Tag: math

On the stochastic differential equation implied by f(σ,Wₜ)

Julien RiposoLeave a comment

To effectively elucidate the causal relationships between various economic processes, it is vital to delineate the evolution of their patterns. For example, recent developments in interest rates highlight the potential correlations among different rates. The onset of global trade tensions, initiated by former President Trump’s policies, has prompted notable adjustments, including reductions in rates by various countries due to decisions made by international institutions such as the European Central Bank (ECB) (see https://www.lesechos.fr/finance-marches/marches-financiers/la-bce-choisit-de-baisser-ses-taux-face-a-lincertitude-economique-2160609). Additionally, the Federal Reserve (FED) is facing pressure to lower its rates in response to these external influences (https://www.marketwatch.com/story/trump-is-furious-that-fed-wont-cut-interest-rates-like-ecb-heres-why-powell-wont-budge-162dfdaa).


A straightforward approach to modeling the evolution of interest rates is through stochastic processes, such as the Ornstein-Uhlenbeck process. Although potential negative rates present a challenge, our focus will be on further exploring multivariate scenarios. Should it be imperative to avoid negative rates, the Heston-White model presents a viable alternative. For a thorough examination of interest rate modeling, refer to the comprehensive work of Damiano Brigo and Fabio Mercurio.

In the following, we are interested in the stochastic differential equation of the form

{\displaystyle \begin{aligned} {\rm d}X_t = -\alpha\,X_t\,{\rm d}t + \dots \end{aligned} } 

where the second term shall be generalized. But what generalization?
We thus introduce the vector stochastic process X_t  of dimension n  (n  interest rates), \alpha  is a n\times n  matrix describing the trends of the vector stochastic processes. 
In other posts of the present blog, the following equation was proposed.

{\displaystyle \begin{aligned} {\rm d}X_t = -\alpha\,X_t\,{\rm d}t + f(\sigma W_t), \end{aligned} }     (1) 

where f  is a function, assumed to be at least continuous on \mathbb{R}  (or \mathbb{R} -Borelian, or "Borealian" to be more precise). In addition, \sigma  is assumed to be some positive number. Finally, W_t  is a vector of standard Wiener processes. The function f  is giving non-linearities and further depenencies 
First, we note that this equation gives

{\displaystyle \begin{aligned} f(\sigma W_t) = {\rm d}X_t + \alpha\,X_t\,{\rm d}t. \end{aligned} } 

This means that f  is a sum of linear forms (i.e. "{\rm d}\dots ") defining some metric of integration. Since {\rm d}X_t  and {\rm d}t  are the only forms which we consider in this equation, then we heuristically we:

{\displaystyle \begin{aligned} f(\sigma W_t) = A_1(X_t,t)\,{\rm d}t + A_2(X_t,t)\,{\rm d}X_t + A_3(X_t,t)\,{\rm d}X_t\,{\rm d}X_t + A_4(X_t,t)\,{\rm d}t\,{\rm d}X_t, \end{aligned} } 

where the A_i 's are Borealian functions of X_t  and t  and we ignore the terms of the form ({\rm d}X_t)^k  with k>2 , and ({\rm d}t)^k  with k>1 . Considering now the fact that f  is only depending on W_t , this means that the term of the form {\rm d}t\,{\rm d}X_t  could be set to zero. Thus we have:

{\displaystyle \begin{aligned} f(\sigma W_t) = A_1(X_t,t)\,{\rm d}t + A_2(X_t,t)\,{\rm d}X_t + A_3(X_t,t)\,{\rm d}X_t\,{\rm d}X_t. \end{aligned} } 

Using again the fact that f  only depends on W_t, we should have

{\displaystyle \begin{aligned} f(\sigma W_t) = \tilde{A}_1(W_t,t)\,{\rm d}t + \tilde{A}_2(W_t,t)\,{\rm d}W_t + \tilde{A}_3(W_t,t)\,{\rm d}W_t\,{\rm d}W_t = \left(\tilde{A}_1(W_t,t)+\tilde{A}_3(W_t,t)\right)\,{\rm d}t + \tilde{A}_2(W_t,t)\,{\rm d}W_t, \end{aligned} } 

where the \tilde{A}_i 's are other Borelian functions but only depending on W_t  (and t ). Repporting to Eq. (1), we then have:

{\displaystyle \begin{aligned} {\rm d}X_t = (-\alpha\,X_t+F(W_t,t))\,{\rm d}t + G(W_t,t)\,{\rm d}W_t. \end{aligned} } 

This (vector) equation turns out to be the most possible general stochastic differential equation related to the function f  introduced in Eq. (1). Note here that F(W_t,t)  is a vector of dimension n  and G(W_t,t)  is a matrix of dimension n\times n , representing the covariance matrix associated with the vector X_t . In fact, this equation is an Itô process.
If the processes only have dependencies in their stochastic terms, we shall set F  to be a vector only depending on time t , i.e. F(W_t,t)\equiv F(t) , so that the final quation of interest is given by:

{\displaystyle \begin{aligned} {\rm d}X_t = (-\alpha\,X_t+F(t))\,{\rm d}t + G(W_t,t)\,{\rm d}W_t. \end{aligned} } 

We integrate this equation by setting:

{\displaystyle \begin{aligned} Y_t = {\rm exp}\,\left(\alpha t\right)\, X_t. \end{aligned} } 

The Itô's lemma gives:

{\displaystyle \begin{aligned} {\rm d}Y_t = \alpha\, {\rm exp}\,\left(\alpha t\right)\, X_t\,{\rm d}t + {\rm exp}\,\left(\alpha t\right)\, \,{\rm d}X_t = {\rm exp}\,\left(\alpha t\right)\,F(t)\,{\rm d}t + {\rm exp}\,\left(\alpha t\right)\,G(W_t,t)\,{\rm d}W_t. \end{aligned} } 

Therefore, integration of this process finally leads to:

{\displaystyle \begin{aligned} X_t = {\rm exp}(-\alpha\,t)\,X_0 + \int_0^t {\rm exp}(-\alpha\,(t-s))\,F(s)\,{\rm d}s + \int_0^t {\rm exp}(-\alpha\,(t-s))\,G(W_s,s)\,{\rm d}W_s. \end{aligned} } 

Now, we note that the only random term is the third one, which has zero expected value. Therefore, we have

{\displaystyle \begin{aligned} X_t \sim \mathcal{N}\left({\rm exp}(-\alpha\,t)\,X_0 + \int_0^t {\rm exp}(-\alpha\,(t-s))\,F(s)\,{\rm d}s,\,\, \int_0^t {\rm exp}(-\alpha\,(t-s))\,G(W_s,s)\,{\rm d}W_s\right). \end{aligned} } 

In words, X_t  is following a normal vector process with covariance \displaystyle \int_0^t {\rm exp}(-\alpha\,(t-s))\,G(W_s,s)\,{\rm d}W_s . It shall be interesting to see in which circumstances the matrix \alpha and vector F may lead to a non-explosive process.