Questions tagged [sde]

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59 questions
5k views

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 ...