The Feynman-Kac Formula for Multi-Dimensional Diffusion Process

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1. On proving the formula

Let (Ω, F, P) be a probability space and let S_t = (S_t^(1), ..., S_t^(n)) be an n-dimensional stochastic process (n positive integer) satisfying the following stochastic differential equation (SDE):

where {W_t^(i) : t>=0} is a standard Wiener process for all i, and

with ρ_ii=1 for all i, and μ, σ are know functions of S_t^(i), for all i, and t, r and Ψ are functions of t and S_T, respectively with t<T.

We consider the following partial differential equation (PDE) problem:

with boundary condition:

Using It?'s formula on the process

prove tha under the filtration F_t, the solution of the PDE is given by:

Sketch of Proof (on your pens!)

Set

and apply the Taylor's expansion and the It?'s lemma on dZ_u, which we can write:

You will obtain:

Bearing in mind with Equation (3), integrate Equation (9) from t to T, then take the expectation by usuing the properties of the It?'s integral, to obtain:

and hence under the filtration F_t, deduce Equation (6). ?

2. Some quick applications of the multi-dimensional case

The Feynman-Kac theorem is a powerful tool in mathematical finance, particularly for pricing derivative securities. It connects partial differential equations (PDEs) with stochastic processes, allowing the use of probabilistic methods to solve problems involving deterministic differential equations. There are many more applications in the multi-dimensional case rather than in the 1D case!

In financial markets, derivatives can depend on multiple underlying assets. For example, a basket option might derive its value from several stocks or indices. In the context of pricing multi-asset options, the multi-dimensional Feynman-Kac theorem can be used to price these options by solving a multi-dimensional PDE that represents the option's value as a function of the underlying asset prices and time. In portfolio optimization, the aim is to manage a "basket" of assets to achieve a desired risk-return profile. The dynamics of the portfolio can be described using multi-dimensional stochastic differential equations. The Feynman-Kac theorem helps in deriving the expected utility of the terminal wealth, facilitating decision-making under uncertainty. In assessing the risk of credit derivatives, such as collateralized debt obligations (CDOs), which involve multiple underlying credit instruments, the multi-dimensional Feynman-Kac framework can help in modeling the joint distribution of credit events and thus in valuing such complex structures. Finally, in multi-factor interest rate models, such as the Heath-Jarrow-Morton (HJM) framework, the future evolution of interest rates depends on several factors. The application of the multi-dimensional Feynman-Kac theorem can help determine bond prices or the pricing of interest rate derivatives within these models.

In physics, in particular quantum mechanics, the Feynman-Kac theorem provides a bridge between quantum mechanics and stochastic processes. It helps in understanding the path integral formulation of quantum mechanics, which describes the evolution of quantum states as sums over histories. By relating Schr?dinger's equation to stochastic differential equations, the theorem allows the computation of quantum mechanical propagators using simulations of Brownian motion. In statistical mechanics, the theorem is used to study systems in equilibrium and non-equilibrium states. For instance, it connects the dynamics of particles with potential fields to diffusion processes, enabling the analysis of phase transitions and the computation of correlation functions. The theorem plays a role in simplifying calculations involving quantum fields. By interpreting field values as averages over stochastic paths, physicists can use methods from probability theory to tackle problems that would otherwise require complex and computationally-intensive methods. Finally, in condensed matter systems, where interactions among many particles are complex, the theorem helps model electron movement in disordered systems and provides insights into phenomena like Anderson localization, where wavefunctions become localized due to disorder.

3. One book recommendation

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