3.14.3 The BSDE Backbone & Nonlinear Feynman–Kac

Basic Prerequisites: BSM · 02 The PDE & Derivation and SC · 04 SDEs & Simulation.

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1. Intuition & Practical Objective

A forward SDE says: given where I start, where do I end up? A backward SDE says the opposite: given where I must end up, what is the value now - and by which control? That inversion is exactly the structure of a hedging problem, and it is why BSDEs are the natural language of this folder.

The classic Feynman–Kac theorem (BSM · 02) already links a linear PDE to an expectation: $u(t,x)=\mathbb{E}[g(X_T)\mid X_t=x]$. Pardoux–Peng (1990) removed the linearity. Replace the expectation by a backward SDE with a driver $f$, and the same link survives in a nonlinear form:

$${\partial }_{t}u+Lu+f(t,u,{\sigma }^{\top }\nabla u)=0⟺Y_t=u(t,X_t),Z_t={\sigma }^{\top }\nabla u(t,X_t).$$

The one-sentence essence:

A BSDE is a Feynman–Kac formula for nonlinear PDEs: the pair $(Y,Z)$ generalises (value, gradient), the driver $f$ generalises the discount/payoff non-linearity, and the entropic risk measure - the only one of page 02’s three with a closed-form dynamics - is exactly the special case $f(z)=-\frac{\gamma }{2}|z{|}^2$ whose dynamics is the exponential transform $e^{-\gamma Y_t}$ being a martingale.

Four things to carry out of this page:

1. The pair $(Y,Z)$, not just $Y$. $Y_t$ is the value (the price, the utility, the risk), and $Z_t$ is the control - in a Markovian setting $Z_t={\sigma }^{\top }\nabla u$, the martingale-representation density, i.e. the hedge in the diffusive scale. The numerical difficulties of deep hedging are almost all difficulties with $Z$.
2. Lipschitz is the classical regime; quadratic is the financial one. Pardoux–Peng need $f$ Lipschitz in $(y,z)$. The entropic driver $-\frac{\gamma }{2}{z}^2$ is quadratic and violates that hypothesis - it needs the Kobylanski theory, and it is precisely the case that matters for hedging. The extra structure (the exponential transform) is what replaces the Lipschitz condition.
3. Nonlinear Feynman–Kac is the computational bridge. It converts “solve a BSDE” into “solve a semilinear PDE” and back - which is why the Deep BSDE and Deep Galerkin solvers of §04 attack the same object from two different directions.
4. $g$-expectations package it. A driver $g$ defines a sublinear (or nonlinear) expectation ${\mathcal{E}}_g[\xi \mid {\mathcal{F}}_t]:=Y_t$. $g\equiv 0$ recovers the ordinary conditional expectation; $g(z)=-(\gamma /2)z^2$ recovers the entropic one. The comparison theorem makes these well-behaved.

The practical objective: write down a BSDE, state the conditions for a solution, take the nonlinear Feynman–Kac correspondence in both directions, solve an exactly-solvable quadratic BSDE by hand, and reproduce it numerically by backward-induction LSMC.

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2. Mathematical Ground Truth & Derivations

2.1 The BSDE

Fix a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0},\mathbb{P})$ carrying a $d$-dimensional Brownian motion $W$, and a terminal condition $\xi \in L^2({\mathcal{F}}_T)$. A BSDE is

$$\boxed{Y_t=\xi +{\int }_t^{T}f(s,Y_s,Z_s)ds-{\int }_t^T{Z}_{s}d{W}_s,0\le t\le T}$$

or, in differential form, $d{Y}_t=-f(t,Y_t,Z_t)dt+Z_{t}d{W}_t$ with $Y_T=\xi$. Two objects are sought: a value $Y$ (continuous, adapted) and a control $Z$ (${\mathbb{R}}^{1\times d}$-valued, adapted). The minus sign in front of the $Z$-integral is a convention; it makes $Z$ the density in the martingale representation of the “hedging error” and is standard in the financial literature.

