# Questions tagged [sde]

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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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### 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 ...
1k views

### 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 ...
183 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 $a_t$ ...
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{...