
$$
\mathbb{E}[dX_tdY_t]
$$
here $X_t,Y_t$ are stochastic ...
0
votes
1answer
255 views
Vasicek model and spot interest rate parametrised by reversion rate
By solving an SDE I want to derive the analytical results for mean and variance of the process of extended Vasicek model.
$$
dr(t) = \left(\eta - \gamma r(t) \right)dt + c dX(t)
$$
where $\gamma$ ...
1
vote
0answers
111 views
How to solve these SDE Problems
Quuestion1.
I make a solution $r(t)$ used by Ito's lemma
$r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$
Is this right?
and I try to make ...
2
votes
0answers
40 views
How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?
I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112:
I am unable to understand:
How does instability arise when we use the Euler scheme on X(t)?
What change does ...
3
votes
1answer
271 views
How to determine the order of convergence of the Euler-Maruyama method?
To make this simple let us consider the Geometric Brownian Motions.
My questions:
1. How can I show that the Euler-Maruyama Method is convergent using GBM?
2. How can I determine the order of ...
3
votes
1answer
97 views
Expected value of stochastic optimization
I have a optimization problem where the SDE is:
$$
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
$$
where $\beta(t)$ and $\theta(t)$ are deterministic ...
3
votes
1answer
153 views
Expectation in a stochastic differential equation
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = \exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
\begin{align}
d(X_t) = \frac{1}{2}X_t dt ...
1
vote
0answers
113 views
Proving Flow Property of Stochastic Differential Equation
I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE:
\begin{equation*}
d X(u)=b(X(u))d ...
0
votes
1answer
367 views
Correlated stock prices and geometric Brownian motion [closed]
I have two uncorrelated stocks which follow geometric Brownian motion, as follows
$$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$
...
3
votes
0answers
46 views
Bounded solution for a SDE
I have this SDE
$$
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))
$$
I've checked some books and I find the solution ...
1
vote
1answer
627 views
How to calculate mean and volatility parameters for Geometric Brownian motion?
Say I have a time series $S_K$ for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{...
6
votes
0answers
1k 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
358 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 ...
3
votes
2answers
361 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:
$$dV = V_tdt+V_S\mu(S,t)dt+\frac{1}{2}V_{SS}\...
2
votes
1answer
188 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ 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)...
4
votes
3answers
408 views
What is the purpose of short rate models?
Just venturing into quantitative finance and studying short rate models (Vasicek, CIR, Hull-White etc.). Wanted to ask a very simple intuitive question. How would a practitioner use these models? I ...
1
vote
1answer
255 views
Dynamics of LIBOR foward rate under T-forward measure
Assume that under the physical measure $\mathbb{P}$ we have for the LIBOR forward rate $L(t):=L(t;S,T) = \frac{1}{T-S}\left(\frac{P(t,S)}{P(t,T)}-1\right)$ that
$$
\mathrm{d}L(t) = L(t)\left(\mu(t)\...
1
vote
0answers
404 views
Time integral of geometric brownian motion
Suppose $S_t$ is a geometric brownian motion. Then how to understand its time integral, i.e., $Y_t=\int_0^{t}S_udu$?
Is $Y_t$ still a stochastic process?
How to compute the expectation of $Y_t$?
...
2
votes
2answers
533 views
Hawkes process intensity solution
Hail to all,
I am struggling to solve the following SDE for intensity:
$d\lambda_t = \kappa(\rho(t) - \lambda_t)dt + \delta dN_t $
I know to expect the solution in the form of
$\lambda_t = c(0)e^{-...
2
votes
1answer
216 views
How to adjust Geometric Brownian Motion to be monotone?
I want to use stochastic process to model subscriber's mobile data consumption as time going in a month. So I think about Geometric Brownian Motion.
However, people's cumulative data consumption ...
0
votes
0answers
145 views
Why do we have to use discretization methods for SDE?
I haven't found the answer for the question above in google. Why can't we just discretize the equation instead of using methods like euler or milstein for the discretization.
0
votes
1answer
470 views
Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]
Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value:
\begin{align*}
\mathbb{E}(S_T\log(S_T)).
\end{align*}
Using a change of measure we can write
...
0
votes
2answers
570 views
CIR discretization Milstein scheme
The CIR model for spot rate $r_t$ is:
$$dr_t=(\eta-\gamma r_t)dt+\sqrt{\alpha r_t} dW_t$$
where $\eta, \gamma, \alpha$ are constants.
How to express this SDE in discrete form using Milstein scheme?
...
1
vote
0answers
146 views
Characteristic function of SDE with coefficients depending upon second coupled SDE
Say we have the following two SDEs driven by the same single Brownian:
$$ dx_t = -0.5\sigma^2g(\psi)^2dt + \sigma g(\psi)dW_t \quad\quad d\psi_t = -(H\psi_t+0.5\sigma^2)dt + \sigma dW_t$$
where $...
2
votes
1answer
587 views
How do you find variance of a sde?
I know how to find the mean of an SDE: write it on integral form, take derivative, solve a simple ODE.
But what to do when we want a variance?
In my case, $$X_{T + \delta t} = X_T + \int_T^{T + \...
3
votes
2answers
214 views
Moment Ito's Process Proof
I have a following stochastic integral - related problem that I have difficulty to solve:
Given
\begin{equation}
dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t
\end{equation}
and the second moment of $...
2
votes
2answers
123 views
How is this SDE interpreted?
I saw this model
$$\frac{dF(t,T)}{F(t,T)} = \sigma(t,T) dW_t + (\exp(e^{-a(T-t)}dJ_t)-1) + \mu_J(t,T)dt$$
to model the forward curve. Rewriting
$$dF(t,T) = \sigma(t,T)F(t,T) dW_t + F(t,T)(\exp(e^{-...
-2
votes
1answer
272 views
How to solve $dX_t = X_t(\sigma_t dW_t + \mu_t dt)$?
Solve the SDE $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$ where $\sigma_t$,$\mu_t$ are deterministic.
Attempted solution
We have $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$
Let $f(x) = \log X$, applying ...
1
vote
1answer
408 views
Stochastic differential equation of a Brownian Motion
I have two questions about Ito's Lemma with respect to calculating SDEs. The examples are simple enough, but I haven't found an answer yet.
Take $W_t$ as a standard Brownian motion and $g(s)$ as some ...
0
votes
0answers
100 views
Approximating an SDE for Volatility Estimation
Consider the SDE
$$
dT(t) = ds(t) + a(s(t) - T(t))dt + \sigma dW(t)
$$
where $s(t)$ is a deterministic function that turns out to be the long-term mean (this SDE is used to model daily temperature, so ...