Existence and uniqueness (Pardoux–Peng 1990). If

$$\xi \in L^2({\mathcal{F}}_T),f\text{is Lipschitz in }(y,z)\text{uniformly in }t,{\int }_0^T\Vert f(t,0,0){\Vert }^{2}dt<\infty ,$$

then there is a unique solution $(Y,Z)$ with $Y$ continuous and both in the appropriate $L^2$ spaces. The proof is a fixed-point argument: the linear BSDE with driver $f^0=<f>dt$ has the explicit solution $Y_t=\mathbb{E}[\xi +{\int }_t^T{f}^{0}ds\mid {\mathcal{F}}_t]$ (a Feynman–Kac identity), and the Picard iteration contracts onto that when $f$ is Lipschitz. The linear case is the one the hub code solves: with $f\equiv 0$, $Y_t=\mathbb{E}[\xi \mid {\mathcal{F}}_t]$ (the price) and $Z_t$ is the martingale-representation density (the hedge). Set $\xi =(S_T-K{)}^{+}$ and you have Black–Scholes written as a BSDE.

2.2 Nonlinear Feynman–Kac

Let the forward process be $d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{W}_t$ with generator $Lu=\mu \cdot \nabla u+\frac{1}{2}\mathrm{tr}(\sigma {\sigma }^{\top }{\nabla }^{2}u)$. Suppose there is a smooth solution $u$ of the semilinear equation

$$\boxed{{\partial }_{t}u+Lu+f(t,u,{\sigma }^{\top }\nabla u)=0,u(T,x)=g(x)}$$

Then the BSDE with terminal $\xi =g(X_T)$ and driver $f$ has the Markovian solution

$$Y_t=u(t,X_t),Z_t={\sigma }^{\top }\nabla u(t,X_t).$$

This is proved with Itô’s lemma on $Y_t=u(t,X_t)$: $d{Y}_t=({\partial }_{t}u+Lu)dt+\nabla u^{\top }\sigma d{W}_t=-fdt+Zd{W}_t$. Reading it backwards is the useful direction in practice: if you can solve the BSDE, you have solved the PDE - and vice versa, which is why Deep BSDE (PDE ⇒ BSDE) and Deep Galerkin (PDE ⇒ residual) are two attacks on the same object.

Two consequences that matter constantly:

• $Z$ is a gradient. In one dimension $Z_t=\sigma {\partial }_{x}u$ - the delta in diffusive units. Learning $Z$ is learning the hedge; the numerical noise in $Z$ (§3 check B) is the numerical noise in the delta.
• The driver is the model’s nonlinearity. $f\equiv 0$ gives Black–Scholes; $f$ linear in $u$ gives discounting; $f$ quadratic in $z$ gives entropic risk; $f$ concave in $z$ gives transaction costs (Leland, §06). The whole zoo of incomplete-market pricing is a choice of driver.

2.3 The entropic (quadratic) driver - the hinge of the folder

Take the driver to be quadratic in $z$:

$$f(t,y,z)=-\frac{\gamma }{2}|z{|}^2(\gamma >0).$$

This is not Lipschitz, so Pardoux–Peng does not apply; but the transform

$$U_t:=e^{-\gamma Y_t}$$

is a martingale. Indeed with $dY=\frac{\gamma }{2}{Z}^{2}dt+ZdW$ Itô gives

$$d{U}_t=-\gamma U_{t}d{Y}_t+\frac{{\gamma }^2}{2}{U}_{t}d⟨Y{⟩}_t=-\gamma U_t(\frac{\gamma }{2}{Z}^{2}dt+Zd{W}_t)+\frac{{\gamma }^2}{2}{U}_t{Z}^{2}dt=-\gamma U_t{Z}_{t}d{W}_t,$$

so $U$ is a local martingale, and under exponential integrability a true one. Hence $U_t=\mathbb{E}[U_T\mid {\mathcal{F}}_t]=\mathbb{E}[e^{-\gamma \xi }\mid {\mathcal{F}}_t]$ and

$$\boxed{Y_t=-\frac{1}{\gamma }\ln \mathbb{E}\left[e^{-\gamma \xi }\right|{\mathcal{F}}_t]}$$

