Questions tagged [sde]

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Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$

Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$....
Mr. Ivan's user avatar
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36 views

Modelling support and resistance using sde

This initiative was sparked by the identification of cointegrated pairs, fitting them to an OU process, and devising an optimal strategy based on the OU process—areas that have already been well ...
lukas kiss's user avatar
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0 answers
23 views

State space equation of CARMA(p,q) processes

Thanks for visting my question:) I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by: for $p>q$ the ...
Valentin's user avatar
0 votes
1 answer
95 views

Maximizing the expected log utility

Let's assume that we have a self-financing portfolio made by $\delta_t$ shares and $M_t$ cash, so that its infinitesimal variation is: $$ dW_t = rM_t \, dt + \delta_t \, dS_t $$ We define $\alpha_t$ ...
Alessandro's user avatar
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1 answer
67 views

Solving the SDE for GBM [closed]

Let's assume that we have the following stochastic differential equation: $dX_t = \mu X_t dt + \sigma X_tdW_t$ and that we have to prove that this is its solution: $X_t = X_0 \exp\left(\left(\mu -{\...
Alessandro's user avatar
2 votes
0 answers
49 views

Benchmark Model for Path-Dependant Monte Carlo Simulations?

As part of my research for my masters thesis, I'm testing out the effectiveness of some different models in Monte Carlo simulations for path dependant options. I will be attempting a model-free ...
Rudy S's user avatar
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72 views

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 ...
Marc Allan's user avatar
1 vote
0 answers
108 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 ...
CasMath's user avatar
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1 answer
164 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 ...
JMNQC's user avatar
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121 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 ...
UNOwen's user avatar
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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-...
Nap D. Lover's user avatar
0 votes
3 answers
194 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 ...
Nic's user avatar
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2 votes
1 answer
92 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 $$\...
Nap D. Lover's user avatar
3 votes
0 answers
67 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}{...
Pefok's user avatar
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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{\...
Pefok's user avatar
  • 163
1 vote
1 answer
58 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{...
John Stevens's user avatar
2 votes
0 answers
59 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} \...
John Stevens's user avatar
1 vote
0 answers
97 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 ...
John Stevens's user avatar
4 votes
0 answers
162 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 ...
Khalil Belghouat's user avatar
5 votes
2 answers
1k 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 ...
Landscape's user avatar
  • 548
1 vote
1 answer
68 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(...
Siron's user avatar
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2 votes
0 answers
178 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 ...
Mr Frog's user avatar
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1 vote
0 answers
59 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 ...
Carl's user avatar
  • 123
3 votes
1 answer
203 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\...
BS.'s user avatar
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0 answers
180 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_{...
Alejandro Andrade's user avatar
1 vote
1 answer
114 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{...
Alejandro Andrade's user avatar
2 votes
2 answers
991 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 ...
Bohdan_'s user avatar
  • 21
2 votes
0 answers
31 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 $k$...
Thomas's user avatar
  • 21
3 votes
2 answers
1k 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 ...
Aian 's user avatar
  • 31
2 votes
1 answer
112 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 \...
robsmith11's user avatar
2 votes
1 answer
387 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 ...
curious123456789's user avatar
1 vote
0 answers
175 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: ...
StefanH's user avatar
  • 201
4 votes
1 answer
387 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 ...
Stochastichelp's user avatar
3 votes
0 answers
79 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. ...
Leguan3000's user avatar
0 votes
0 answers
86 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 ...
Ratanna's user avatar
3 votes
1 answer
266 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)=...
Martin_Gale's user avatar
0 votes
2 answers
224 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 $df = f_tdt+f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt$) ...
user13232877's user avatar
1 vote
0 answers
82 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 ...
user13232877's user avatar
0 votes
1 answer
184 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 ...
user144410's user avatar
2 votes
0 answers
143 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 ...
RedLapm's user avatar
  • 33
5 votes
2 answers
342 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: $$...
Jan Stuller's user avatar
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4 votes
0 answers
279 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{\...
fwd_T's user avatar
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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 ...
Question Anxiety's user avatar
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 ...
Jan Stuller's user avatar
  • 5,998
3 votes
2 answers
699 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 ...
RedZoro's user avatar
  • 33
2 votes
0 answers
32 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_{...
Victor's user avatar
  • 509
3 votes
1 answer
385 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 $\...
Pinnochio Da Firenze's user avatar
0 votes
0 answers
228 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 ...
Hunger Learn's user avatar
2 votes
1 answer
882 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 ...
TmSmth's user avatar
  • 385
1 vote
3 answers
719 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 + \...
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