# Questions tagged [differential-equations]

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### Infinitesimal Generators and Expectation of First Hitting Time as Solution of Differential Equation

I've been learning about Linear Diffusions and how their infinitesimal generators can be used to relate expectations and deterministic differential equations. Let $X$ be an one-dimensional diffusion ...
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### Term structure equation in the Vasicek model

Consider the SDE $$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$ Let $$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r).$$ (F can be interpreted as price of a zero coupon bond with ...
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### Need to solve the stochastic differential equation of Vasicek Model

How to solve the stochastic differential equation of the Vasicek model for the analysis of credit risk? I search in the article "The Distribution of loan portfolio value" (Vasicek) but he doesn't ...
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Given the CIR process $\ dX_t = (a − bX_t ) dt + \sigma \sqrt{X_t}dW_t$ - I want to show that its Milstein scheme is $\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\... 0answers 473 views ### Differential Sortino Ratio I'm attempting to optimize a reinforcement learning system to maximize risk adjusted returns. I have currently defined the reward as the differential Sharpe ratio at each step: the influence of the ... 0answers 31 views ### Implicit Scheme for Cox-Ingersoll-Ross Model PDE I am considering the PDE for the price of a bond$V(r,t)$with maturity$T$under the Cox-Ingersoll-Ross model, $$V_t+\frac12\sigma^2rV_{rr}+\nu(\theta-r)V_r-rV=0\quad r>0, t\in(0,1)$$ with ... 0answers 23 views ### Backward differential equation with binomial tree I'm trying to understand/solve the following question but I honestly don't know what it's even asking about. I've included my attempt following the picture of the question. I would approximate the ... 0answers 151 views ### CIR model. Is there a closed-form solution or even a good proxy of analytical solution? Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE \begin{equation} dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1} \end{equation} ? Notice that$\{r_t\}$is our ... 0answers 45 views ### Differential Equation of Type Ricatti as part of Short Rate Model I currently despair of the following solution of a differiental equation (Ricatti Type) as part of a short rate model: $$B_t=\frac{1}{2}aB^2+bB-1$$ First I am "guessing" a particular solution $$... 0answers 54 views ### Computing squared returns given differential equation for prices I am looking for general advice on how to start tackling the problem below. My background in math is fairly bad when it comes to stochastic differential equations, but if you have any recommendations ... 0answers 62 views ### Probability distributions as solutions to differential equations As far as what I can tell, the popularity of the Black-Scholes-Merton model partly stems from the fact that it formulates the value of a derivative in a differential form in which the solution has a ... 0answers 70 views ### Boundary condition of lookback option This is a well know conclusion of the boundary condition of lookback option. Here$$\dfrac{d S_t}{S_t} = (\mu - D)dt + \sigma ... 0answers 30 views ### Some confusion on american put pde Suppose $$L(v) = \dfrac{\partial v}{\partial t} + rS\dfrac{\partial v}{\partial S} + \dfrac{1}{2}\sigma^2S^2\dfrac{\partial^2 v}{\partial S^2} -rv$$ is Black-Scholes operator. ... 0answers 39 views ### How to derive put option from Black-Scholes equation? The Question is as follows: The diffusion equation is: I have tried attempting this question by making some change of variables and separating the cumulative distributive function but I get stuck ... 0answers 26 views ### Ito's formula with a random jump measure Suppose all processes and functions defined are nice enough such that all the following definitions make sense. On a probability space$(\Omega,\mathcal{F},\mathbb{P})$equipped with a filtration$\...
I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...