The quadratic BSDE is the exponential transform of a conditional expectation. This is why entropic risk is the tractable case: the nonlinearity is entirely absorbed by the exponential, and the remaining objects are ordinary conditional expectations. Two immediate corollaries:

• The comparison/level structure. For a Gaussian terminal $\xi \sim N(m,s^2)$ the closed form is exact: $Y_t=m_t-\frac{\gamma }{2}{s}_t^2$, and in general $Y_0=m-\frac{\gamma }{2}{s}^2+O({\gamma }^2)$ - the mean–variance hedge is the small-risk-aversion limit (§02 §2.4).
• This is the indifference price. ${\rho }_{\gamma }(\xi )=\frac{1}{\gamma }\ln \mathbb{E}[e^{\gamma \xi }]$ (a loss with $+\gamma$) is the exponential-utility indifference price of §02 §2.3; the sign of $\gamma$ in the driver is the sign convention between “value of a liability” and “risk measure of a payoff”.

2.4 The exactly solvable example (used in §3)

Restrict to arithmetic Brownian motion, $d{X}_t=\sigma d{W}_t$ (so $L=\frac{{\sigma }^2}{2}{\partial }_x^2$), and take the terminal condition $\xi =X_T^2$ (a quadratic, hence unbounded, payoff - deliberately: it shows the method is not confined to bounded claims). With $f(z)=-\frac{1}{2}{z}^2$ (i.e. $\gamma =1$), the nonlinear Feynman–Kac equation is

$${\partial }_{t}u+\frac{{\sigma }^2}{2}{u}_{xx}-\frac{{\sigma }^2}{2}(u_x{)}^2=0,u(T,x)=x^2.$$

Guess $u(t,x)=a(\tau )+b(\tau )x^2$ with $\tau =T-t$; substituting gives $b^{\prime }=-2{\sigma }^2{b}^2$, $a^{\prime }={\sigma }^{2}b$, $b(0)=1,a(0)=0$. Separating: $b=1/(1+2{\sigma }^2\tau )$ and $a=\frac{1}{2}\ln (1+2{\sigma }^2\tau )$. Hence

$$\boxed{Y_t=\frac{1}{2}\ln (1+2{\sigma }^2\tau )+\frac{X_t^2}{1+2{\sigma }^2\tau },Z_t=\sigma {\partial }_{x}u=\frac{2\sigma X_t}{1+2{\sigma }^2\tau },\tau =T-t}$$

Two sanity limits: at $\tau =0$, $Y_T=X_T^2=\xi$ ✓; as $\sigma \to 0$, $Y_t\to X_t^2$ ✓ (no randomness, no risk). Everything in §3 checks against these.

2.5 $g$-expectations and the comparison theorem

A driver $g$ (state-independent, $g(t,0,0)=0$) defines the $g$-expectation ${\mathcal{E}}_g[\xi \mid {\mathcal{F}}_t]:=Y_t$ of the BSDE with driver $g$ and terminal $\xi$. Properties follow from the comparison theorem:

$$f_1\le f_2\text{pointwise and }{\xi }_1\le {\xi }_2⟹Y_t^1\le Y_t^2\text{a.s. for all }t.$$

• $g\equiv 0$: ${\mathcal{E}}_0[\xi \mid {\mathcal{F}}_t]=\mathbb{E}[\xi \mid {\mathcal{F}}_t]$ - the ordinary expectation.
• $g$ convex in $z$: ${\mathcal{E}}_g$ is sublinear (monotone, translation-invariant, convex) and its dual is a set of measures - the BSDE face of the robust representation of §02.
• $g(z)=-\frac{\gamma }{2}{z}^2$ (concave): ${\mathcal{E}}_g$ is superlinear, and equals the entropic transform above.

