Questions tagged [sde]

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Affine Jump Diffusion

I'm currently looking into affine jump-diffusions. I would like to get to know the literature better and I know the paper by Duffie, Pan, and Singleton (2000) is a very celebrated paper. Although I ...
1 vote
0 answers
90 views

Stochastic volatility with jumps [closed]

I'm reading the Duffie, Pan, and Singleton (2000) paper now and I've stumbled upon something that seems to me as an inconsistency. Whenever I look up the SVJJ model, I see that its log-transform is ...
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0 votes
1 answer
106 views

How calculate expectation and variation of stochastic integral Based on Heston model?

I was calculated Heston volatility model. But I think it is wrong. $dS_t = \mu dt + \sqrt V_t dW_t^s$ $dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$. $dW^s_t dW^v_t = \rho dt$ take integral to ...
  • 43
2 votes
0 answers
108 views

Is the time derivative of asset returns expressible as an SDE?

Consider the following SDE for $(S_t)_{t\geq 0}$ under $\mathbb{Q}$, \begin{equation} \mathop{dS_t}=S_t\left(r\mathop{dt}+\sigma(t,S_t)\mathop{dW_t}\right), \end{equation} which (in Langevin form) may ...
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1 vote
0 answers
42 views

References for path-dependent GBMs or continuous time analog of discrete time filters

Consider a path-dependent GBM model for a stock price: $$dS_t = \mu(t, S_.)S_tdt + \sigma(t, S_.) S_t dB_t,$$ where $\mu, \sigma : [0,\infty)\times C_{[0,\infty)}\to \mathbb{R}$ are previsible path-...
0 votes
3 answers
145 views

Simulate from a SDE where drift and diffusion terms are matrices using Yuima in R

I'm trying to implement an SDE in R using Yuima. Most of examples and literature show how to implement and how the math works for SDE where drift and diffusion terms are scalar. What if I want to ...
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2 votes
1 answer
65 views

Filtering SDE for Heston Volatility

Consider a GBM model with Heston volatility: $$dS_t = \mu S_t dt + \sqrt{V_t} S_t dB_t^1$$ $$dV_t = \kappa(\theta-V_t)dt+\xi \sqrt{V_t}dB_t^2,$$ where $(B_t^1, B_t^2)$ is a correlated BM. Let $$\...
3 votes
0 answers
58 views

Feymann Kac pde with correlated process

I have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
  • 163
1 vote
0 answers
92 views

Feymann Kac for multidimensional pde

I Have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{\...
  • 163
1 vote
1 answer
55 views

Dynamics of discounted prices (multi-dimensional)

My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
2 votes
0 answers
57 views

Munk (2011) exercise 3.6

I'm trying to solve the exercise in Munk (2011). The exercise reads: "Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
1 vote
0 answers
92 views

Analytical expression for SDE

I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
4 votes
0 answers
139 views

optimal stopping time problem

I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
5 votes
2 answers
827 views

Dynamics of FX rate

I've see a couple of places where a FX rate, denoted $X$, such as EURUSD (quoted as "the number of USD needed to buy 1 EUR") is modeled with a diffusion process / Geometric Brownian Motion ...
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1 vote
1 answer
57 views

SDE linear combination of stock price

Assume that $X_t$ is a process with dynamics $dX_t = \sigma X_t dW_t$ is where $W_t$ is a standard Brownian motion. Given two deterministic functions $p(t)$ and $q(t)$, compute $\mathbb{E}[p(t)X(t)+q(...
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2 votes
0 answers
128 views

How did Bachelier characterize the Brownian motion?

The model for a stock price $$ dS_t=\mu dt + \sigma dB_t $$ where $B_t$ is a Brownian motion on $(\Omega, \mathcal{F},P)$, is commonly attributed to the work that Bachelier has carried out in his PhD ...
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1 vote
0 answers
54 views

Is there a closed form solution to the following system of SDEs?

Suppose we have the system \begin{align} dr_t=\alpha_r(x_t-r_t)dt+\sigma_rdW_t^r\\ dx_t=\alpha_x(\bar{x}-x_t)dt+\sigma_xdW_t^x\\ \end{align} As this system is affine, I believe there should be an easy ...
  • 123
3 votes
1 answer
194 views

Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$

Assume $(X_t)_{t\geq 0}$ follows an SDE of the form: $$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$ where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
  • 165
0 votes
0 answers
136 views

Simulating sum of squared brownian motions process

I'm trying to simulate the following stochastic process: \begin{equation} R_t = \sum_{i=1}^nB_{i,t}^2 \end{equation} which has the following dynamics: \begin{equation} \begin{aligned} dR_t = \sum_{...
1 vote
1 answer
109 views

Show that the solution to a SDE is strong

I have the following SDE \begin{equation} dX_t = - \frac{1}{1+t}X_t dt + \frac{1}{1+t}dB_t \end{equation} that has the solution: \begin{equation} \begin{aligned} X_t = \frac{X_0 + B_t}{1+t} = \frac{...
2 votes
2 answers
584 views

Solving SDE using integration factor and Ito's lemma [closed]

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 ...
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2 votes
0 answers
29 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 $k$...
  • 21
3 votes
2 answers
730 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 ...
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2 votes
1 answer
104 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
1 answer
260 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
0 answers
105 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: ...
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4 votes
1 answer
315 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
0 answers
76 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
0 answers
84 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
1 answer
230 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
2 answers
176 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
0 answers
78 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
1 answer
161 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
0 answers
141 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 ...
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5 votes
2 answers
301 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: $$...
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4 votes
0 answers
223 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{\...
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0 votes
0 answers
81 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
0 answers
52 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 ...
  • 5,513
3 votes
2 answers
557 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 ...
  • 33
2 votes
0 answers
31 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_{...
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3 votes
1 answer
321 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
0 answers
196 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
1 answer
732 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 ...
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1 vote
3 answers
579 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 + \...
user avatar
2 votes
1 answer
4k 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{...
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6 votes
1 answer
363 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(...
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3 votes
1 answer
267 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 {\...
  • 33
2 votes
1 answer
184 views

Problem of stochastic differential equation (SDE)

Please help to answer this stochastic differential equation (SDE). Thank you very much.
5 votes
1 answer
219 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
0 answers
335 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+\...