Using Feynman–Kac for a Time-Dependent Drift and Quadratic Payoff

Summary

The document sets up a terminal-value PDE with a time-dependent drift, constant diffusion, and a quadratic payoff, then asks how to evaluate its Feynman–Kac representation. It identifies the associated stochastic differential equation and expresses the terminal state as the current value plus drift and Brownian increments.

The central calculation would require finding the conditional expectation of the squared terminal state, accounting for both its conditional mean and variance. The document does not provide that calculation or a resulting PDE solution. Its value is as a focused stochastic calculus exercise; it gives no numerical example, verification, or discussion of the time interval’s restrictions around the singular drift.

Key ideas

• The document relates a terminal-value PDE to a diffusion process through Feynman–Kac.
• The drift depends on time and becomes singular as time approaches one.
• The payoff is the square of the process at the terminal time.
• Evaluating the conditional second moment is the unresolved step in the document.

Tags

# Feynman-Kac formula for $\mu(t,x)=-\frac{1}{1-t}, \sigma(t,x)=1$ and $g(t,x)=x^2$


# Feynman-Kac formula for $\mu(t,x)=-\frac{1}{1-t}, \sigma(t,x)=1$ and $g(t,x)=x^2$












Consider the following PDE on $[0,T]\times \mathbb{R}$: $$ \begin{cases} \dfrac{\partial F}{\partial t}+\mu(t,x) \dfrac{\partial F}{\partial x}+ \frac12 \sigma^2(t,x)\dfrac{\partial^2 F}{\partial x^2} = 0 \\ F(T,x)=g(x) \end{cases} $$ with $\mu(t,x)=-\frac{1}{1-t}, \ \sigma(t,x)=1, \ g(x)=x^2$.

Question: What is the solution of the problem?

I know that if $X=\{X_t: t \geq 0\}$ satisfies $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$ then (by Feynman-Kac's theorem) $$F(t,x)=\mathbb{E}^Q[g(X_T)|X_t=x]$$

I know that $$X_T=X_t-\int_{t}^T\frac{1}{1-s}ds+\int_{t}^TdW_s$$ but then I don't know how to deal with $(X_T)^2$. How can I get the solution?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.