The financial reading: a nonlinear expectation is a set of priors (model uncertainty), and choosing a driver is choosing how much the desk distrusts its own model. This is the BSDE-theoretic root of the distributionally-robust hedge of 06.

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3. Computational Implementation - a quadratic BSDE solved by hand and by LSMC

Three independent checks on the exactly-solvable example of §2.4: (A) the closed form satisfies the nonlinear PDE to machine precision; (B) a backward-induction LSMC solver - regression for $Y$ and the martingale-representation regression for $Z$ - reproduces both $Y_0$ and $Z_0$; (C) the Gaussian closed form $m-\frac{\gamma }{2}{s}^2$ is verified against a Monte-Carlo entropic transform.

Reading the output.

• (A) The closed form is a genuine solution. Its maximum PDE residual over $3000$ random collocation points is $7.2\times 10^{-8}$ - finite-difference truncation error, not model error (the step is $h=10^{-5}$, so $O(h^2)$ discretisation of a smooth function sits at that scale). The nonlinear Feynman–Kac correspondence ${\partial }_{t}u+\frac{{\sigma }^2}{2}{u}_{xx}-\frac{{\sigma }^2}{2}(u_x{)}^2=0$ is satisfied.
• (B) The LSMC solver works, and the error is in the control. $Y_0$ is recovered with a bias of $-0.9$ to $-1.9\times 10^{-3}$ on a value of $0.9644$ ($\approx 0.2\%$) shrinking slowly; $Z_0$ is recovered with a bias of $\pm (3\text{–}14)\times 10^{-3}$ on $0.3704$ - comparable in relative terms but far noisier across seeds. This is the canonical BSDE error signature: the backward regression computes the value functional (smooth, well-conditioned) to high accuracy, and the control (a gradient, hence differentiation of a noisy object) to much lower accuracy. Every deep-hedging implementation inherits this asymmetry.
• (C) The Gaussian closed form is exact and the MC agrees. $\xi \sim N(1.04,0.1632)$ with $\gamma =1$: the Monte-Carlo entropic transform gives $0.957976$ against $m-\frac{\gamma }{2}{s}^2=0.958400$ - a $0.04\%$ gap, within the Monte-Carlo and log-bias error of an estimator whose integrand is $\log$-normal. This confirms the mean-minus-half-risk-aversion-times-variance identity, i.e. the exact bridge from the entropic BSDE to the variance-optimal hedge of page 01.

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4. Failure Modes & First-Principles Breakdowns

