## Search found 114 matches

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April 23rd, 2020, 2:16 am
Forum: Numerical Methods Forum
Topic: How to solve this quasi-variational inequality (QVI) numerically?
Replies: 6
Views: 733

### Re: How to solve this quasi-variational inequality (QVI) numerically?

Yes, i t is a minimisation problem at each time level. I hope this is a non-trival example: $$0=\min[ \frac{\partial{h(t,s)}}{\partial{t}}+ \min \limits_{a\in A} (\mathbb{E}(h(t,s+\xi a)-h(t,s-\xi a))) ; h(t,s)-b; -h(t,s)-b]$$ where $\mathbb{E}$ is the expectation operator with respect to the B...
April 22nd, 2020, 5:52 am
Forum: Numerical Methods Forum
Topic: How to solve this quasi-variational inequality (QVI) numerically?
Replies: 6
Views: 733

### Re: How to solve this quasi-variational inequality (QVI) numerically?

Maybe the QVI above can be simplified as below: $$0=min( \frac{\partial{h}}{\partial{t}}+\phi_1 h; \phi_2 h; \phi_3 h)$$ where $\phi_1,\phi_2$, and $\phi_3$ indicate some combinations of  operators (e.g. differential operator or expectation operator) on an unknown function $h$. Is there...
April 22nd, 2020, 1:05 am
Forum: Numerical Methods Forum
Topic: How to solve this quasi-variational inequality (QVI) numerically?
Replies: 6
Views: 733

### Re: How to solve this quasi-variational inequality (QVI) numerically?

Sorry,I have updated with a full link
April 21st, 2020, 10:58 am
Forum: Numerical Methods Forum
Topic: How to solve this quasi-variational inequality (QVI) numerically?
Replies: 6
Views: 733

### How to solve this quasi-variational inequality (QVI) numerically?

<a href= https://i.ibb.co/zbWQyPg/QVI.jpg " /> I found this equation from Eq (10) in the paper Hedge and Speculate: Replicating Option Payoffs with Limit and Market Orders by Alvaro Cartea, Luhui Gan, and Sebastian Jaimungal.   I think I can solve the first equation which is for the use of limit or...
April 13th, 2020, 10:11 am
Forum: Numerical Methods Forum
Topic: How to solve this ODE?
Replies: 9
Views: 1014

Thank you so much for your suggestions. I think I can solve this ODE numerically with (1) A psuedo time term is added to the equation as below: $$min[rV(x)-\frac{\sigma^2}{2} \frac{d^2V(x)}{dx^2}-\mu(\theta-x) \frac{dV(x)}{dx}+\frac{dV(x)}{dt}, V(x)-(x-c)]=0$$ (2) boundary conditions: $\frac {\pa... March 30th, 2020, 10:06 am Forum: Numerical Methods Forum Topic: How to solve this ODE? Replies: 9 Views: 1014 ### Re: How to solve this ODE? Thank you all for your suggestions. Following Alan’s suggestions to split the integration limits, I can do the integration by a change of variable method to go around the singularity point. However, I am still not sure on the below two questions: (1) Boundary conditions It seems that as \(x\rig... March 27th, 2020, 12:35 pm Forum: Numerical Methods Forum Topic: How to solve this ODE? Replies: 9 Views: 1014 ### How to solve this ODE? The following ODE is obtained from the Ornstein-Uhlenbeck process. I read it from the paper Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit.  \frac{\sigma^2}{2} \frac{d^2u(x)}{dx^2}+\mu(\theta-x) \frac{du(x)}{dx}=ru(x)  The paper does not provide the boundary conditions ... February 18th, 2020, 10:02 am Forum: Technical Forum Topic: Is Ito’s lemma applicable to a diffusion process with transition probability? Replies: 2 Views: 1627 ### Re: Is Ito’s lemma applicable to a diffusion process with transition probability? Thank you, Alan. February 14th, 2020, 8:15 am Forum: Technical Forum Topic: Is Ito’s lemma applicable to a diffusion process with transition probability? Replies: 2 Views: 1627 ### Is Ito’s lemma applicable to a diffusion process with transition probability? I want to model a continuous variable \(X_t$ by a stochastic process. With probability $1-q(X_t)dt$ at an infinitesimal period $dt$, it is a diffusion process. However, with probability $q(X_t)dt$, $X_t$ may jump to $Y_t$. The probability density function of $Y_t$ is $p(Y_t)$. If I am...
November 21st, 2019, 12:28 am
Forum: General Forum
Topic: How to make ito's forumula for jump-diffusion a martingale
Replies: 3
Views: 1911

### Re: How to make ito's forumula for jump-diffusion a martingale

Thank you so much Alan. That's my also my understanding of this problem before reading page 26 of http://people.ucalgary.ca/~aswish/JumpProcesses.pdf, based on which after vanishing the $dt$ term, there should be an additional $-E [Z_{N_t}] \eta_t f'(X_t)$  term in the $f(t,x)$ PIDE.
November 20th, 2019, 7:41 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 2173

### Re: What’s Ito’s lemma for Poisson process in this function?

Thank you Alan. Maybe I confuse it with something. I have made a new thread for this problem.
November 20th, 2019, 7:40 am
Forum: General Forum
Topic: How to make ito's forumula for jump-diffusion a martingale
Replies: 3
Views: 1911

### How to make ito's forumula for jump-diffusion a martingale

$dY_t=Z_{N_t} dN_t$ is a compound Poisson process with intensity $\lambda$ and $Z_{N_t}$ is a random variable for the jump size. $dW_t$ is Brownian motion. The jump diffusion process $X_t$ is defined as $$dX_t=\nu_t dt+u_t dW_t + \eta_t dY_t$$ So the Ito's lemma for this jump diffusio...
November 19th, 2019, 1:17 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 2173

### Re: What’s Ito’s lemma for Poisson process in this function?

Thank you. I have corrected the typo. However, in $dY_t=Z_{N_t} dN_t$, $Z_{N_t}$ is a random variable, indicating the random jump size. I think it makes sense to include an expectation w.r.t.  $Z_{N_t}$ in the drift term, as in page 26 of  http://people.ucalgary.ca/~aswish/JumpProcesses.pdf ...
November 18th, 2019, 8:53 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 2173

### Re: What’s Ito’s lemma for Poisson process in this function?

Thank you Alan. May I ask one more question? In page 645 of your given file, there is an Ito formula for the compensated compund Poisson process: $$f(Y_t)=f(0)+\int_0^t \!(f(Y_s)-f(Y_s-))(dN_s-\lambda ds)+\lambda \int_0^t \!(f(Y_s)-f(Y_s-))ds$$ where $dY_t=Z_{N_t} dN_t$ is a compound Poisson proc...
November 16th, 2019, 10:25 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 2173

### What’s Ito’s lemma for Poisson process in this function?

If $N_t$ is a Poisson process with intensity $\lambda$,and $dX_t=\delta dN_t$, $q_t=-N_t$, then the Ito's lemma for function $H(X_t)$ should be $$dH(X_t)=[H(X_t+\delta)-H(X_t)]dN_t$$ For the function   $H(X_t,q_t)$, why it is not something like this?  dH(X_t,q_t)=[H(X_t+\delta,q_t)-H...
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