# Questions tagged [sde]

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59 questions
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### Strictly local martingales: what is the intuition behind them?

A process $X_t$ is a local martingale if for each increasing sequence of stopping times $\{\tau_k,k=1,2,...\}$ the stopped process is a martingale. All true martingales are local martingales, but the ...
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### Processes used in quant finance

What are the main stochastic processes (and their SDE) used in quant finance? For example to model currency prices, stock prices, etc.
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### Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?

I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process: $$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$ with the ...
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### Itô diffusion processes in finance with unknown distribution at a terminal value

In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
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### Shortcomings of generalized Brownian motion for asset price modelling

I'm simply interested on hearing some views on which shortcomings arise by using the (multidimensional) SDE $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$ as a model for asset prices. I know ...
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### How to do a Brownian Bridge with quasi-random numbers in the Heston model?

I'm required to use the Euler Monte Carlo method to compute the option price under Heston model settings. I know from some paper that the convergence is volatile for the Heston model with a plain ...
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### 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 ...
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### 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 ...
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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^{-... 2answers 111 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 $$dF(t,T) = \sigma(t,T)F(t,T) dW_t + F(t,T)(\exp(e^{-... 1answer 420 views ### Methods of SDE Calibration There is somewhere summary of methods that can be used to estimate parameters of SDE? I currently using MLE and regression due to linear dependence between samples. I searching for something ... 1answer 102 views ### Meaning of w in SDE I'm missing meaning of w in typical SDE like dX_t(w) = f_t(X_t(w)) + \sigma(X_t(w))dW_t, in context of w \in F_{xxx}. Does it mean that both w is one of events that could happen before ... 1answer 79 views ### Expected value of stochastic optimization I have a optimization problem where the SDE is:$$ 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 $$where \beta(t) and \theta(t) are deterministic ... 1answer 178 views ### How to adjust Geometric Brownian Motion to be monotone? I want to use stochastic process to model subscriber's mobile data consumption as time going in a month. So I think about Geometric Brownian Motion. However, people's cumulative data consumption ... 3answers 234 views ### Understanding the HJM drift condition's dimensions In an HJM model the forward rate dynamics follow$$ df_t(T) =a_t(f_t(T))dt+b_t(f_t(T))dW_t $$where W_t is a d-dimensional brownian motion, b_t takes values in \mathbb{R}^{d\times d} and a_t ... 2answers 202 views ### Transformation into Martingale If f is some function of BV on \mathbb{R} and dZ_t = f(W_t)dW_t + \mu_t dt (W_t is a 1-dimensional standard Brownian Motion), then what choice of real valued function F makes: \begin{... 0answers 37 views ### How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability? I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112: I am unable to understand: How does instability arise when we use the Euler scheme on X(t)? What change does ... 1answer 146 views ### Two papers - two different solutions of the Ornstein-Uhlenbeck process Bernal 2016 says that the solution of$$ 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)... 1answer 296 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 ... 4answers 802 views ### Exploding Libor Rates in Libor Market Model I have implemented the Libor Market Model in Matlab. When I generate a number of paths, I notice that some of them explode. Does anybody have an idea what could cause this? I already tried solving ... 1answer 171 views ### Question about the stochastic differential equation in the Merton model in the following stochastic differential equation merton model we have $$\frac{ds}{s}=(\alpha-\lambda k)dt+\sigma dW+dq$$ where$\alpha$is the instantaneous expected return on the stock;$\sigma^2$... 1answer 168 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 $$\mathrm{d}L(t) = L(t)\left(\mu(t)\... 1answer 260 views ### Prove that E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t] Let T > 0. Let (\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P) be a filtered probability space where \mathscr F_t = \sigma(W_u, u \in [0,t]) where W_t is standard Brownian ... 1answer 131 views ### Bracket-Notation in SDEs I often come across the following notation in my script, and I have not found it anywhere else. While our lecturer insists it is of utmost importance to write this way in his exams, he yet failed to ... 1answer 93 views ### How to calculate mean and volatility parameters for Geometric Brownian motion? Say I have a time series S_K for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{... 1answer 118 views ### 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} ... 1answer 596 views ### Integration in the Hull-White SDE I'm stuck in solving the SDE in Hull-White interest rate model. I do not have a thorough background in math (only Real Analysis during my blissful undergrad years), so I am having trouble ... 1answer 265 views ### Geometric Brownian Motion: d(S) vs. d(ln(S)) I am quoting from "Tools for Computational Finance, 5th Edition" [Seydel]. I wonder whether the histogram of simulations of the first (yellow) SDE makes sense... especially given that Seydel (... 0answers 60 views ### How to solve these SDE Problems Quuestion1. I make a solutionr(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)$Is this right? and I try to make ... 0answers 58 views ### Proving Flow Property of Stochastic Differential Equation I am trying to show that$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 ... 0answers 336 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$Y_t$? ... 0answers 135 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$...
Maybe this is the right place for my question: I have a system of coupled FBSDEs in 3 dimensions as follows (in cartesian coordinates):  \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
I'm working on a model in which there are two processes, $H$ and $L$, and the final variable to model starts as $H$ and then whenever a jump occurs, an instance of the $L$ processes starts and ...