1. Applying Pardoux–Peng to a quadratic driver. The theorem needs $f$ Lipschitz in $z$; $f=-\frac{\gamma }{2}{z}^2$ is not. Existence for quadratic drivers needs bounded terminal data (or exponential moments) and the Kobylanski machinery, and uniqueness can fail without them. “There is a BSDE theory” does not mean “your BSDE is covered by the textbook theorem”.
2. Confusing $Z$ with the delta. $Z_t={\sigma }^{\top }\nabla u$: it is the delta scaled by the volatility, and in multiple dimensions it is a row vector contracted with the diffusion matrix. Reporting $Z$ as “the delta” silently inserts a factor $\sigma S$ (the hub code reports both: $Z_0=10.919$ and $Z_0/(\sigma S_0)=0.546$). Get the scaling wrong and a correct solver still “validates” against a correctly-scaled wrong benchmark.
3. Treating the driver as arbitrary. Different drivers give different prices, not just different numerical behaviour: $f\equiv 0$ prices at $\mathbb{E}[\xi ]$; the entropic driver prices at the indifference price. A sign error in $f$ (or in $\gamma$) flips “risk-averse” into “risk-seeking” and is easy to miss because the code still runs.
4. Using the wrong measure in the forward simulation. The BSDE is stated under $\mathbb{Q}$ (or under $\mathbb{P}$ with a driver that absorbs the risk premium). Simulating $X$ with the physical drift and then pricing with a risk-neutral BSDE gives a systematic bias that no amount of Monte Carlo removes. The drift belongs in the driver, and the driver must be stated. See SC · 05 Girsanov & Risk-Neutral.
5. Ignoring the terminal integrability required for the exponential transform. The identity $Y_t=-\frac{1}{\gamma }\ln \mathbb{E}[e^{-\gamma \xi }\mid {\mathcal{F}}_t]$ requires $\mathbb{E}[e^{-\gamma \xi }]<\infty$. For heavy-tailed or unbounded-below payoffs (short calls, quadratic payoffs on log-normal spots) the exponential does not integrate, the transform fails, and the “solution” is $-\infty$. §3 uses a quadratic payoff on an additive Brownian motion precisely because the exponential moment exists there; on a geometric Brownian motion the same payoff would explode the entropic transform.
6. Forgetting that the comparison theorem is strict about the conditions. $f_1\le f_2$ and ${\xi }_1\le {\xi }_2$ give $Y^1\le Y^2$ only under the same integrability regime; for quadratic drivers the ordering can fail without boundedness. Bulk monotonicity intuitions (“a riskier payoff has a higher value”) are theorems with hypotheses, not axioms.
7. Using LSMC with a poor regressor basis. The check-B bias comes from the regression family, not from the number of paths: a basis that cannot represent $E[Y_{t_{i+1}}\mid X_{t_i}]$ produces a bias, not noise, and increasing the path count converges confidently to the wrong answer. Always fit on a basis that contains the true conditional-mean family (here $\{1,x,x^2\}$, which is exact for the quadratic example), or validate with an independent method. See NM · 03 Monte-Carlo Pricing.
8. Believing the value is “the answer”. $Y_0$ is one number; the strategy is the whole process $Z_t$, and it is $Z$ that a desk trades. A solver validated only on $Y_0$ has validated only a diagnostic, not the deliverable.

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5. References

• Pardoux, E. & Peng, S. (1990), Adapted solution of a backward stochastic differential equation, Systems & Control Letters 14, 55–61
• Kobylanski, M. (2000), Backward stochastic differential equations and partial differential equations with quadratic growth, Annals of Probability 28(2), 558–602
• Peng, S. (1997/2004), Monotonic limit theorem of BSDE and nonlinear decomposition theorem, and Nonlinear expectations and nonlinear Markov chains
• Buehler, H., Gonon, L., Teichmann, J., Wood, B. (2019), Deep Hedging, Quantitative Finance 19(8), 1271–1291
• Gobet, E., Lemor, J.-P., Warin, X. (2005), A regression-based Monte Carlo method to solve backward stochastic differential equations, Annals of Applied Probability 15(3), 2172–2202
• Karatzas, I. & Shreve, S., Brownian Motion and Stochastic Calculus

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6. Connected Graph Bridges

• Back: 02 · Convex Risk & the Deep-Hedging Objective · 01 · From Zero
• Forward: 04 · Deep BSDE & Deep Galerkin · Index Hub
• Theory base: Stochastic Calculus & Itô’s Lemma · SC · 03 Itô Integral & Doeblin · SC · 04 SDEs & Simulation · SC · 05 Girsanov & Risk-Neutral · PMT · 05 Martingales · PMT · 04 Conditional Expectation
• The linear special case: BSM · 02 The PDE & Derivation · BSM · 03 Pricing Formulas
• Numerical neighbours: NM · 03 Monte-Carlo Pricing · NM · 04 Variance Reduction · NM · 03 Monte Carlo (Foundations)
• Risk-measure counterpart: ES · 03 Coherent Risk Measures