# Questions tagged [differential-equations]

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### Approximating Sharpe and Sortino ratios from Exponential moving averages

So I've been studying the paper "Learning To Trade via Direct Reinforcement" Moody and Saffell (2001) which describes in detail how to use exponential moving estimates (EMAs) of returns at ...
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### Black Scholes to Heat Equation

Equation (2) was derived by setting r=0 in the Black-Scholes equation for the Bachelier model (1). Can someone please help me understand all the steps for how we get from the heat equation under time ...
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### 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 $$dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}$$ ? Notice that $\{r_t\}$ is our ...
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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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### Finding a PDE for an option $V(t,r(t),S(t))$

I have 2 approaches in my mind for finding a pde of an option that depends both on the short rate as well as the stock price- $V(t,r(t),S(t)$. Are these equivalent? Find a hedging portfolio by ...
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### What are the advantages and limitations of predicting future stock prices using stochastic differential equations?

Recently I came across the following stochastic differential equation that "predicts" the value of a given stock: dS_t = \mu S_t dt + \sigma S_tdW_t \\ S_t(0) =S_0 \end{...
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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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### Finding B(t) in the Vasicek model relating to the bond equation, more specifcally from the initial condition

In the Vasicek model for derving bond prices, we have the ODE $$\frac{dB}{dt}=\gamma B-1$$ which gives rise to the general solution $$B(t)=C_1 e^{\gamma t}+C_2$$My problem is that we have the "initial"...
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### 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 ...
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### 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 ...
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### 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 ...
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### Why do we have to use discretization methods for SDE?

I haven't found the answer for the question above in google. Why can't we just discretize the equation instead of using methods like euler or milstein for the discretization.
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### How to solve this PDE using Feynman-Kac?

I have the following problem right now: solve $$F_t(t,x) + rxF_x(t,x) + \frac{\sigma^2}{2}F_{xx}(t,x) = rF(t,x), \\ F(T,x) = (x - K)^2.$$ How do I solve this? There exists a theorem to solve this, ...
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Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $... 1answer 449 views ### Modelling EUR/USD with Ornstein-Uhlenbeck + jumps? I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process: $$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$ with the parameters$\...
Given the PDE $$\frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2} = 0$$ with condition $F(T,x) = x^2$, one can use the Feynman-Kac formula to arrive at F(t,x) =...